id sid tid token lemma pos ejpam-4877 1 1 european european PROPN ejpam-4877 1 2 journal journal PROPN ejpam-4877 1 3 of of ADP ejpam-4877 1 4 pure pure ADJ ejpam-4877 1 5 and and CCONJ ejpam-4877 1 6 applied apply VERB ejpam-4877 1 7 mathematics mathematic NOUN ejpam-4877 1 8 vol vol NOUN ejpam-4877 1 9 . . PUNCT ejpam-4877 2 1 16 16 NUM ejpam-4877 2 2 , , PUNCT ejpam-4877 2 3 no no INTJ ejpam-4877 2 4 . . NOUN ejpam-4877 2 5 4 4 NUM ejpam-4877 2 6 , , PUNCT ejpam-4877 2 7 2023 2023 NUM ejpam-4877 2 8 , , PUNCT ejpam-4877 2 9 2597 2597 NUM ejpam-4877 2 10 - - SYM ejpam-4877 2 11 2612 2612 NUM ejpam-4877 3 1 issn issn PROPN ejpam-4877 3 2 1307 1307 NUM ejpam-4877 3 3 - - SYM ejpam-4877 3 4 5543 5543 NUM ejpam-4877 3 5 – – PUNCT ejpam-4877 3 6 ejpam.com ejpam.com X ejpam-4877 3 7 published publish VERB ejpam-4877 3 8 by by ADP ejpam-4877 3 9 new new PROPN ejpam-4877 3 10 york york PROPN ejpam-4877 3 11 business business PROPN ejpam-4877 3 12 global global PROPN ejpam-4877 3 13 grundy grundy PROPN ejpam-4877 3 14 total total NOUN ejpam-4877 3 15 hop hop NOUN ejpam-4877 3 16 dominating dominate VERB ejpam-4877 3 17 sequences sequence NOUN ejpam-4877 3 18 in in ADP ejpam-4877 3 19 graphs graph NOUN ejpam-4877 3 20 javier javier PROPN ejpam-4877 3 21 a. a. PROPN ejpam-4877 3 22 hassan1,∗ hassan1,∗ PROPN ejpam-4877 3 23 , , PUNCT ejpam-4877 3 24 sergio sergio PROPN ejpam-4877 3 25 r. r. PROPN ejpam-4877 3 26 canoy canoy PROPN ejpam-4877 3 27 , , PUNCT ejpam-4877 3 28 jr.2,3 jr.2,3 PROPN ejpam-4877 3 29 1mathematics 1mathematics NUM ejpam-4877 3 30 and and CCONJ ejpam-4877 3 31 sciences sciences PROPN ejpam-4877 3 32 department department PROPN ejpam-4877 3 33 , , PUNCT ejpam-4877 3 34 college college NOUN ejpam-4877 3 35 of of ADP ejpam-4877 3 36 arts art NOUN ejpam-4877 3 37 and and CCONJ ejpam-4877 3 38 sciences sciences PROPN ejpam-4877 3 39 msu msu PROPN ejpam-4877 3 40 tawi tawi PROPN ejpam-4877 3 41 - - PUNCT ejpam-4877 3 42 tawi tawi PROPN ejpam-4877 3 43 college college PROPN ejpam-4877 3 44 of of ADP ejpam-4877 3 45 technology technology NOUN ejpam-4877 3 46 and and CCONJ ejpam-4877 3 47 oceanography oceanography NOUN ejpam-4877 3 48 bongao bongao NOUN ejpam-4877 3 49 , , PUNCT ejpam-4877 3 50 tawi tawi NOUN ejpam-4877 3 51 - - PUNCT ejpam-4877 3 52 tawi tawi NOUN ejpam-4877 3 53 , , PUNCT ejpam-4877 3 54 philippines philippine NOUN ejpam-4877 3 55 2department 2department NUM ejpam-4877 3 56 of of ADP ejpam-4877 3 57 mathematics mathematic NOUN ejpam-4877 3 58 and and CCONJ ejpam-4877 3 59 statistics statistic NOUN ejpam-4877 3 60 , , PUNCT ejpam-4877 3 61 college college NOUN ejpam-4877 3 62 of of ADP ejpam-4877 3 63 science science NOUN ejpam-4877 3 64 and and CCONJ ejpam-4877 3 65 mathematics mathematic NOUN ejpam-4877 3 66 msu msu PROPN ejpam-4877 3 67 - - PUNCT ejpam-4877 3 68 iligan iligan PROPN ejpam-4877 3 69 institute institute PROPN ejpam-4877 3 70 of of ADP ejpam-4877 3 71 technology technology PROPN ejpam-4877 3 72 , , PUNCT ejpam-4877 3 73 9200 9200 NUM ejpam-4877 3 74 iligan iligan ADJ ejpam-4877 3 75 city city NOUN ejpam-4877 3 76 , , PUNCT ejpam-4877 3 77 philippines philippine NOUN ejpam-4877 3 78 3 3 NUM ejpam-4877 3 79 center center NOUN ejpam-4877 3 80 for for ADP ejpam-4877 3 81 mathematical mathematical ADJ ejpam-4877 3 82 and and CCONJ ejpam-4877 3 83 theoretical theoretical ADJ ejpam-4877 3 84 physical physical ADJ ejpam-4877 3 85 sciences science NOUN ejpam-4877 3 86 , , PUNCT ejpam-4877 3 87 prism prism NOUN ejpam-4877 3 88 msu msu PROPN ejpam-4877 3 89 - - PUNCT ejpam-4877 3 90 iligan iligan PROPN ejpam-4877 3 91 institute institute PROPN ejpam-4877 3 92 of of ADP ejpam-4877 3 93 technology technology PROPN ejpam-4877 3 94 , , PUNCT ejpam-4877 3 95 9200 9200 NUM ejpam-4877 3 96 iligan iligan ADJ ejpam-4877 3 97 city city NOUN ejpam-4877 3 98 , , PUNCT ejpam-4877 3 99 philippines philippine NOUN ejpam-4877 3 100 abstract abstract ADJ ejpam-4877 3 101 . . PUNCT ejpam-4877 4 1 let let VERB ejpam-4877 4 2 g g PROPN ejpam-4877 4 3 = = SYM ejpam-4877 4 4 ( ( PUNCT ejpam-4877 4 5 v v NOUN ejpam-4877 4 6 ( ( PUNCT ejpam-4877 4 7 g g NOUN ejpam-4877 4 8 ) ) PUNCT ejpam-4877 4 9 , , PUNCT ejpam-4877 4 10 e(g e(g PROPN ejpam-4877 4 11 ) ) PUNCT ejpam-4877 4 12 ) ) PUNCT ejpam-4877 5 1 be be AUX ejpam-4877 5 2 an an DET ejpam-4877 5 3 undirected undirected ADJ ejpam-4877 5 4 graph graph NOUN ejpam-4877 5 5 with with ADP ejpam-4877 5 6 γ(c γ(c PROPN ejpam-4877 5 7 ) ) PUNCT ejpam-4877 5 8 ̸= ̸= PROPN ejpam-4877 5 9 1 1 NUM ejpam-4877 5 10 for for ADP ejpam-4877 5 11 each each DET ejpam-4877 5 12 component component NOUN ejpam-4877 5 13 c c PROPN ejpam-4877 5 14 of of ADP ejpam-4877 5 15 g. g. PROPN ejpam-4877 5 16 let let VERB ejpam-4877 5 17 s s AUX ejpam-4877 5 18 = = PUNCT ejpam-4877 5 19 ( ( PUNCT ejpam-4877 5 20 v1 v1 PROPN ejpam-4877 5 21 , , PUNCT ejpam-4877 5 22 v2 v2 PROPN ejpam-4877 5 23 , , PUNCT ejpam-4877 5 24 · · PUNCT ejpam-4877 5 25 · · PUNCT ejpam-4877 5 26 · · PUNCT ejpam-4877 5 27 , , PUNCT ejpam-4877 5 28 vk vk AUX ejpam-4877 5 29 ) ) PUNCT ejpam-4877 5 30 be be AUX ejpam-4877 5 31 a a DET ejpam-4877 5 32 sequence sequence NOUN ejpam-4877 5 33 of of ADP ejpam-4877 5 34 distint distint ADJ ejpam-4877 5 35 vertices vertex NOUN ejpam-4877 5 36 of of ADP ejpam-4877 5 37 a a DET ejpam-4877 5 38 graph graph NOUN ejpam-4877 5 39 g g NOUN ejpam-4877 5 40 , , PUNCT ejpam-4877 5 41 and and CCONJ ejpam-4877 5 42 let let VERB ejpam-4877 5 43 ŝ ŝ X ejpam-4877 5 44 = = SYM ejpam-4877 5 45 { { PUNCT ejpam-4877 5 46 v1 v1 PROPN ejpam-4877 5 47 , , PUNCT ejpam-4877 5 48 v2 v2 PROPN ejpam-4877 5 49 , , PUNCT ejpam-4877 5 50 . . PUNCT ejpam-4877 5 51 . . PUNCT ejpam-4877 6 1 . . PUNCT ejpam-4877 7 1 , , PUNCT ejpam-4877 7 2 vk vk ADP ejpam-4877 7 3 } } PUNCT ejpam-4877 7 4 . . PUNCT ejpam-4877 8 1 then then ADV ejpam-4877 8 2 s s VERB ejpam-4877 8 3 is be AUX ejpam-4877 8 4 a a DET ejpam-4877 8 5 legal legal ADJ ejpam-4877 8 6 open open ADJ ejpam-4877 8 7 hop hop NOUN ejpam-4877 8 8 neighborhood neighborhood NOUN ejpam-4877 8 9 sequence sequence NOUN ejpam-4877 8 10 if if SCONJ ejpam-4877 8 11 n2 n2 ADJ ejpam-4877 8 12 g(vi g(vi NUM ejpam-4877 8 13 ) ) PUNCT ejpam-4877 8 14 \ \ NOUN ejpam-4877 8 15 ⋃i−1 ⋃i−1 NOUN ejpam-4877 8 16 j=1 j=1 PROPN ejpam-4877 8 17 n n CCONJ ejpam-4877 8 18 2 2 NUM ejpam-4877 8 19 g(vj g(vj NOUN ejpam-4877 8 20 ) ) PUNCT ejpam-4877 8 21 ̸= ̸= PROPN ejpam-4877 8 22 ∅ ∅ NOUN ejpam-4877 8 23 for for ADP ejpam-4877 8 24 every every DET ejpam-4877 8 25 i i PROPN ejpam-4877 8 26 ∈ ∈ PROPN ejpam-4877 8 27 { { PUNCT ejpam-4877 8 28 2 2 NUM ejpam-4877 8 29 , , PUNCT ejpam-4877 8 30 . . PUNCT ejpam-4877 8 31 . . PUNCT ejpam-4877 8 32 . . PUNCT ejpam-4877 9 1 , , PUNCT ejpam-4877 9 2 k k X ejpam-4877 9 3 } } PUNCT ejpam-4877 9 4 . . PUNCT ejpam-4877 10 1 if if SCONJ ejpam-4877 10 2 , , PUNCT ejpam-4877 10 3 in in ADP ejpam-4877 10 4 addition addition NOUN ejpam-4877 10 5 , , PUNCT ejpam-4877 10 6 ŝ ŝ X ejpam-4877 10 7 is be AUX ejpam-4877 10 8 a a DET ejpam-4877 10 9 total total ADJ ejpam-4877 10 10 hop hop NOUN ejpam-4877 10 11 dominating dominating NOUN ejpam-4877 10 12 set set NOUN ejpam-4877 10 13 of of ADP ejpam-4877 10 14 g g PROPN ejpam-4877 10 15 , , PUNCT ejpam-4877 10 16 then then ADV ejpam-4877 10 17 s s VERB ejpam-4877 10 18 is be AUX ejpam-4877 10 19 a a DET ejpam-4877 10 20 grundy grundy PROPN ejpam-4877 10 21 total total NOUN ejpam-4877 10 22 hop hop NOUN ejpam-4877 10 23 dominating dominating NOUN ejpam-4877 10 24 sequence sequence NOUN ejpam-4877 10 25 . . PUNCT ejpam-4877 11 1 the the DET ejpam-4877 11 2 maximum maximum ADJ ejpam-4877 11 3 length length NOUN ejpam-4877 11 4 of of ADP ejpam-4877 11 5 a a DET ejpam-4877 11 6 grundy grundy PROPN ejpam-4877 11 7 total total NOUN ejpam-4877 11 8 hop hop NOUN ejpam-4877 11 9 dominating dominating NOUN ejpam-4877 11 10 sequence sequence NOUN ejpam-4877 11 11 in in ADP ejpam-4877 11 12 a a DET ejpam-4877 11 13 graph graph NOUN ejpam-4877 11 14 g g NOUN ejpam-4877 11 15 , , PUNCT ejpam-4877 11 16 denoted denote VERB ejpam-4877 11 17 by by ADP ejpam-4877 11 18 γth γth ADJ ejpam-4877 11 19 gr(g gr(g PROPN ejpam-4877 11 20 ) ) PUNCT ejpam-4877 11 21 , , PUNCT ejpam-4877 11 22 is be AUX ejpam-4877 11 23 the the DET ejpam-4877 11 24 grundy grundy PROPN ejpam-4877 11 25 total total NOUN ejpam-4877 11 26 hop hop NOUN ejpam-4877 11 27 domination domination NOUN ejpam-4877 11 28 number number NOUN ejpam-4877 11 29 of of ADP ejpam-4877 11 30 g. g. PROPN ejpam-4877 11 31 in in ADP ejpam-4877 11 32 this this DET ejpam-4877 11 33 paper paper NOUN ejpam-4877 11 34 , , PUNCT ejpam-4877 11 35 we we PRON ejpam-4877 11 36 show show VERB ejpam-4877 11 37 that that SCONJ ejpam-4877 11 38 the the DET ejpam-4877 11 39 grundy grundy PROPN ejpam-4877 11 40 total total NOUN ejpam-4877 11 41 hop hop NOUN ejpam-4877 11 42 domination domination NOUN ejpam-4877 11 43 number number NOUN ejpam-4877 11 44 of of ADP ejpam-4877 11 45 a a DET ejpam-4877 11 46 graph graph NOUN ejpam-4877 11 47 g g NOUN ejpam-4877 11 48 is be AUX ejpam-4877 11 49 between between ADP ejpam-4877 11 50 the the DET ejpam-4877 11 51 total total ADJ ejpam-4877 11 52 hop hop NOUN ejpam-4877 11 53 domination domination NOUN ejpam-4877 11 54 number number NOUN ejpam-4877 11 55 and and CCONJ ejpam-4877 11 56 twice twice DET ejpam-4877 11 57 the the DET ejpam-4877 11 58 grundy grundy PROPN ejpam-4877 11 59 hop hop PROPN ejpam-4877 11 60 domination domination NOUN ejpam-4877 11 61 number number NOUN ejpam-4877 11 62 of of ADP ejpam-4877 11 63 g. g. PROPN ejpam-4877 11 64 moreover moreover ADV ejpam-4877 11 65 , , PUNCT ejpam-4877 11 66 determine determine VERB ejpam-4877 11 67 values value NOUN ejpam-4877 11 68 or or CCONJ ejpam-4877 11 69 bounds bound NOUN ejpam-4877 11 70 of of ADP ejpam-4877 11 71 the the DET ejpam-4877 11 72 grundy grundy PROPN ejpam-4877 11 73 total total NOUN ejpam-4877 11 74 hop hop NOUN ejpam-4877 11 75 domination domination NOUN ejpam-4877 11 76 number number NOUN ejpam-4877 11 77 of of ADP ejpam-4877 11 78 some some DET ejpam-4877 11 79 graphs graph NOUN ejpam-4877 11 80 . . PUNCT ejpam-4877 12 1 2020 2020 NUM ejpam-4877 12 2 mathematics mathematic NOUN ejpam-4877 12 3 subject subject NOUN ejpam-4877 12 4 classifications classification NOUN ejpam-4877 12 5 : : PUNCT ejpam-4877 12 6 05c69 05c69 X ejpam-4877 12 7 key key ADJ ejpam-4877 12 8 words word NOUN ejpam-4877 12 9 and and CCONJ ejpam-4877 12 10 phrases phrase NOUN ejpam-4877 12 11 : : PUNCT ejpam-4877 12 12 total total ADJ ejpam-4877 12 13 hop hop NOUN ejpam-4877 12 14 domination domination NOUN ejpam-4877 12 15 , , PUNCT ejpam-4877 12 16 total total ADJ ejpam-4877 12 17 hop hop NOUN ejpam-4877 12 18 domination domination NOUN ejpam-4877 12 19 number number NOUN ejpam-4877 12 20 , , PUNCT ejpam-4877 12 21 open open ADJ ejpam-4877 12 22 hop hop NOUN ejpam-4877 12 23 neighborhood neighborhood NOUN ejpam-4877 12 24 sequence sequence NOUN ejpam-4877 12 25 , , PUNCT ejpam-4877 12 26 grundy grundy PROPN ejpam-4877 12 27 total total NOUN ejpam-4877 12 28 hop hop NOUN ejpam-4877 12 29 dominating dominating NOUN ejpam-4877 12 30 sequence sequence NOUN ejpam-4877 12 31 , , PUNCT ejpam-4877 12 32 grundy grundy PROPN ejpam-4877 12 33 total total NOUN ejpam-4877 12 34 hop hop NOUN ejpam-4877 12 35 domination domination NOUN ejpam-4877 12 36 number number NOUN ejpam-4877 12 37 1 1 NUM ejpam-4877 12 38 . . PUNCT ejpam-4877 13 1 introduction introduction NOUN ejpam-4877 13 2 domination domination NOUN ejpam-4877 13 3 has have AUX ejpam-4877 13 4 attracted attract VERB ejpam-4877 13 5 many many ADJ ejpam-4877 13 6 researchers researcher NOUN ejpam-4877 13 7 because because SCONJ ejpam-4877 13 8 of of ADP ejpam-4877 13 9 its its PRON ejpam-4877 13 10 nice nice ADJ ejpam-4877 13 11 applications application NOUN ejpam-4877 13 12 in in ADP ejpam-4877 13 13 various various ADJ ejpam-4877 13 14 fields field NOUN ejpam-4877 13 15 and and CCONJ ejpam-4877 13 16 in in ADP ejpam-4877 13 17 networks network NOUN ejpam-4877 13 18 . . PUNCT ejpam-4877 14 1 a a DET ejpam-4877 14 2 number number NOUN ejpam-4877 14 3 of of ADP ejpam-4877 14 4 variations variation NOUN ejpam-4877 14 5 of of ADP ejpam-4877 14 6 the the DET ejpam-4877 14 7 domination domination NOUN ejpam-4877 14 8 concept concept NOUN ejpam-4877 14 9 ( ( PUNCT ejpam-4877 14 10 see see VERB ejpam-4877 14 11 for for ADP ejpam-4877 14 12 example example NOUN ejpam-4877 14 13 , , PUNCT ejpam-4877 14 14 [ [ X ejpam-4877 14 15 7 7 NUM ejpam-4877 14 16 , , PUNCT ejpam-4877 14 17 18 18 NUM ejpam-4877 14 18 , , PUNCT ejpam-4877 14 19 19 19 NUM ejpam-4877 14 20 , , PUNCT ejpam-4877 14 21 21 21 NUM ejpam-4877 14 22 ] ] PUNCT ejpam-4877 14 23 ) ) PUNCT ejpam-4877 14 24 have have AUX ejpam-4877 14 25 been be AUX ejpam-4877 14 26 introduced introduce VERB ejpam-4877 14 27 and and CCONJ ejpam-4877 14 28 studied study VERB ejpam-4877 14 29 . . PUNCT ejpam-4877 15 1 recently recently ADV ejpam-4877 15 2 , , PUNCT ejpam-4877 15 3 hop hop NOUN ejpam-4877 15 4 domination domination NOUN ejpam-4877 15 5 was be AUX ejpam-4877 15 6 defined define VERB ejpam-4877 15 7 and and CCONJ ejpam-4877 15 8 studied study VERB ejpam-4877 15 9 by by ADP ejpam-4877 15 10 natarajan natarajan PROPN ejpam-4877 15 11 and and CCONJ ejpam-4877 15 12 ayyaswamy ayyaswamy ADV ejpam-4877 15 13 in in ADP ejpam-4877 15 14 [ [ X ejpam-4877 15 15 17 17 NUM ejpam-4877 15 16 ] ] PUNCT ejpam-4877 15 17 . . PUNCT ejpam-4877 16 1 from from ADP ejpam-4877 16 2 then then ADV ejpam-4877 16 3 on on ADP ejpam-4877 16 4 a a DET ejpam-4877 16 5 lot lot NOUN ejpam-4877 16 6 of of ADP ejpam-4877 16 7 investigations investigation NOUN ejpam-4877 16 8 of of ADP ejpam-4877 16 9 the the DET ejpam-4877 16 10 concept concept NOUN ejpam-4877 16 11 and and CCONJ ejpam-4877 16 12 some some PRON ejpam-4877 16 13 of of ADP ejpam-4877 16 14 its its PRON ejpam-4877 16 15 variants variant NOUN ejpam-4877 16 16 have have AUX ejpam-4877 16 17 been be AUX ejpam-4877 16 18 done do VERB ejpam-4877 16 19 ( ( PUNCT ejpam-4877 16 20 see see VERB ejpam-4877 16 21 [ [ X ejpam-4877 16 22 1 1 NUM ejpam-4877 16 23 , , PUNCT ejpam-4877 16 24 2 2 NUM ejpam-4877 16 25 , , PUNCT ejpam-4877 16 26 8–15 8–15 VERB ejpam-4877 16 27 , , PUNCT ejpam-4877 16 28 20 20 NUM ejpam-4877 16 29 ] ] PUNCT ejpam-4877 16 30 ) ) PUNCT ejpam-4877 16 31 . . PUNCT ejpam-4877 17 1 in in ADP ejpam-4877 17 2 2014 2014 NUM ejpam-4877 17 3 , , PUNCT ejpam-4877 17 4 bresar bresar VERB ejpam-4877 17 5 et et PROPN ejpam-4877 17 6 al al PROPN ejpam-4877 17 7 . . PUNCT ejpam-4877 18 1 [ [ X ejpam-4877 18 2 4 4 X ejpam-4877 18 3 ] ] PUNCT ejpam-4877 18 4 introduced introduce VERB ejpam-4877 18 5 another another DET ejpam-4877 18 6 concept concept NOUN ejpam-4877 18 7 called call VERB ejpam-4877 18 8 grundy grundy PROPN ejpam-4877 18 9 dominating dominating NOUN ejpam-4877 18 10 sequence sequence NOUN ejpam-4877 18 11 in in ADP ejpam-4877 18 12 a a DET ejpam-4877 18 13 graph graph NOUN ejpam-4877 18 14 . . PUNCT ejpam-4877 19 1 the the DET ejpam-4877 19 2 newly newly ADV ejpam-4877 19 3 defined define VERB ejpam-4877 19 4 concept concept NOUN ejpam-4877 19 5 has have AUX ejpam-4877 19 6 subsequently subsequently ADV ejpam-4877 19 7 attracted attract VERB ejpam-4877 19 8 other other ADJ ejpam-4877 19 9 researchers researcher NOUN ejpam-4877 19 10 in in ADP ejpam-4877 19 11 the the DET ejpam-4877 19 12 area area NOUN ejpam-4877 19 13 to to PART ejpam-4877 19 14 study study VERB ejpam-4877 19 15 and and CCONJ ejpam-4877 19 16 generate generate VERB ejpam-4877 19 17 more more ADV ejpam-4877 19 18 interesting interesting ADJ ejpam-4877 19 19 results result NOUN ejpam-4877 19 20 ( ( PUNCT ejpam-4877 19 21 see see VERB ejpam-4877 19 22 [ [ X ejpam-4877 19 23 6 6 NUM ejpam-4877 19 24 , , PUNCT ejpam-4877 19 25 16 16 NUM ejpam-4877 19 26 ] ] PUNCT ejpam-4877 19 27 ) ) PUNCT ejpam-4877 19 28 on on ADP ejpam-4877 19 29 it it PRON ejpam-4877 19 30 . . PUNCT ejpam-4877 20 1 ∗corresponding ∗corresponde VERB ejpam-4877 20 2 author author NOUN ejpam-4877 20 3 . . PUNCT ejpam-4877 21 1 doi doi PROPN ejpam-4877 21 2 : : PUNCT ejpam-4877 21 3 https://doi.org/10.29020/nybg.ejpam.v16i4.4877 https://doi.org/10.29020/nybg.ejpam.v16i4.4877 PROPN ejpam-4877 21 4 email email NOUN ejpam-4877 21 5 addresses address NOUN ejpam-4877 21 6 : : PUNCT ejpam-4877 21 7 javierhassan@msutawi-tawi.edu.ph javierhassan@msutawi-tawi.edu.ph PROPN ejpam-4877 21 8 ( ( PUNCT ejpam-4877 21 9 j. j. PROPN ejpam-4877 21 10 hassan hassan PROPN ejpam-4877 21 11 ) ) PUNCT ejpam-4877 21 12 , , PUNCT ejpam-4877 21 13 sergio.canoy@g.msuiit.edu.ph sergio.canoy@g.msuiit.edu.ph PROPN ejpam-4877 21 14 ( ( PUNCT ejpam-4877 21 15 s. s. PROPN ejpam-4877 21 16 canoy canoy PROPN ejpam-4877 21 17 ) ) PUNCT ejpam-4877 21 18 https://www.ejpam.com https://www.ejpam.com X ejpam-4877 22 1 2597 2597 NUM ejpam-4877 23 1 © © PROPN ejpam-4877 23 2 2023 2023 NUM ejpam-4877 23 3 ejpam ejpam NOUN ejpam-4877 23 4 all all DET ejpam-4877 23 5 rights right NOUN ejpam-4877 23 6 reserved reserve VERB ejpam-4877 23 7 . . PUNCT ejpam-4877 24 1 j.a j.a PROPN ejpam-4877 24 2 . . PROPN ejpam-4877 24 3 hassan hassan PROPN ejpam-4877 24 4 , , PUNCT ejpam-4877 24 5 s. s. PROPN ejpam-4877 24 6 canoy canoy PROPN ejpam-4877 24 7 / / SYM ejpam-4877 24 8 eur eur PROPN ejpam-4877 24 9 . . PUNCT ejpam-4877 25 1 j. j. PROPN ejpam-4877 25 2 pure pure PROPN ejpam-4877 25 3 appl appl PROPN ejpam-4877 25 4 . . PROPN ejpam-4877 25 5 math math PROPN ejpam-4877 25 6 , , PUNCT ejpam-4877 25 7 16 16 NUM ejpam-4877 25 8 ( ( PUNCT ejpam-4877 25 9 4 4 NUM ejpam-4877 25 10 ) ) PUNCT ejpam-4877 25 11 ( ( PUNCT ejpam-4877 25 12 2023 2023 NUM ejpam-4877 25 13 ) ) PUNCT ejpam-4877 25 14 , , PUNCT ejpam-4877 25 15 2597 2597 NUM ejpam-4877 25 16 - - SYM ejpam-4877 25 17 2612 2612 NUM ejpam-4877 25 18 2598 2598 NUM ejpam-4877 25 19 in in ADP ejpam-4877 25 20 2016 2016 NUM ejpam-4877 25 21 , , PUNCT ejpam-4877 25 22 the the DET ejpam-4877 25 23 concept concept NOUN ejpam-4877 25 24 of of ADP ejpam-4877 25 25 grundy grundy PROPN ejpam-4877 25 26 total total ADJ ejpam-4877 25 27 domination domination NOUN ejpam-4877 25 28 in in ADP ejpam-4877 25 29 graphs graph NOUN ejpam-4877 25 30 was be AUX ejpam-4877 25 31 investigated investigate VERB ejpam-4877 25 32 by by ADP ejpam-4877 25 33 bresar bresar VERB ejpam-4877 25 34 et et PROPN ejpam-4877 25 35 al al PROPN ejpam-4877 25 36 . . PUNCT ejpam-4877 26 1 [ [ X ejpam-4877 26 2 5 5 NUM ejpam-4877 26 3 ] ] PUNCT ejpam-4877 26 4 . . PUNCT ejpam-4877 27 1 bresar bresar VERB ejpam-4877 27 2 [ [ X ejpam-4877 27 3 3 3 NUM ejpam-4877 27 4 ] ] PUNCT ejpam-4877 27 5 studied study VERB ejpam-4877 27 6 further far ADV ejpam-4877 27 7 the the DET ejpam-4877 27 8 concept concept NOUN ejpam-4877 27 9 on on ADP ejpam-4877 27 10 the the DET ejpam-4877 27 11 product product NOUN ejpam-4877 27 12 of of ADP ejpam-4877 27 13 graphs graph NOUN ejpam-4877 27 14 . . PUNCT ejpam-4877 28 1 in in ADP ejpam-4877 28 2 this this DET ejpam-4877 28 3 study study NOUN ejpam-4877 28 4 , , PUNCT ejpam-4877 28 5 the the DET ejpam-4877 28 6 concept concept NOUN ejpam-4877 28 7 of of ADP ejpam-4877 28 8 grundy grundy PROPN ejpam-4877 28 9 total total PROPN ejpam-4877 28 10 hop hop NOUN ejpam-4877 28 11 domination domination NOUN ejpam-4877 28 12 number number NOUN ejpam-4877 28 13 of of ADP ejpam-4877 28 14 a a DET ejpam-4877 28 15 graph graph NOUN ejpam-4877 28 16 will will AUX ejpam-4877 28 17 be be AUX ejpam-4877 28 18 introduced introduce VERB ejpam-4877 28 19 and and CCONJ ejpam-4877 28 20 investigated investigate VERB ejpam-4877 28 21 . . PUNCT ejpam-4877 29 1 its its PRON ejpam-4877 29 2 relationship relationship NOUN ejpam-4877 29 3 with with ADP ejpam-4877 29 4 total total ADJ ejpam-4877 29 5 hop hop NOUN ejpam-4877 29 6 domination domination NOUN ejpam-4877 29 7 , , PUNCT ejpam-4877 29 8 and and CCONJ ejpam-4877 29 9 grundy grundy PROPN ejpam-4877 29 10 hop hop PROPN ejpam-4877 29 11 domination domination NOUN ejpam-4877 29 12 numbers number NOUN ejpam-4877 29 13 of of ADP ejpam-4877 29 14 a a DET ejpam-4877 29 15 graph graph NOUN ejpam-4877 29 16 will will AUX ejpam-4877 29 17 be be AUX ejpam-4877 29 18 given give VERB ejpam-4877 29 19 . . PUNCT ejpam-4877 30 1 bounds bound NOUN ejpam-4877 30 2 for for ADP ejpam-4877 30 3 the the DET ejpam-4877 30 4 parameter parameter NOUN ejpam-4877 30 5 will will AUX ejpam-4877 30 6 be be AUX ejpam-4877 30 7 determined determine VERB ejpam-4877 30 8 for for SCONJ ejpam-4877 30 9 the the DET ejpam-4877 30 10 shadow shadow NOUN ejpam-4877 30 11 graph graph NOUN ejpam-4877 30 12 as as ADV ejpam-4877 30 13 well well ADV ejpam-4877 30 14 as as ADP ejpam-4877 30 15 the the DET ejpam-4877 30 16 join join NOUN ejpam-4877 30 17 and and CCONJ ejpam-4877 30 18 the the DET ejpam-4877 30 19 corona corona NOUN ejpam-4877 30 20 of of ADP ejpam-4877 30 21 two two NUM ejpam-4877 30 22 graphs graph NOUN ejpam-4877 30 23 . . PUNCT ejpam-4877 31 1 2 2 X ejpam-4877 31 2 . . X ejpam-4877 31 3 terminology terminology NOUN ejpam-4877 31 4 and and CCONJ ejpam-4877 31 5 notation notation NOUN ejpam-4877 31 6 two two NUM ejpam-4877 31 7 vertices vertex NOUN ejpam-4877 31 8 u u NOUN ejpam-4877 31 9 , , PUNCT ejpam-4877 31 10 v v NOUN ejpam-4877 31 11 of of ADP ejpam-4877 31 12 a a DET ejpam-4877 31 13 graph graph NOUN ejpam-4877 31 14 g g NOUN ejpam-4877 31 15 are be AUX ejpam-4877 31 16 adjacent adjacent ADJ ejpam-4877 31 17 , , PUNCT ejpam-4877 31 18 or or CCONJ ejpam-4877 31 19 neighbors neighbor NOUN ejpam-4877 31 20 , , PUNCT ejpam-4877 31 21 if if SCONJ ejpam-4877 31 22 uv uv NOUN ejpam-4877 31 23 is be AUX ejpam-4877 31 24 an an DET ejpam-4877 31 25 edge edge NOUN ejpam-4877 31 26 of of ADP ejpam-4877 31 27 g. g. PROPN ejpam-4877 31 28 moreover moreover ADV ejpam-4877 31 29 , , PUNCT ejpam-4877 31 30 an an DET ejpam-4877 31 31 edge edge NOUN ejpam-4877 31 32 uv uv NOUN ejpam-4877 31 33 of of ADP ejpam-4877 31 34 g g PROPN ejpam-4877 31 35 is be AUX ejpam-4877 31 36 incident incident NOUN ejpam-4877 31 37 to to ADP ejpam-4877 31 38 two two NUM ejpam-4877 31 39 vertices vertex NOUN ejpam-4877 31 40 u u NOUN ejpam-4877 31 41 , , PUNCT ejpam-4877 31 42 v v NOUN ejpam-4877 31 43 of of ADP ejpam-4877 31 44 g. g. PROPN ejpam-4877 31 45 the the DET ejpam-4877 31 46 set set NOUN ejpam-4877 31 47 of of ADP ejpam-4877 31 48 neighbors neighbor NOUN ejpam-4877 31 49 of of ADP ejpam-4877 31 50 a a DET ejpam-4877 31 51 vertex vertex NOUN ejpam-4877 31 52 u u NOUN ejpam-4877 31 53 in in ADP ejpam-4877 31 54 g g NOUN ejpam-4877 31 55 , , PUNCT ejpam-4877 31 56 denoted denote VERB ejpam-4877 31 57 by by ADP ejpam-4877 31 58 ng(u ng(u NOUN ejpam-4877 31 59 ) ) PUNCT ejpam-4877 31 60 , , PUNCT ejpam-4877 31 61 is be AUX ejpam-4877 31 62 called call VERB ejpam-4877 31 63 the the DET ejpam-4877 31 64 open open ADJ ejpam-4877 31 65 neighborhood neighborhood NOUN ejpam-4877 31 66 of of ADP ejpam-4877 31 67 u u PROPN ejpam-4877 31 68 in in ADP ejpam-4877 31 69 g. g. PROPN ejpam-4877 31 70 the the DET ejpam-4877 31 71 closed close VERB ejpam-4877 31 72 neighborhood neighborhood NOUN ejpam-4877 31 73 of of ADP ejpam-4877 31 74 u u NOUN ejpam-4877 31 75 in in ADP ejpam-4877 31 76 g g PROPN ejpam-4877 31 77 is be AUX ejpam-4877 31 78 the the DET ejpam-4877 31 79 set set NOUN ejpam-4877 31 80 ng[u ng[u PROPN ejpam-4877 31 81 ] ] X ejpam-4877 31 82 = = SYM ejpam-4877 31 83 ng(u ng(u PROPN ejpam-4877 31 84 ) ) PUNCT ejpam-4877 31 85 ∪ ∪ NOUN ejpam-4877 31 86 { { PUNCT ejpam-4877 31 87 u u NOUN ejpam-4877 31 88 } } PUNCT ejpam-4877 31 89 . . PUNCT ejpam-4877 32 1 if if SCONJ ejpam-4877 32 2 x x PROPN ejpam-4877 32 3 ⊆ ⊆ NUM ejpam-4877 32 4 v v X ejpam-4877 32 5 ( ( PUNCT ejpam-4877 32 6 g g NOUN ejpam-4877 32 7 ) ) PUNCT ejpam-4877 32 8 , , PUNCT ejpam-4877 32 9 the the DET ejpam-4877 32 10 open open ADJ ejpam-4877 32 11 neighborhood neighborhood NOUN ejpam-4877 32 12 of of ADP ejpam-4877 32 13 x x PUNCT ejpam-4877 32 14 in in ADP ejpam-4877 32 15 g g PROPN ejpam-4877 32 16 is be AUX ejpam-4877 32 17 the the DET ejpam-4877 32 18 set set NOUN ejpam-4877 32 19 ng(x ng(x NUM ejpam-4877 32 20 ) ) PUNCT ejpam-4877 33 1 = = SYM ejpam-4877 33 2 ⋃ ⋃ NOUN ejpam-4877 33 3 u∈x u∈x NOUN ejpam-4877 33 4 ng(u ng(u NOUN ejpam-4877 33 5 ) ) PUNCT ejpam-4877 33 6 . . PUNCT ejpam-4877 34 1 the the DET ejpam-4877 34 2 closed closed ADJ ejpam-4877 34 3 neighborhood neighborhood NOUN ejpam-4877 34 4 of of ADP ejpam-4877 34 5 x x PUNCT ejpam-4877 34 6 in in ADP ejpam-4877 34 7 g g PROPN ejpam-4877 34 8 is be AUX ejpam-4877 34 9 the the DET ejpam-4877 34 10 set set NOUN ejpam-4877 34 11 ng[x ng[x PROPN ejpam-4877 34 12 ] ] X ejpam-4877 34 13 = = PUNCT ejpam-4877 34 14 ng(x ng(x X ejpam-4877 34 15 ) ) PUNCT ejpam-4877 34 16 ∪x ∪x X ejpam-4877 34 17 . . PUNCT ejpam-4877 35 1 let let VERB ejpam-4877 35 2 g g PRON ejpam-4877 35 3 be be AUX ejpam-4877 35 4 a a DET ejpam-4877 35 5 graph graph NOUN ejpam-4877 35 6 . . PUNCT ejpam-4877 36 1 a a DET ejpam-4877 36 2 set set NOUN ejpam-4877 36 3 d d NOUN ejpam-4877 36 4 ⊆ ⊆ NUM ejpam-4877 36 5 v v ADP ejpam-4877 36 6 ( ( PUNCT ejpam-4877 36 7 g g NOUN ejpam-4877 36 8 ) ) PUNCT ejpam-4877 36 9 is be AUX ejpam-4877 36 10 a a DET ejpam-4877 36 11 total total ADJ ejpam-4877 36 12 dominating dominating NOUN ejpam-4877 36 13 set set NOUN ejpam-4877 36 14 of of ADP ejpam-4877 36 15 g g PROPN ejpam-4877 36 16 if if SCONJ ejpam-4877 36 17 for for ADP ejpam-4877 36 18 every every DET ejpam-4877 36 19 v v NUM ejpam-4877 36 20 ∈ ∈ NOUN ejpam-4877 36 21 v v NOUN ejpam-4877 36 22 ( ( PUNCT ejpam-4877 36 23 g g NOUN ejpam-4877 36 24 ) ) PUNCT ejpam-4877 36 25 , , PUNCT ejpam-4877 36 26 there there PRON ejpam-4877 36 27 exists exist VERB ejpam-4877 36 28 u u NOUN ejpam-4877 36 29 ∈ ∈ PROPN ejpam-4877 36 30 d d ADP ejpam-4877 36 31 such such ADJ ejpam-4877 36 32 that that DET ejpam-4877 36 33 uv uv PROPN ejpam-4877 36 34 ∈ ∈ PROPN ejpam-4877 36 35 e(g e(g PROPN ejpam-4877 36 36 ) ) PUNCT ejpam-4877 36 37 , , PUNCT ejpam-4877 36 38 that that ADV ejpam-4877 36 39 is is ADV ejpam-4877 36 40 , , PUNCT ejpam-4877 36 41 ng(d ng(d PUNCT ejpam-4877 36 42 ) ) PUNCT ejpam-4877 36 43 = = SYM ejpam-4877 36 44 v v X ejpam-4877 36 45 ( ( PUNCT ejpam-4877 36 46 g g NOUN ejpam-4877 36 47 ) ) PUNCT ejpam-4877 36 48 . . PUNCT ejpam-4877 37 1 the the DET ejpam-4877 37 2 total total ADJ ejpam-4877 37 3 domination domination NOUN ejpam-4877 37 4 number number NOUN ejpam-4877 37 5 of of ADP ejpam-4877 37 6 g g NOUN ejpam-4877 37 7 , , PUNCT ejpam-4877 37 8 denoted denote VERB ejpam-4877 37 9 by by ADP ejpam-4877 37 10 γt(g γt(g NOUN ejpam-4877 37 11 ) ) PUNCT ejpam-4877 37 12 , , PUNCT ejpam-4877 37 13 is be AUX ejpam-4877 37 14 the the DET ejpam-4877 37 15 minimum minimum ADJ ejpam-4877 37 16 cardinality cardinality NOUN ejpam-4877 37 17 of of ADP ejpam-4877 37 18 a a DET ejpam-4877 37 19 total total ADJ ejpam-4877 37 20 dominating dominating NOUN ejpam-4877 37 21 set set NOUN ejpam-4877 37 22 of of ADP ejpam-4877 37 23 g. g. PROPN ejpam-4877 37 24 any any DET ejpam-4877 37 25 total total ADJ ejpam-4877 37 26 dominating dominating NOUN ejpam-4877 37 27 set set VERB ejpam-4877 37 28 with with ADP ejpam-4877 37 29 cardinality cardinality NOUN ejpam-4877 37 30 equal equal ADJ ejpam-4877 37 31 to to ADP ejpam-4877 37 32 γt(g γt(g NOUN ejpam-4877 37 33 ) ) PUNCT ejpam-4877 37 34 is be AUX ejpam-4877 37 35 called call VERB ejpam-4877 37 36 a a DET ejpam-4877 37 37 γt γt NOUN ejpam-4877 37 38 - - NOUN ejpam-4877 37 39 set set NOUN ejpam-4877 37 40 . . PUNCT ejpam-4877 38 1 let let VERB ejpam-4877 38 2 s s PRON ejpam-4877 38 3 = = PUNCT ejpam-4877 38 4 ( ( PUNCT ejpam-4877 38 5 v1 v1 PROPN ejpam-4877 38 6 , , PUNCT ejpam-4877 38 7 v2 v2 PROPN ejpam-4877 38 8 , , PUNCT ejpam-4877 38 9 · · PUNCT ejpam-4877 38 10 · · PUNCT ejpam-4877 38 11 · · PUNCT ejpam-4877 38 12 , , PUNCT ejpam-4877 38 13 vk vk AUX ejpam-4877 38 14 ) ) PUNCT ejpam-4877 38 15 be be AUX ejpam-4877 38 16 a a DET ejpam-4877 38 17 sequence sequence NOUN ejpam-4877 38 18 of of ADP ejpam-4877 38 19 distint distint ADJ ejpam-4877 38 20 vertices vertex NOUN ejpam-4877 38 21 of of ADP ejpam-4877 38 22 a a DET ejpam-4877 38 23 graph graph NOUN ejpam-4877 38 24 g g NOUN ejpam-4877 38 25 , , PUNCT ejpam-4877 38 26 and and CCONJ ejpam-4877 38 27 let let VERB ejpam-4877 38 28 ŝ ŝ X ejpam-4877 38 29 = = SYM ejpam-4877 38 30 { { PUNCT ejpam-4877 38 31 v1 v1 PROPN ejpam-4877 38 32 , , PUNCT ejpam-4877 38 33 v2 v2 PROPN ejpam-4877 38 34 , , PUNCT ejpam-4877 38 35 . . PUNCT ejpam-4877 38 36 . . PUNCT ejpam-4877 39 1 . . PUNCT ejpam-4877 40 1 , , PUNCT ejpam-4877 40 2 vk vk ADP ejpam-4877 40 3 } } PUNCT ejpam-4877 40 4 . . PUNCT ejpam-4877 41 1 then then ADV ejpam-4877 41 2 s s VERB ejpam-4877 41 3 is be AUX ejpam-4877 41 4 a a DET ejpam-4877 41 5 legal legal ADJ ejpam-4877 41 6 open open ADJ ejpam-4877 41 7 neighborhood neighborhood NOUN ejpam-4877 41 8 sequence sequence NOUN ejpam-4877 41 9 if if SCONJ ejpam-4877 41 10 ng(vi)\ ng(vi)\ NOUN ejpam-4877 41 11 ⋃i−1 ⋃i−1 NOUN ejpam-4877 41 12 j=1ng(vj j=1ng(vj NOUN ejpam-4877 41 13 ) ) PUNCT ejpam-4877 41 14 ̸= ̸= PROPN ejpam-4877 41 15 ∅ ∅ NOUN ejpam-4877 41 16 for for ADP ejpam-4877 41 17 every every DET ejpam-4877 41 18 i i PROPN ejpam-4877 41 19 ∈ ∈ PROPN ejpam-4877 41 20 { { PUNCT ejpam-4877 41 21 2 2 NUM ejpam-4877 41 22 , , PUNCT ejpam-4877 41 23 . . PUNCT ejpam-4877 41 24 . . PUNCT ejpam-4877 41 25 . . PUNCT ejpam-4877 42 1 , , PUNCT ejpam-4877 42 2 k k X ejpam-4877 42 3 } } PUNCT ejpam-4877 42 4 . . PUNCT ejpam-4877 43 1 if if SCONJ ejpam-4877 43 2 , , PUNCT ejpam-4877 43 3 in in ADP ejpam-4877 43 4 addition addition NOUN ejpam-4877 43 5 , , PUNCT ejpam-4877 43 6 ŝ ŝ X ejpam-4877 43 7 is be AUX ejpam-4877 43 8 a a DET ejpam-4877 43 9 total total ADJ ejpam-4877 43 10 dominating dominating NOUN ejpam-4877 43 11 set set NOUN ejpam-4877 43 12 of of ADP ejpam-4877 43 13 g g PROPN ejpam-4877 43 14 , , PUNCT ejpam-4877 43 15 then then ADV ejpam-4877 43 16 s s VERB ejpam-4877 43 17 is be AUX ejpam-4877 43 18 called call VERB ejpam-4877 43 19 a a DET ejpam-4877 43 20 grundy grundy PROPN ejpam-4877 43 21 total total ADJ ejpam-4877 43 22 dominating dominating NOUN ejpam-4877 43 23 sequence sequence NOUN ejpam-4877 43 24 . . PUNCT ejpam-4877 44 1 the the DET ejpam-4877 44 2 maximum maximum ADJ ejpam-4877 44 3 length length NOUN ejpam-4877 44 4 of of ADP ejpam-4877 44 5 a a DET ejpam-4877 44 6 grundy grundy PROPN ejpam-4877 44 7 total total ADJ ejpam-4877 44 8 dominating dominating NOUN ejpam-4877 44 9 sequence sequence NOUN ejpam-4877 44 10 in in ADP ejpam-4877 44 11 a a DET ejpam-4877 44 12 graph graph NOUN ejpam-4877 44 13 g g NOUN ejpam-4877 44 14 is be AUX ejpam-4877 44 15 called call VERB ejpam-4877 44 16 the the DET ejpam-4877 44 17 grundy grundy PROPN ejpam-4877 44 18 total total NOUN ejpam-4877 44 19 domination domination NOUN ejpam-4877 44 20 number number NOUN ejpam-4877 44 21 of of ADP ejpam-4877 44 22 g g NOUN ejpam-4877 44 23 , , PUNCT ejpam-4877 44 24 and and CCONJ ejpam-4877 44 25 is be AUX ejpam-4877 44 26 denoted denote VERB ejpam-4877 44 27 by by ADP ejpam-4877 44 28 γtgr(g γtgr(g PROPN ejpam-4877 44 29 ) ) PUNCT ejpam-4877 44 30 . . PUNCT ejpam-4877 45 1 a a DET ejpam-4877 45 2 vertex vertex NOUN ejpam-4877 45 3 v v NOUN ejpam-4877 45 4 in in ADP ejpam-4877 45 5 g g PROPN ejpam-4877 45 6 is be AUX ejpam-4877 45 7 a a DET ejpam-4877 45 8 hop hop NOUN ejpam-4877 45 9 neighbor neighbor NOUN ejpam-4877 45 10 of of ADP ejpam-4877 45 11 vertex vertex NOUN ejpam-4877 45 12 u u NOUN ejpam-4877 45 13 in in ADP ejpam-4877 45 14 g g PROPN ejpam-4877 45 15 if if SCONJ ejpam-4877 45 16 dg(u dg(u NOUN ejpam-4877 45 17 , , PUNCT ejpam-4877 45 18 v v NOUN ejpam-4877 45 19 ) ) PUNCT ejpam-4877 45 20 = = SYM ejpam-4877 45 21 2 2 X ejpam-4877 45 22 . . X ejpam-4877 46 1 the the DET ejpam-4877 46 2 set set ADJ ejpam-4877 46 3 n2 n2 ADJ ejpam-4877 46 4 g(u g(u PROPN ejpam-4877 46 5 ) ) PUNCT ejpam-4877 46 6 = = PRON ejpam-4877 46 7 { { PUNCT ejpam-4877 46 8 v v NUM ejpam-4877 46 9 ∈ ∈ NOUN ejpam-4877 46 10 v v NOUN ejpam-4877 46 11 ( ( PUNCT ejpam-4877 46 12 g g NOUN ejpam-4877 46 13 ) ) PUNCT ejpam-4877 46 14 : : PUNCT ejpam-4877 46 15 dg(v dg(v X ejpam-4877 46 16 , , PUNCT ejpam-4877 46 17 u u NOUN ejpam-4877 46 18 ) ) PUNCT ejpam-4877 46 19 = = SYM ejpam-4877 46 20 2 2 X ejpam-4877 46 21 } } PUNCT ejpam-4877 46 22 is be AUX ejpam-4877 46 23 called call VERB ejpam-4877 46 24 the the DET ejpam-4877 46 25 open open ADJ ejpam-4877 46 26 hop hop NOUN ejpam-4877 46 27 neighborhood neighborhood NOUN ejpam-4877 46 28 of of ADP ejpam-4877 46 29 u. u. PROPN ejpam-4877 46 30 the the DET ejpam-4877 46 31 closed closed ADJ ejpam-4877 46 32 hop hop NOUN ejpam-4877 46 33 neighborhood neighborhood NOUN ejpam-4877 46 34 of of ADP ejpam-4877 46 35 u u PROPN ejpam-4877 46 36 in in ADP ejpam-4877 46 37 g g PROPN ejpam-4877 46 38 is be AUX ejpam-4877 46 39 given give VERB ejpam-4877 46 40 by by ADP ejpam-4877 46 41 n2 n2 PROPN ejpam-4877 46 42 g[u g[u PROPN ejpam-4877 46 43 ] ] X ejpam-4877 46 44 = = SYM ejpam-4877 46 45 n2 n2 ADJ ejpam-4877 46 46 g(u g(u PROPN ejpam-4877 46 47 ) ) PUNCT ejpam-4877 46 48 ∪ ∪ NOUN ejpam-4877 46 49 { { PUNCT ejpam-4877 46 50 u u NOUN ejpam-4877 46 51 } } PUNCT ejpam-4877 46 52 . . PUNCT ejpam-4877 47 1 the the DET ejpam-4877 47 2 open open ADJ ejpam-4877 47 3 hop hop NOUN ejpam-4877 47 4 neighborhood neighborhood NOUN ejpam-4877 47 5 of of ADP ejpam-4877 47 6 x x PROPN ejpam-4877 47 7 ⊆ ⊆ NUM ejpam-4877 47 8 v v ADP ejpam-4877 47 9 ( ( PUNCT ejpam-4877 47 10 g g NOUN ejpam-4877 47 11 ) ) PUNCT ejpam-4877 47 12 is be AUX ejpam-4877 47 13 the the DET ejpam-4877 47 14 set set ADJ ejpam-4877 47 15 n2 n2 ADJ ejpam-4877 47 16 g(x g(x NOUN ejpam-4877 47 17 ) ) PUNCT ejpam-4877 48 1 = = SYM ejpam-4877 48 2 ⋃ ⋃ NOUN ejpam-4877 48 3 u∈x u∈x ADJ ejpam-4877 48 4 n2 n2 NOUN ejpam-4877 48 5 g(u g(u PROPN ejpam-4877 48 6 ) ) PUNCT ejpam-4877 48 7 . . PUNCT ejpam-4877 49 1 the the DET ejpam-4877 49 2 closed closed ADJ ejpam-4877 49 3 hop hop NOUN ejpam-4877 49 4 neighborhood neighborhood NOUN ejpam-4877 49 5 of of ADP ejpam-4877 49 6 x x PUNCT ejpam-4877 49 7 in in ADP ejpam-4877 49 8 g g PROPN ejpam-4877 49 9 is be AUX ejpam-4877 49 10 the the DET ejpam-4877 49 11 set set ADJ ejpam-4877 49 12 n2 n2 NOUN ejpam-4877 49 13 g[x g[x PROPN ejpam-4877 49 14 ] ] X ejpam-4877 49 15 = = SYM ejpam-4877 49 16 n2 n2 PROPN ejpam-4877 49 17 g(x g(x NOUN ejpam-4877 49 18 ) ) PUNCT ejpam-4877 49 19 ∪x ∪x NUM ejpam-4877 49 20 . . PUNCT ejpam-4877 50 1 a a DET ejpam-4877 50 2 set set NOUN ejpam-4877 50 3 s s NOUN ejpam-4877 50 4 ⊆ ⊆ NUM ejpam-4877 50 5 v v NOUN ejpam-4877 50 6 ( ( PUNCT ejpam-4877 50 7 g g NOUN ejpam-4877 50 8 ) ) PUNCT ejpam-4877 50 9 is be AUX ejpam-4877 50 10 a a DET ejpam-4877 50 11 hop hop NOUN ejpam-4877 50 12 dominating dominating NOUN ejpam-4877 50 13 set set NOUN ejpam-4877 50 14 of of ADP ejpam-4877 50 15 g g PROPN ejpam-4877 50 16 if if SCONJ ejpam-4877 50 17 n2 n2 ADJ ejpam-4877 50 18 g[s g[s PROPN ejpam-4877 50 19 ] ] X ejpam-4877 50 20 = = SYM ejpam-4877 50 21 v v NOUN ejpam-4877 50 22 ( ( PUNCT ejpam-4877 50 23 g g NOUN ejpam-4877 50 24 ) ) PUNCT ejpam-4877 50 25 , , PUNCT ejpam-4877 50 26 that that ADV ejpam-4877 50 27 is is ADV ejpam-4877 50 28 , , PUNCT ejpam-4877 50 29 for for ADP ejpam-4877 50 30 every every DET ejpam-4877 50 31 v v NUM ejpam-4877 50 32 ∈ ∈ NOUN ejpam-4877 50 33 v v NOUN ejpam-4877 50 34 ( ( PUNCT ejpam-4877 50 35 g)\s g)\s NOUN ejpam-4877 50 36 , , PUNCT ejpam-4877 50 37 there there PRON ejpam-4877 50 38 exists exist VERB ejpam-4877 50 39 u u PROPN ejpam-4877 50 40 ∈ ∈ PROPN ejpam-4877 50 41 s s VERB ejpam-4877 50 42 such such ADJ ejpam-4877 50 43 that that DET ejpam-4877 50 44 dg(u dg(u ADJ ejpam-4877 50 45 , , PUNCT ejpam-4877 50 46 v v NOUN ejpam-4877 50 47 ) ) PUNCT ejpam-4877 51 1 = = SYM ejpam-4877 51 2 2 2 X ejpam-4877 51 3 . . PUNCT ejpam-4877 52 1 the the DET ejpam-4877 52 2 minimum minimum ADJ ejpam-4877 52 3 cardinality cardinality NOUN ejpam-4877 52 4 among among ADP ejpam-4877 52 5 all all DET ejpam-4877 52 6 hop hop NOUN ejpam-4877 52 7 dominating dominating NOUN ejpam-4877 52 8 sets set NOUN ejpam-4877 52 9 of of ADP ejpam-4877 52 10 g g NOUN ejpam-4877 52 11 , , PUNCT ejpam-4877 52 12 denoted denote VERB ejpam-4877 52 13 by by ADP ejpam-4877 52 14 γh(g γh(g NOUN ejpam-4877 52 15 ) ) PUNCT ejpam-4877 52 16 , , PUNCT ejpam-4877 52 17 is be AUX ejpam-4877 52 18 called call VERB ejpam-4877 52 19 the the DET ejpam-4877 52 20 hop hop NOUN ejpam-4877 52 21 domination domination NOUN ejpam-4877 52 22 number number NOUN ejpam-4877 52 23 of of ADP ejpam-4877 52 24 g. g. PROPN ejpam-4877 52 25 any any DET ejpam-4877 52 26 hop hop NOUN ejpam-4877 52 27 dominating dominating NOUN ejpam-4877 52 28 set set VERB ejpam-4877 52 29 with with ADP ejpam-4877 52 30 cardinality cardinality NOUN ejpam-4877 52 31 equal equal ADJ ejpam-4877 52 32 to to ADP ejpam-4877 52 33 γh(g γh(g NOUN ejpam-4877 52 34 ) ) PUNCT ejpam-4877 52 35 is be AUX ejpam-4877 52 36 called call VERB ejpam-4877 52 37 a a DET ejpam-4877 52 38 γh γh ADV ejpam-4877 52 39 - - PUNCT ejpam-4877 52 40 set set NOUN ejpam-4877 52 41 . . PUNCT ejpam-4877 53 1 let let VERB ejpam-4877 53 2 s s PRON ejpam-4877 53 3 = = PUNCT ejpam-4877 53 4 ( ( PUNCT ejpam-4877 53 5 v1 v1 PROPN ejpam-4877 53 6 , , PUNCT ejpam-4877 53 7 v2 v2 PROPN ejpam-4877 53 8 , , PUNCT ejpam-4877 53 9 · · PUNCT ejpam-4877 53 10 · · PUNCT ejpam-4877 53 11 · · PUNCT ejpam-4877 53 12 , , PUNCT ejpam-4877 53 13 vk vk AUX ejpam-4877 53 14 ) ) PUNCT ejpam-4877 53 15 be be AUX ejpam-4877 53 16 a a DET ejpam-4877 53 17 sequence sequence NOUN ejpam-4877 53 18 of of ADP ejpam-4877 53 19 distinct distinct ADJ ejpam-4877 53 20 vertices vertex NOUN ejpam-4877 53 21 of of ADP ejpam-4877 53 22 g g NOUN ejpam-4877 53 23 and and CCONJ ejpam-4877 53 24 let let VERB ejpam-4877 53 25 ŝ ŝ X ejpam-4877 53 26 = = SYM ejpam-4877 53 27 { { PUNCT ejpam-4877 53 28 v1 v1 PROPN ejpam-4877 53 29 , , PUNCT ejpam-4877 53 30 · · PUNCT ejpam-4877 53 31 · · PUNCT ejpam-4877 53 32 · · PUNCT ejpam-4877 53 33 , , PUNCT ejpam-4877 53 34 vk vk ADP ejpam-4877 53 35 } } PUNCT ejpam-4877 53 36 . . PUNCT ejpam-4877 54 1 then then ADV ejpam-4877 54 2 s s VERB ejpam-4877 54 3 is be AUX ejpam-4877 54 4 a a DET ejpam-4877 54 5 legal legal ADJ ejpam-4877 54 6 closed close VERB ejpam-4877 54 7 hop hop NOUN ejpam-4877 54 8 neighborhood neighborhood NOUN ejpam-4877 54 9 sequence sequence NOUN ejpam-4877 54 10 of of ADP ejpam-4877 54 11 g g PROPN ejpam-4877 54 12 if if SCONJ ejpam-4877 54 13 n2 n2 PROPN ejpam-4877 54 14 g[vi g[vi PROPN ejpam-4877 54 15 ] ] PUNCT ejpam-4877 54 16 \ \ X ejpam-4877 55 1 ∪i−1 ∪i−1 PUNCT ejpam-4877 55 2 j=1n j=1n PROPN ejpam-4877 55 3 2 2 NUM ejpam-4877 55 4 g[vj g[vj PROPN ejpam-4877 55 5 ] ] PUNCT ejpam-4877 55 6 ̸= ̸= PROPN ejpam-4877 55 7 ∅ ∅ NOUN ejpam-4877 55 8 for for ADP ejpam-4877 55 9 each each DET ejpam-4877 55 10 i i PRON ejpam-4877 55 11 ∈ ∈ PROPN ejpam-4877 55 12 { { PUNCT ejpam-4877 55 13 2 2 NUM ejpam-4877 55 14 , , PUNCT ejpam-4877 55 15 · · PUNCT ejpam-4877 55 16 · · PUNCT ejpam-4877 55 17 · · PUNCT ejpam-4877 55 18 , , PUNCT ejpam-4877 55 19 k k NOUN ejpam-4877 55 20 } } PUNCT ejpam-4877 55 21 . . PUNCT ejpam-4877 56 1 if if SCONJ ejpam-4877 56 2 , , PUNCT ejpam-4877 56 3 in in ADP ejpam-4877 56 4 addition addition NOUN ejpam-4877 56 5 , , PUNCT ejpam-4877 56 6 ŝ ŝ X ejpam-4877 56 7 is be AUX ejpam-4877 56 8 a a DET ejpam-4877 56 9 hop hop NOUN ejpam-4877 56 10 dominating dominating NOUN ejpam-4877 56 11 set set NOUN ejpam-4877 56 12 of of ADP ejpam-4877 56 13 g g PROPN ejpam-4877 56 14 , , PUNCT ejpam-4877 56 15 then then ADV ejpam-4877 56 16 s s VERB ejpam-4877 56 17 is be AUX ejpam-4877 56 18 called call VERB ejpam-4877 56 19 a a DET ejpam-4877 56 20 grundy grundy PROPN ejpam-4877 56 21 hop hop NOUN ejpam-4877 56 22 dominating dominating NOUN ejpam-4877 56 23 sequence sequence NOUN ejpam-4877 56 24 . . PUNCT ejpam-4877 57 1 the the DET ejpam-4877 57 2 maximum maximum ADJ ejpam-4877 57 3 length length NOUN ejpam-4877 57 4 of of ADP ejpam-4877 57 5 a a DET ejpam-4877 57 6 grundy grundy PROPN ejpam-4877 57 7 hop hop NOUN ejpam-4877 57 8 dominating dominating NOUN ejpam-4877 57 9 sequence sequence NOUN ejpam-4877 57 10 in in ADP ejpam-4877 57 11 a a DET ejpam-4877 57 12 graph graph NOUN ejpam-4877 57 13 g g NOUN ejpam-4877 57 14 , , PUNCT ejpam-4877 57 15 denoted denote VERB ejpam-4877 57 16 by by ADP ejpam-4877 57 17 γhgr(g γhgr(g PROPN ejpam-4877 57 18 ) ) PUNCT ejpam-4877 57 19 , , PUNCT ejpam-4877 57 20 is be AUX ejpam-4877 57 21 called call VERB ejpam-4877 57 22 the the DET ejpam-4877 57 23 grundy grundy PROPN ejpam-4877 57 24 hop hop PROPN ejpam-4877 57 25 domination domination NOUN ejpam-4877 57 26 number number NOUN ejpam-4877 57 27 of of ADP ejpam-4877 57 28 g. g. PROPN ejpam-4877 57 29 any any DET ejpam-4877 57 30 grundy grundy PROPN ejpam-4877 57 31 hop hop NOUN ejpam-4877 57 32 dominating dominating NOUN ejpam-4877 57 33 sequence sequence NOUN ejpam-4877 57 34 s s PART ejpam-4877 57 35 with with ADP ejpam-4877 57 36 |ŝ| |ŝ| PROPN ejpam-4877 57 37 = = SYM ejpam-4877 57 38 γhgr(g γhgr(g PROPN ejpam-4877 57 39 ) ) PUNCT ejpam-4877 57 40 is be AUX ejpam-4877 57 41 called call VERB ejpam-4877 57 42 a a DET ejpam-4877 57 43 maximum maximum ADJ ejpam-4877 57 44 j.a j.a PROPN ejpam-4877 57 45 . . PUNCT ejpam-4877 58 1 hassan hassan PROPN ejpam-4877 58 2 , , PUNCT ejpam-4877 58 3 s. s. PROPN ejpam-4877 58 4 canoy canoy PROPN ejpam-4877 58 5 / / SYM ejpam-4877 58 6 eur eur PROPN ejpam-4877 58 7 . . PUNCT ejpam-4877 59 1 j. j. PROPN ejpam-4877 59 2 pure pure PROPN ejpam-4877 59 3 appl appl PROPN ejpam-4877 59 4 . . PROPN ejpam-4877 59 5 math math PROPN ejpam-4877 59 6 , , PUNCT ejpam-4877 59 7 16 16 NUM ejpam-4877 59 8 ( ( PUNCT ejpam-4877 59 9 4 4 NUM ejpam-4877 59 10 ) ) PUNCT ejpam-4877 59 11 ( ( PUNCT ejpam-4877 59 12 2023 2023 NUM ejpam-4877 59 13 ) ) PUNCT ejpam-4877 59 14 , , PUNCT ejpam-4877 59 15 2597 2597 NUM ejpam-4877 59 16 - - SYM ejpam-4877 59 17 2612 2612 NUM ejpam-4877 59 18 2599 2599 NUM ejpam-4877 59 19 grundy grundy PROPN ejpam-4877 59 20 hop hop NOUN ejpam-4877 59 21 dominating dominating NOUN ejpam-4877 59 22 sequence sequence NOUN ejpam-4877 59 23 or or CCONJ ejpam-4877 59 24 a a DET ejpam-4877 59 25 γhgr γhgr ADJ ejpam-4877 59 26 - - PUNCT ejpam-4877 59 27 sequence sequence NOUN ejpam-4877 59 28 of of ADP ejpam-4877 59 29 g. g. PROPN ejpam-4877 59 30 in in ADP ejpam-4877 59 31 this this DET ejpam-4877 59 32 case case NOUN ejpam-4877 59 33 , , PUNCT ejpam-4877 59 34 we we PRON ejpam-4877 59 35 call call VERB ejpam-4877 59 36 ŝ ŝ NOUN ejpam-4877 59 37 a a DET ejpam-4877 59 38 γhgr γhgr VERB ejpam-4877 59 39 - - PUNCT ejpam-4877 59 40 set set NOUN ejpam-4877 59 41 of of ADP ejpam-4877 59 42 g. g. PROPN ejpam-4877 59 43 a a DET ejpam-4877 59 44 subset subset NOUN ejpam-4877 59 45 s s NOUN ejpam-4877 59 46 of of ADP ejpam-4877 59 47 v v NOUN ejpam-4877 59 48 ( ( PUNCT ejpam-4877 59 49 g g NOUN ejpam-4877 59 50 ) ) PUNCT ejpam-4877 59 51 is be AUX ejpam-4877 59 52 a a DET ejpam-4877 59 53 total total ADJ ejpam-4877 59 54 hop hop NOUN ejpam-4877 59 55 dominating dominating NOUN ejpam-4877 59 56 set set NOUN ejpam-4877 59 57 of of ADP ejpam-4877 59 58 g g PROPN ejpam-4877 59 59 if if SCONJ ejpam-4877 59 60 for for ADP ejpam-4877 59 61 every every DET ejpam-4877 59 62 v v NUM ejpam-4877 59 63 ∈ ∈ NOUN ejpam-4877 59 64 v v NOUN ejpam-4877 59 65 ( ( PUNCT ejpam-4877 59 66 g g NOUN ejpam-4877 59 67 ) ) PUNCT ejpam-4877 59 68 , , PUNCT ejpam-4877 59 69 there there PRON ejpam-4877 59 70 exists exist VERB ejpam-4877 59 71 u u PROPN ejpam-4877 59 72 ∈ ∈ PROPN ejpam-4877 59 73 s s VERB ejpam-4877 59 74 such such ADJ ejpam-4877 59 75 that that DET ejpam-4877 59 76 dg(u dg(u ADJ ejpam-4877 59 77 , , PUNCT ejpam-4877 59 78 v v NOUN ejpam-4877 59 79 ) ) PUNCT ejpam-4877 60 1 = = SYM ejpam-4877 60 2 2 2 X ejpam-4877 60 3 . . X ejpam-4877 60 4 the the DET ejpam-4877 60 5 smallest small ADJ ejpam-4877 60 6 cardinality cardinality NOUN ejpam-4877 60 7 of of ADP ejpam-4877 60 8 a a DET ejpam-4877 60 9 total total ADJ ejpam-4877 60 10 hop hop NOUN ejpam-4877 60 11 dominating dominating NOUN ejpam-4877 60 12 set set NOUN ejpam-4877 60 13 of of ADP ejpam-4877 60 14 g g PROPN ejpam-4877 60 15 denoted denote VERB ejpam-4877 60 16 by by ADP ejpam-4877 60 17 γth(g γth(g NOUN ejpam-4877 60 18 ) ) PUNCT ejpam-4877 60 19 , , PUNCT ejpam-4877 60 20 is be AUX ejpam-4877 60 21 called call VERB ejpam-4877 60 22 the the DET ejpam-4877 60 23 total total ADJ ejpam-4877 60 24 hop hop NOUN ejpam-4877 60 25 domination domination NOUN ejpam-4877 60 26 number number NOUN ejpam-4877 60 27 of of ADP ejpam-4877 60 28 g. g. PROPN ejpam-4877 60 29 any any DET ejpam-4877 60 30 hop hop NOUN ejpam-4877 60 31 dominating dominating NOUN ejpam-4877 60 32 set set VERB ejpam-4877 60 33 with with ADP ejpam-4877 60 34 cardinality cardinality NOUN ejpam-4877 60 35 equal equal ADJ ejpam-4877 60 36 to to ADP ejpam-4877 60 37 γth(g γth(g NOUN ejpam-4877 60 38 ) ) PUNCT ejpam-4877 60 39 is be AUX ejpam-4877 60 40 called call VERB ejpam-4877 60 41 a a DET ejpam-4877 60 42 γth γth NOUN ejpam-4877 60 43 - - PUNCT ejpam-4877 60 44 set set NOUN ejpam-4877 60 45 . . PUNCT ejpam-4877 61 1 let let VERB ejpam-4877 61 2 g g NOUN ejpam-4877 61 3 be be AUX ejpam-4877 61 4 any any DET ejpam-4877 61 5 graph graph NOUN ejpam-4877 61 6 with with ADP ejpam-4877 61 7 γ(c γ(c PROPN ejpam-4877 61 8 ) ) PUNCT ejpam-4877 61 9 ̸= ̸= PROPN ejpam-4877 61 10 1 1 NUM ejpam-4877 61 11 for for ADP ejpam-4877 61 12 each each DET ejpam-4877 61 13 component component NOUN ejpam-4877 61 14 c c PROPN ejpam-4877 61 15 of of ADP ejpam-4877 61 16 g. g. PROPN ejpam-4877 61 17 let let VERB ejpam-4877 61 18 s s AUX ejpam-4877 61 19 = = PUNCT ejpam-4877 61 20 ( ( PUNCT ejpam-4877 61 21 v1 v1 PROPN ejpam-4877 61 22 , , PUNCT ejpam-4877 61 23 v2 v2 PROPN ejpam-4877 61 24 , , PUNCT ejpam-4877 61 25 · · PUNCT ejpam-4877 61 26 · · PUNCT ejpam-4877 61 27 · · PUNCT ejpam-4877 61 28 , , PUNCT ejpam-4877 61 29 vk vk AUX ejpam-4877 61 30 ) ) PUNCT ejpam-4877 61 31 be be AUX ejpam-4877 61 32 a a DET ejpam-4877 61 33 sequence sequence NOUN ejpam-4877 61 34 of of ADP ejpam-4877 61 35 distint distint ADJ ejpam-4877 61 36 vertices vertex NOUN ejpam-4877 61 37 of of ADP ejpam-4877 61 38 a a DET ejpam-4877 61 39 graph graph NOUN ejpam-4877 61 40 g g NOUN ejpam-4877 61 41 , , PUNCT ejpam-4877 61 42 and and CCONJ ejpam-4877 61 43 let let VERB ejpam-4877 61 44 ŝ ŝ X ejpam-4877 61 45 = = SYM ejpam-4877 61 46 { { PUNCT ejpam-4877 61 47 v1 v1 PROPN ejpam-4877 61 48 , , PUNCT ejpam-4877 61 49 v2 v2 PROPN ejpam-4877 61 50 , , PUNCT ejpam-4877 61 51 . . PUNCT ejpam-4877 61 52 . . PUNCT ejpam-4877 62 1 . . PUNCT ejpam-4877 63 1 , , PUNCT ejpam-4877 63 2 vk vk ADP ejpam-4877 63 3 } } PUNCT ejpam-4877 63 4 . . PUNCT ejpam-4877 64 1 then then ADV ejpam-4877 64 2 s s VERB ejpam-4877 64 3 is be AUX ejpam-4877 64 4 a a DET ejpam-4877 64 5 legal legal ADJ ejpam-4877 64 6 open open ADJ ejpam-4877 64 7 hop hop NOUN ejpam-4877 64 8 neighborhood neighborhood NOUN ejpam-4877 64 9 sequence sequence NOUN ejpam-4877 64 10 if if SCONJ ejpam-4877 64 11 n2 n2 ADJ ejpam-4877 64 12 g(vi)\ g(vi)\ NOUN ejpam-4877 64 13 ⋃i−1 ⋃i−1 NOUN ejpam-4877 64 14 j=1n j=1n VERB ejpam-4877 64 15 2 2 NUM ejpam-4877 64 16 g(vj g(vj NOUN ejpam-4877 64 17 ) ) PUNCT ejpam-4877 64 18 ̸= ̸= NOUN ejpam-4877 64 19 ∅ ∅ NOUN ejpam-4877 64 20 for for ADP ejpam-4877 64 21 every every DET ejpam-4877 64 22 i i PROPN ejpam-4877 64 23 ∈ ∈ PROPN ejpam-4877 64 24 { { PUNCT ejpam-4877 64 25 2 2 NUM ejpam-4877 64 26 , , PUNCT ejpam-4877 64 27 . . PUNCT ejpam-4877 64 28 . . PUNCT ejpam-4877 64 29 . . PUNCT ejpam-4877 65 1 , , PUNCT ejpam-4877 65 2 k k X ejpam-4877 65 3 } } PUNCT ejpam-4877 65 4 . . PUNCT ejpam-4877 66 1 if if SCONJ ejpam-4877 66 2 , , PUNCT ejpam-4877 66 3 in in ADP ejpam-4877 66 4 addition addition NOUN ejpam-4877 66 5 , , PUNCT ejpam-4877 66 6 ŝ ŝ X ejpam-4877 66 7 is be AUX ejpam-4877 66 8 a a DET ejpam-4877 66 9 total total ADJ ejpam-4877 66 10 hop hop NOUN ejpam-4877 66 11 dominating dominating NOUN ejpam-4877 66 12 set set NOUN ejpam-4877 66 13 of of ADP ejpam-4877 66 14 g g PROPN ejpam-4877 66 15 , , PUNCT ejpam-4877 66 16 then then ADV ejpam-4877 66 17 s s VERB ejpam-4877 66 18 is be AUX ejpam-4877 66 19 called call VERB ejpam-4877 66 20 a a DET ejpam-4877 66 21 grundy grundy PROPN ejpam-4877 66 22 total total NOUN ejpam-4877 66 23 hop hop NOUN ejpam-4877 66 24 dominating dominating NOUN ejpam-4877 66 25 sequence sequence NOUN ejpam-4877 66 26 . . PUNCT ejpam-4877 67 1 the the DET ejpam-4877 67 2 maximum maximum ADJ ejpam-4877 67 3 length length NOUN ejpam-4877 67 4 of of ADP ejpam-4877 67 5 a a DET ejpam-4877 67 6 grundy grundy PROPN ejpam-4877 67 7 total total NOUN ejpam-4877 67 8 hop hop NOUN ejpam-4877 67 9 dominating dominating NOUN ejpam-4877 67 10 sequence sequence NOUN ejpam-4877 67 11 in in ADP ejpam-4877 67 12 a a DET ejpam-4877 67 13 graph graph NOUN ejpam-4877 67 14 g g NOUN ejpam-4877 67 15 is be AUX ejpam-4877 67 16 called call VERB ejpam-4877 67 17 the the DET ejpam-4877 67 18 grundy grundy PROPN ejpam-4877 67 19 total total NOUN ejpam-4877 67 20 hop hop NOUN ejpam-4877 67 21 domination domination NOUN ejpam-4877 67 22 number number NOUN ejpam-4877 67 23 of of ADP ejpam-4877 67 24 g g NOUN ejpam-4877 67 25 , , PUNCT ejpam-4877 67 26 and and CCONJ ejpam-4877 67 27 is be AUX ejpam-4877 67 28 denoted denote VERB ejpam-4877 67 29 by by ADP ejpam-4877 67 30 γthgr(g γthgr(g PROPN ejpam-4877 67 31 ) ) PUNCT ejpam-4877 67 32 . . PUNCT ejpam-4877 68 1 any any DET ejpam-4877 68 2 grundy grundy PROPN ejpam-4877 68 3 total total NOUN ejpam-4877 68 4 hop hop NOUN ejpam-4877 68 5 dominating dominating NOUN ejpam-4877 68 6 sequence sequence NOUN ejpam-4877 68 7 s s PART ejpam-4877 68 8 with with ADP ejpam-4877 68 9 |ŝ| |ŝ| PROPN ejpam-4877 68 10 = = SYM ejpam-4877 68 11 γthgr(g γthgr(g PROPN ejpam-4877 68 12 ) ) PUNCT ejpam-4877 68 13 is be AUX ejpam-4877 68 14 called call VERB ejpam-4877 68 15 a a DET ejpam-4877 68 16 maximum maximum ADJ ejpam-4877 68 17 grundy grundy PROPN ejpam-4877 68 18 total total NOUN ejpam-4877 68 19 hop hop NOUN ejpam-4877 68 20 dominating dominating NOUN ejpam-4877 68 21 sequence sequence NOUN ejpam-4877 68 22 or or CCONJ ejpam-4877 68 23 a a DET ejpam-4877 68 24 γthgr γthgr VERB ejpam-4877 68 25 - - PUNCT ejpam-4877 68 26 sequence sequence NOUN ejpam-4877 68 27 of of ADP ejpam-4877 68 28 g. g. PROPN ejpam-4877 68 29 in in ADP ejpam-4877 68 30 this this DET ejpam-4877 68 31 case case NOUN ejpam-4877 68 32 , , PUNCT ejpam-4877 68 33 we we PRON ejpam-4877 68 34 call call VERB ejpam-4877 68 35 ŝ ŝ NOUN ejpam-4877 68 36 a a DET ejpam-4877 68 37 γthgr γthgr NOUN ejpam-4877 68 38 - - PUNCT ejpam-4877 68 39 set set NOUN ejpam-4877 68 40 of of ADP ejpam-4877 68 41 g. g. PROPN ejpam-4877 68 42 a a DET ejpam-4877 68 43 legal legal ADJ ejpam-4877 68 44 open open ADJ ejpam-4877 68 45 hop hop NOUN ejpam-4877 68 46 neighborhood neighborhood NOUN ejpam-4877 68 47 sequence sequence NOUN ejpam-4877 68 48 s s PART ejpam-4877 68 49 = = PUNCT ejpam-4877 68 50 ( ( PUNCT ejpam-4877 68 51 v1 v1 PROPN ejpam-4877 68 52 , , PUNCT ejpam-4877 68 53 v2 v2 PROPN ejpam-4877 68 54 , , PUNCT ejpam-4877 68 55 · · PUNCT ejpam-4877 68 56 · · PUNCT ejpam-4877 68 57 · · PUNCT ejpam-4877 68 58 , , PUNCT ejpam-4877 68 59 vk vk PROPN ejpam-4877 68 60 ) ) PUNCT ejpam-4877 68 61 with with ADP ejpam-4877 68 62 maximum maximum ADJ ejpam-4877 68 63 length length NOUN ejpam-4877 68 64 , , PUNCT ejpam-4877 68 65 i.e. i.e. X ejpam-4877 68 66 , , PUNCT ejpam-4877 68 67 k k NOUN ejpam-4877 68 68 = = PUNCT ejpam-4877 68 69 max{p max{p NOUN ejpam-4877 68 70 ∈ ∈ PROPN ejpam-4877 68 71 n n NOUN ejpam-4877 68 72 : : PUNCT ejpam-4877 68 73 ∃ ∃ PROPN ejpam-4877 68 74 a a DET ejpam-4877 68 75 legal legal ADJ ejpam-4877 68 76 open open ADJ ejpam-4877 68 77 hop hop NOUN ejpam-4877 68 78 neighborhood neighborhood NOUN ejpam-4877 68 79 sequence sequence NOUN ejpam-4877 68 80 ( ( PUNCT ejpam-4877 68 81 x1 x1 PROPN ejpam-4877 68 82 , , PUNCT ejpam-4877 68 83 · · PUNCT ejpam-4877 68 84 · · PUNCT ejpam-4877 68 85 · · PUNCT ejpam-4877 68 86 , , PUNCT ejpam-4877 68 87 xp xp X ejpam-4877 68 88 ) ) PUNCT ejpam-4877 68 89 of of ADP ejpam-4877 68 90 g g NOUN ejpam-4877 68 91 } } PUNCT ejpam-4877 68 92 , , PUNCT ejpam-4877 68 93 will will AUX ejpam-4877 68 94 be be AUX ejpam-4877 68 95 referred refer VERB ejpam-4877 68 96 to to ADP ejpam-4877 68 97 as as ADP ejpam-4877 68 98 a a DET ejpam-4877 68 99 maximum maximum ADJ ejpam-4877 68 100 legal legal ADJ ejpam-4877 68 101 open open ADJ ejpam-4877 68 102 hop hop NOUN ejpam-4877 68 103 neighborhood neighborhood NOUN ejpam-4877 68 104 sequence sequence NOUN ejpam-4877 68 105 . . PUNCT ejpam-4877 69 1 we we PRON ejpam-4877 69 2 say say VERB ejpam-4877 69 3 that that DET ejpam-4877 69 4 vertex vertex NOUN ejpam-4877 69 5 vi vi PROPN ejpam-4877 69 6 hop hop NOUN ejpam-4877 69 7 - - PUNCT ejpam-4877 69 8 footprints footprint NOUN ejpam-4877 69 9 the the DET ejpam-4877 69 10 vertices vertex NOUN ejpam-4877 69 11 from from ADP ejpam-4877 69 12 n2 n2 PROPN ejpam-4877 69 13 g[vi g[vi PROPN ejpam-4877 69 14 ] ] PUNCT ejpam-4877 69 15 \ \ NOUN ejpam-4877 69 16 ∪i ∪i PUNCT ejpam-4877 69 17 j=1n j=1n PROPN ejpam-4877 69 18 2 2 NUM ejpam-4877 69 19 g[vj g[vj PROPN ejpam-4877 69 20 ] ] PUNCT ejpam-4877 69 21 ( ( PUNCT ejpam-4877 69 22 resp resp NOUN ejpam-4877 69 23 . . PUNCT ejpam-4877 70 1 n2 n2 ADJ ejpam-4877 70 2 g(vi g(vi NUM ejpam-4877 70 3 ) ) PUNCT ejpam-4877 70 4 \ \ NOUN ejpam-4877 71 1 ∪i ∪i PUNCT ejpam-4877 71 2 j=1n j=1n VERB ejpam-4877 71 3 2 2 NUM ejpam-4877 71 4 g(vj g(vj NOUN ejpam-4877 71 5 ) ) PUNCT ejpam-4877 71 6 ) ) PUNCT ejpam-4877 71 7 , , PUNCT ejpam-4877 71 8 and and CCONJ ejpam-4877 71 9 that that DET ejpam-4877 71 10 vi vi PROPN ejpam-4877 71 11 is be AUX ejpam-4877 71 12 their their PRON ejpam-4877 71 13 hop hop NOUN ejpam-4877 71 14 - - PUNCT ejpam-4877 71 15 footprinter footprinter NOUN ejpam-4877 71 16 . . PUNCT ejpam-4877 72 1 two two NUM ejpam-4877 72 2 sequences sequence NOUN ejpam-4877 72 3 s s NOUN ejpam-4877 72 4 and and CCONJ ejpam-4877 72 5 s′ s′ ADJ ejpam-4877 72 6 in in ADP ejpam-4877 72 7 g g PROPN ejpam-4877 72 8 are be AUX ejpam-4877 72 9 loh loh NOUN ejpam-4877 72 10 - - PUNCT ejpam-4877 72 11 identical identical ADJ ejpam-4877 72 12 if if SCONJ ejpam-4877 72 13 they they PRON ejpam-4877 72 14 are be AUX ejpam-4877 72 15 legal legal ADJ ejpam-4877 72 16 open open ADJ ejpam-4877 72 17 hop hop NOUN ejpam-4877 72 18 neighborhood neighborhood NOUN ejpam-4877 72 19 sequences sequence NOUN ejpam-4877 72 20 ( ( PUNCT ejpam-4877 72 21 or or CCONJ ejpam-4877 72 22 grundy grundy PROPN ejpam-4877 72 23 total total ADJ ejpam-4877 72 24 hop hop NOUN ejpam-4877 72 25 dominating dominating NOUN ejpam-4877 72 26 sequences sequence NOUN ejpam-4877 72 27 ) ) PUNCT ejpam-4877 72 28 and and CCONJ ejpam-4877 72 29 ŝ ŝ X ejpam-4877 72 30 = = SYM ejpam-4877 72 31 ŝ′ ŝ′ PROPN ejpam-4877 72 32 ( ( PUNCT ejpam-4877 72 33 i.e. i.e. X ejpam-4877 72 34 , , PUNCT ejpam-4877 72 35 one one NUM ejpam-4877 72 36 is be AUX ejpam-4877 72 37 a a DET ejpam-4877 72 38 reaarrangement reaarrangement NOUN ejpam-4877 72 39 of of ADP ejpam-4877 72 40 the the DET ejpam-4877 72 41 terms term NOUN ejpam-4877 72 42 of of ADP ejpam-4877 72 43 the the DET ejpam-4877 72 44 other other ADJ ejpam-4877 72 45 ) ) PUNCT ejpam-4877 72 46 . . PUNCT ejpam-4877 73 1 a a DET ejpam-4877 73 2 sequence sequence NOUN ejpam-4877 73 3 s s PART ejpam-4877 73 4 = = PUNCT ejpam-4877 73 5 ( ( PUNCT ejpam-4877 73 6 v1 v1 PROPN ejpam-4877 73 7 , , PUNCT ejpam-4877 73 8 v2 v2 PROPN ejpam-4877 73 9 , , PUNCT ejpam-4877 73 10 · · PUNCT ejpam-4877 73 11 · · PUNCT ejpam-4877 73 12 · · PUNCT ejpam-4877 73 13 , , PUNCT ejpam-4877 73 14 vk vk PROPN ejpam-4877 73 15 ) ) PUNCT ejpam-4877 73 16 of of ADP ejpam-4877 73 17 distinct distinct ADJ ejpam-4877 73 18 vertices vertex NOUN ejpam-4877 73 19 of of ADP ejpam-4877 73 20 a a DET ejpam-4877 73 21 graph graph NOUN ejpam-4877 73 22 g g NOUN ejpam-4877 73 23 is be AUX ejpam-4877 73 24 a a DET ejpam-4877 73 25 co co ADJ ejpam-4877 73 26 - - ADJ ejpam-4877 73 27 legal legal ADJ ejpam-4877 73 28 open open ADJ ejpam-4877 73 29 neighborhood neighborhood NOUN ejpam-4877 73 30 sequence sequence NOUN ejpam-4877 73 31 in in ADP ejpam-4877 73 32 g g PROPN ejpam-4877 73 33 if if SCONJ ejpam-4877 73 34 [ [ X ejpam-4877 73 35 v v X ejpam-4877 73 36 ( ( PUNCT ejpam-4877 73 37 g g NOUN ejpam-4877 73 38 ) ) PUNCT ejpam-4877 73 39 \ \ PUNCT ejpam-4877 74 1 ng[vi ng[vi PROPN ejpam-4877 74 2 ] ] X ejpam-4877 74 3 ] ] PUNCT ejpam-4877 74 4 \ \ X ejpam-4877 74 5 ∪i−1 ∪i−1 PROPN ejpam-4877 74 6 j=1[v j=1[v X ejpam-4877 74 7 ( ( PUNCT ejpam-4877 74 8 g g NOUN ejpam-4877 74 9 ) ) PUNCT ejpam-4877 74 10 \ \ PUNCT ejpam-4877 75 1 ng[vj ng[vj PROPN ejpam-4877 75 2 ] ] X ejpam-4877 75 3 ] ] X ejpam-4877 75 4 ̸= ̸= NOUN ejpam-4877 75 5 ∅ ∅ NOUN ejpam-4877 75 6 for for ADP ejpam-4877 75 7 each each DET ejpam-4877 75 8 i i PRON ejpam-4877 75 9 ∈ ∈ PROPN ejpam-4877 75 10 { { PUNCT ejpam-4877 75 11 2 2 NUM ejpam-4877 75 12 , , PUNCT ejpam-4877 75 13 . . PUNCT ejpam-4877 75 14 . . PUNCT ejpam-4877 75 15 . . PUNCT ejpam-4877 76 1 , , PUNCT ejpam-4877 76 2 k k X ejpam-4877 76 3 } } PUNCT ejpam-4877 76 4 . . PUNCT ejpam-4877 77 1 a a DET ejpam-4877 77 2 co co ADJ ejpam-4877 77 3 - - ADJ ejpam-4877 77 4 legal legal ADJ ejpam-4877 77 5 open open ADJ ejpam-4877 77 6 neighborhood neighborhood NOUN ejpam-4877 77 7 sequence sequence NOUN ejpam-4877 77 8 s s PART ejpam-4877 77 9 = = PUNCT ejpam-4877 77 10 ( ( PUNCT ejpam-4877 77 11 v1 v1 PROPN ejpam-4877 77 12 , , PUNCT ejpam-4877 77 13 v2 v2 NOUN ejpam-4877 77 14 , , PUNCT ejpam-4877 77 15 . . PUNCT ejpam-4877 77 16 . . PUNCT ejpam-4877 77 17 . . PUNCT ejpam-4877 78 1 , , PUNCT ejpam-4877 78 2 vk vk PROPN ejpam-4877 78 3 ) ) PUNCT ejpam-4877 78 4 is be AUX ejpam-4877 78 5 a a DET ejpam-4877 78 6 co co ADJ ejpam-4877 78 7 - - ADJ ejpam-4877 78 8 grundy grundy ADJ ejpam-4877 78 9 total total ADJ ejpam-4877 78 10 dominating dominating NOUN ejpam-4877 78 11 sequence sequence NOUN ejpam-4877 78 12 if if SCONJ ejpam-4877 78 13 v v X ejpam-4877 78 14 ( ( PUNCT ejpam-4877 78 15 g g NOUN ejpam-4877 78 16 ) ) PUNCT ejpam-4877 78 17 = = PUNCT ejpam-4877 79 1 ∪k ∪k NUM ejpam-4877 79 2 i=1[v i=1[v X ejpam-4877 79 3 ( ( PUNCT ejpam-4877 79 4 g g NOUN ejpam-4877 79 5 ) ) PUNCT ejpam-4877 79 6 \ng[vi \ng[vi PROPN ejpam-4877 79 7 ] ] X ejpam-4877 79 8 ] ] PUNCT ejpam-4877 79 9 . . PUNCT ejpam-4877 80 1 the the DET ejpam-4877 80 2 maximum maximum ADJ ejpam-4877 80 3 length length NOUN ejpam-4877 80 4 of of ADP ejpam-4877 80 5 a a DET ejpam-4877 80 6 cogrundy cogrundy NOUN ejpam-4877 80 7 total total ADJ ejpam-4877 80 8 dominating dominating NOUN ejpam-4877 80 9 sequence sequence NOUN ejpam-4877 80 10 in in ADP ejpam-4877 80 11 a a DET ejpam-4877 80 12 graph graph NOUN ejpam-4877 80 13 g g NOUN ejpam-4877 80 14 is be AUX ejpam-4877 80 15 called call VERB ejpam-4877 80 16 the the DET ejpam-4877 80 17 co co NOUN ejpam-4877 80 18 - - ADJ ejpam-4877 80 19 grundy grundy ADJ ejpam-4877 80 20 total total ADJ ejpam-4877 80 21 domination domination NOUN ejpam-4877 80 22 number number NOUN ejpam-4877 80 23 of of ADP ejpam-4877 80 24 g g NOUN ejpam-4877 80 25 , , PUNCT ejpam-4877 80 26 and and CCONJ ejpam-4877 80 27 is be AUX ejpam-4877 80 28 denoted denote VERB ejpam-4877 80 29 by by ADP ejpam-4877 80 30 γtcogr(g γtcogr(g PROPN ejpam-4877 80 31 ) ) PUNCT ejpam-4877 80 32 . . PUNCT ejpam-4877 81 1 let let VERB ejpam-4877 81 2 s1 s1 PROPN ejpam-4877 81 3 = = SYM ejpam-4877 81 4 ( ( PUNCT ejpam-4877 81 5 v1 v1 PROPN ejpam-4877 81 6 , , PUNCT ejpam-4877 81 7 · · PUNCT ejpam-4877 81 8 · · PUNCT ejpam-4877 81 9 · · PUNCT ejpam-4877 81 10 , , PUNCT ejpam-4877 81 11 vn vn PROPN ejpam-4877 81 12 ) ) PUNCT ejpam-4877 81 13 and and CCONJ ejpam-4877 81 14 s2 s2 NOUN ejpam-4877 81 15 = = SYM ejpam-4877 81 16 ( ( PUNCT ejpam-4877 81 17 u1 u1 PROPN ejpam-4877 81 18 , , PUNCT ejpam-4877 81 19 · · PUNCT ejpam-4877 81 20 · · PUNCT ejpam-4877 81 21 · · PUNCT ejpam-4877 81 22 , , PUNCT ejpam-4877 81 23 um um INTJ ejpam-4877 81 24 ) ) PUNCT ejpam-4877 81 25 , , PUNCT ejpam-4877 81 26 n n CCONJ ejpam-4877 81 27 , , PUNCT ejpam-4877 81 28 m m VERB ejpam-4877 81 29 ≥ ≥ NOUN ejpam-4877 81 30 1 1 NUM ejpam-4877 81 31 be be AUX ejpam-4877 81 32 two two NUM ejpam-4877 81 33 sequences sequence NOUN ejpam-4877 81 34 of of ADP ejpam-4877 81 35 distinct distinct ADJ ejpam-4877 81 36 vertices vertex NOUN ejpam-4877 81 37 of of ADP ejpam-4877 81 38 g. g. PROPN ejpam-4877 81 39 the the DET ejpam-4877 81 40 concatenation concatenation NOUN ejpam-4877 81 41 of of ADP ejpam-4877 81 42 s1 s1 PROPN ejpam-4877 81 43 and and CCONJ ejpam-4877 81 44 s2 s2 PROPN ejpam-4877 81 45 , , PUNCT ejpam-4877 81 46 denoted denote VERB ejpam-4877 81 47 by by ADP ejpam-4877 81 48 s1 s1 PROPN ejpam-4877 81 49 ⊕ ⊕ PROPN ejpam-4877 81 50 s2 s2 PROPN ejpam-4877 81 51 , , PUNCT ejpam-4877 81 52 is be AUX ejpam-4877 81 53 the the DET ejpam-4877 81 54 sequence sequence NOUN ejpam-4877 81 55 given give VERB ejpam-4877 81 56 by by ADP ejpam-4877 81 57 s1 s1 PROPN ejpam-4877 81 58 ⊕ ⊕ PROPN ejpam-4877 81 59 s2 s2 PROPN ejpam-4877 81 60 = = SYM ejpam-4877 81 61 ( ( PUNCT ejpam-4877 81 62 v1 v1 PROPN ejpam-4877 81 63 , , PUNCT ejpam-4877 81 64 · · PUNCT ejpam-4877 81 65 · · PUNCT ejpam-4877 81 66 · · PUNCT ejpam-4877 81 67 , , PUNCT ejpam-4877 81 68 vn vn PROPN ejpam-4877 81 69 , , PUNCT ejpam-4877 81 70 u1 u1 NOUN ejpam-4877 81 71 , , PUNCT ejpam-4877 81 72 · · PUNCT ejpam-4877 81 73 · · PUNCT ejpam-4877 81 74 · · PUNCT ejpam-4877 81 75 , , PUNCT ejpam-4877 81 76 um um INTJ ejpam-4877 81 77 ) ) PUNCT ejpam-4877 81 78 . . PUNCT ejpam-4877 82 1 the the DET ejpam-4877 82 2 shadow shadow NOUN ejpam-4877 82 3 graph graph NOUN ejpam-4877 82 4 s(g s(g PROPN ejpam-4877 82 5 ) ) PUNCT ejpam-4877 82 6 of of ADP ejpam-4877 82 7 a a DET ejpam-4877 82 8 graph graph NOUN ejpam-4877 82 9 g g NOUN ejpam-4877 82 10 is be AUX ejpam-4877 82 11 constructed construct VERB ejpam-4877 82 12 by by ADP ejpam-4877 82 13 taking take VERB ejpam-4877 82 14 two two NUM ejpam-4877 82 15 copies copy NOUN ejpam-4877 82 16 of of ADP ejpam-4877 82 17 g g NOUN ejpam-4877 82 18 , , PUNCT ejpam-4877 82 19 say say VERB ejpam-4877 82 20 g1 g1 PROPN ejpam-4877 82 21 and and CCONJ ejpam-4877 82 22 g2 g2 PROPN ejpam-4877 82 23 and and CCONJ ejpam-4877 82 24 joining join VERB ejpam-4877 82 25 each each DET ejpam-4877 82 26 vertex vertex NOUN ejpam-4877 82 27 u u PROPN ejpam-4877 82 28 ∈ ∈ PROPN ejpam-4877 82 29 g1 g1 PROPN ejpam-4877 82 30 to to ADP ejpam-4877 82 31 the the DET ejpam-4877 82 32 neighbors neighbor NOUN ejpam-4877 82 33 of of ADP ejpam-4877 82 34 the the DET ejpam-4877 82 35 corresponding corresponding ADJ ejpam-4877 82 36 vertex vertex NOUN ejpam-4877 82 37 u′ u′ PROPN ejpam-4877 82 38 ∈ ∈ PROPN ejpam-4877 82 39 g2 g2 PROPN ejpam-4877 82 40 . . PUNCT ejpam-4877 83 1 let let VERB ejpam-4877 83 2 g g NOUN ejpam-4877 83 3 and and CCONJ ejpam-4877 83 4 h h NOUN ejpam-4877 83 5 be be VERB ejpam-4877 83 6 any any DET ejpam-4877 83 7 two two NUM ejpam-4877 83 8 graphs graph NOUN ejpam-4877 83 9 . . PUNCT ejpam-4877 84 1 the the DET ejpam-4877 84 2 join join NOUN ejpam-4877 84 3 of of ADP ejpam-4877 84 4 g g PROPN ejpam-4877 84 5 and and CCONJ ejpam-4877 84 6 h h NOUN ejpam-4877 84 7 , , PUNCT ejpam-4877 84 8 denoted denote VERB ejpam-4877 84 9 by by ADP ejpam-4877 84 10 g+h g+h PROPN ejpam-4877 84 11 is be AUX ejpam-4877 84 12 the the DET ejpam-4877 84 13 graph graph NOUN ejpam-4877 84 14 with with ADP ejpam-4877 84 15 vertex vertex NOUN ejpam-4877 84 16 set set VERB ejpam-4877 84 17 v v NOUN ejpam-4877 84 18 ( ( PUNCT ejpam-4877 84 19 g+h g+h NOUN ejpam-4877 84 20 ) ) PUNCT ejpam-4877 85 1 = = SYM ejpam-4877 85 2 v v X ejpam-4877 85 3 ( ( PUNCT ejpam-4877 85 4 g)∪v g)∪v NOUN ejpam-4877 85 5 ( ( PUNCT ejpam-4877 85 6 h h NOUN ejpam-4877 85 7 ) ) PUNCT ejpam-4877 85 8 and and CCONJ ejpam-4877 85 9 edge edge NOUN ejpam-4877 85 10 set set VERB ejpam-4877 85 11 e(g+h e(g+h NUM ejpam-4877 85 12 ) ) PUNCT ejpam-4877 86 1 = = SYM ejpam-4877 86 2 e(g)∪e(h)∪{uv e(g)∪e(h)∪{uv X ejpam-4877 86 3 : : PUNCT ejpam-4877 86 4 u u PROPN ejpam-4877 86 5 ∈ ∈ PROPN ejpam-4877 86 6 v v NOUN ejpam-4877 86 7 ( ( PUNCT ejpam-4877 86 8 g g NOUN ejpam-4877 86 9 ) ) PUNCT ejpam-4877 86 10 , , PUNCT ejpam-4877 86 11 v v X ejpam-4877 86 12 ∈ ∈ PROPN ejpam-4877 86 13 v v NOUN ejpam-4877 86 14 ( ( PUNCT ejpam-4877 86 15 h h NOUN ejpam-4877 86 16 ) ) PUNCT ejpam-4877 86 17 } } PUNCT ejpam-4877 86 18 . . PUNCT ejpam-4877 87 1 the the DET ejpam-4877 87 2 corona corona NOUN ejpam-4877 87 3 g g PROPN ejpam-4877 87 4 and and CCONJ ejpam-4877 87 5 h h NOUN ejpam-4877 87 6 , , PUNCT ejpam-4877 87 7 denoted denote VERB ejpam-4877 87 8 by by ADP ejpam-4877 87 9 g g PROPN ejpam-4877 87 10 ◦ ◦ NOUN ejpam-4877 87 11 h h NOUN ejpam-4877 87 12 , , PUNCT ejpam-4877 87 13 the the DET ejpam-4877 87 14 graph graph NOUN ejpam-4877 87 15 obtained obtain VERB ejpam-4877 87 16 by by ADP ejpam-4877 87 17 taking take VERB ejpam-4877 87 18 one one NUM ejpam-4877 87 19 copy copy NOUN ejpam-4877 87 20 of of ADP ejpam-4877 87 21 g g PROPN ejpam-4877 87 22 and and CCONJ ejpam-4877 87 23 |v |v PROPN ejpam-4877 87 24 ( ( PUNCT ejpam-4877 87 25 g)| g)| NOUN ejpam-4877 87 26 copies copy NOUN ejpam-4877 87 27 of of ADP ejpam-4877 87 28 h h NOUN ejpam-4877 87 29 , , PUNCT ejpam-4877 87 30 and and CCONJ ejpam-4877 87 31 then then ADV ejpam-4877 87 32 joining join VERB ejpam-4877 87 33 the the DET ejpam-4877 87 34 ith ith PROPN ejpam-4877 87 35 vertex vertex NOUN ejpam-4877 87 36 of of ADP ejpam-4877 87 37 g g NOUN ejpam-4877 87 38 to to ADP ejpam-4877 87 39 every every DET ejpam-4877 87 40 vertex vertex NOUN ejpam-4877 87 41 of of ADP ejpam-4877 87 42 the the DET ejpam-4877 87 43 ith ith PROPN ejpam-4877 87 44 copy copy NOUN ejpam-4877 87 45 of of ADP ejpam-4877 87 46 h. h. PROPN ejpam-4877 87 47 we we PRON ejpam-4877 87 48 denote denote VERB ejpam-4877 87 49 by by ADP ejpam-4877 87 50 hv hv PROPN ejpam-4877 87 51 the the DET ejpam-4877 87 52 copy copy NOUN ejpam-4877 87 53 of of ADP ejpam-4877 87 54 h h NOUN ejpam-4877 87 55 in in ADP ejpam-4877 87 56 g g PROPN ejpam-4877 87 57 ◦ ◦ NOUN ejpam-4877 87 58 h h NOUN ejpam-4877 87 59 corresponding correspond VERB ejpam-4877 87 60 to to ADP ejpam-4877 87 61 the the DET ejpam-4877 87 62 vertex vertex NOUN ejpam-4877 87 63 v v ADP ejpam-4877 87 64 ∈ ∈ PROPN ejpam-4877 87 65 g g NOUN ejpam-4877 87 66 and and CCONJ ejpam-4877 87 67 write write VERB ejpam-4877 87 68 v v ADP ejpam-4877 87 69 + + PROPN ejpam-4877 87 70 hv hv NOUN ejpam-4877 87 71 for for ADP ejpam-4877 87 72 ⟨{v}⟩+hv ⟨{v}⟩+hv PROPN ejpam-4877 87 73 . . PUNCT ejpam-4877 88 1 j.a j.a PROPN ejpam-4877 88 2 . . PROPN ejpam-4877 88 3 hassan hassan PROPN ejpam-4877 88 4 , , PUNCT ejpam-4877 88 5 s. s. PROPN ejpam-4877 88 6 canoy canoy PROPN ejpam-4877 88 7 / / SYM ejpam-4877 88 8 eur eur PROPN ejpam-4877 88 9 . . PUNCT ejpam-4877 89 1 j. j. PROPN ejpam-4877 89 2 pure pure PROPN ejpam-4877 89 3 appl appl PROPN ejpam-4877 89 4 . . PROPN ejpam-4877 89 5 math math PROPN ejpam-4877 89 6 , , PUNCT ejpam-4877 89 7 16 16 NUM ejpam-4877 89 8 ( ( PUNCT ejpam-4877 89 9 4 4 NUM ejpam-4877 89 10 ) ) PUNCT ejpam-4877 89 11 ( ( PUNCT ejpam-4877 89 12 2023 2023 NUM ejpam-4877 89 13 ) ) PUNCT ejpam-4877 89 14 , , PUNCT ejpam-4877 89 15 2597 2597 NUM ejpam-4877 89 16 - - SYM ejpam-4877 89 17 2612 2612 NUM ejpam-4877 89 18 2600 2600 NUM ejpam-4877 89 19 3 3 NUM ejpam-4877 89 20 . . PUNCT ejpam-4877 89 21 results result NOUN ejpam-4877 89 22 theorem theorem VERB ejpam-4877 89 23 1 1 X ejpam-4877 89 24 . . PUNCT ejpam-4877 90 1 let let VERB ejpam-4877 90 2 g g PRON ejpam-4877 90 3 be be AUX ejpam-4877 90 4 a a DET ejpam-4877 90 5 graph graph NOUN ejpam-4877 90 6 of of ADP ejpam-4877 90 7 order order NOUN ejpam-4877 90 8 n n PRON ejpam-4877 90 9 with with ADP ejpam-4877 90 10 γ(c γ(c PROPN ejpam-4877 90 11 ) ) PUNCT ejpam-4877 90 12 ̸= ̸= PROPN ejpam-4877 90 13 1 1 NUM ejpam-4877 90 14 for for ADP ejpam-4877 90 15 each each DET ejpam-4877 90 16 component component NOUN ejpam-4877 90 17 c c PROPN ejpam-4877 90 18 of of ADP ejpam-4877 90 19 g. g. PROPN ejpam-4877 90 20 then then ADV ejpam-4877 90 21 the the DET ejpam-4877 90 22 following follow VERB ejpam-4877 90 23 statements statement NOUN ejpam-4877 90 24 hold hold VERB ejpam-4877 90 25 . . PUNCT ejpam-4877 91 1 ( ( PUNCT ejpam-4877 91 2 i i NOUN ejpam-4877 91 3 ) ) PUNCT ejpam-4877 91 4 if if SCONJ ejpam-4877 91 5 γth(g γth(g NOUN ejpam-4877 91 6 ) ) PUNCT ejpam-4877 91 7 = = SYM ejpam-4877 91 8 t t PROPN ejpam-4877 91 9 and and CCONJ ejpam-4877 91 10 d d NOUN ejpam-4877 91 11 = = PUNCT ejpam-4877 91 12 { { PUNCT ejpam-4877 91 13 u1 u1 NOUN ejpam-4877 91 14 , , PUNCT ejpam-4877 91 15 u2 u2 NOUN ejpam-4877 91 16 , , PUNCT ejpam-4877 91 17 . . PUNCT ejpam-4877 91 18 . . PUNCT ejpam-4877 91 19 . . PUNCT ejpam-4877 92 1 , , PUNCT ejpam-4877 92 2 ut ut PROPN ejpam-4877 92 3 } } PUNCT ejpam-4877 92 4 is be AUX ejpam-4877 92 5 a a DET ejpam-4877 92 6 minimum minimum ADJ ejpam-4877 92 7 total total ADJ ejpam-4877 92 8 hop hop NOUN ejpam-4877 92 9 dominating dominating NOUN ejpam-4877 92 10 set set NOUN ejpam-4877 92 11 of of ADP ejpam-4877 92 12 g g NOUN ejpam-4877 92 13 , , PUNCT ejpam-4877 92 14 then then ADV ejpam-4877 92 15 s s VERB ejpam-4877 92 16 = = PUNCT ejpam-4877 92 17 ( ( PUNCT ejpam-4877 92 18 u1 u1 PROPN ejpam-4877 92 19 , , PUNCT ejpam-4877 92 20 u2 u2 PROPN ejpam-4877 92 21 , , PUNCT ejpam-4877 92 22 · · PUNCT ejpam-4877 92 23 · · PUNCT ejpam-4877 92 24 · · PUNCT ejpam-4877 92 25 , , PUNCT ejpam-4877 92 26 ut ut PROPN ejpam-4877 92 27 ) ) PUNCT ejpam-4877 92 28 is be AUX ejpam-4877 92 29 a a DET ejpam-4877 92 30 grundy grundy PROPN ejpam-4877 92 31 total total NOUN ejpam-4877 92 32 hop hop NOUN ejpam-4877 92 33 dominating dominating NOUN ejpam-4877 92 34 sequence sequence NOUN ejpam-4877 92 35 . . PUNCT ejpam-4877 93 1 in in ADP ejpam-4877 93 2 particular particular ADJ ejpam-4877 93 3 , , PUNCT ejpam-4877 93 4 γth(g γth(g NOUN ejpam-4877 93 5 ) ) PUNCT ejpam-4877 93 6 ≤ ≤ NOUN ejpam-4877 93 7 γthgr(g γthgr(g PROPN ejpam-4877 93 8 ) ) PUNCT ejpam-4877 93 9 . . PUNCT ejpam-4877 94 1 ( ( PUNCT ejpam-4877 94 2 ii ii X ejpam-4877 94 3 ) ) PUNCT ejpam-4877 94 4 if if SCONJ ejpam-4877 94 5 s s VERB ejpam-4877 94 6 = = PUNCT ejpam-4877 94 7 ( ( PUNCT ejpam-4877 94 8 u1 u1 PROPN ejpam-4877 94 9 , , PUNCT ejpam-4877 94 10 u2 u2 PROPN ejpam-4877 94 11 , , PUNCT ejpam-4877 94 12 · · PUNCT ejpam-4877 94 13 · · PUNCT ejpam-4877 94 14 · · PUNCT ejpam-4877 94 15 , , PUNCT ejpam-4877 94 16 us we PRON ejpam-4877 94 17 ) ) PUNCT ejpam-4877 94 18 is be AUX ejpam-4877 94 19 a a DET ejpam-4877 94 20 minimum minimum ADJ ejpam-4877 94 21 grundy grundy PROPN ejpam-4877 94 22 total total NOUN ejpam-4877 94 23 hop hop NOUN ejpam-4877 94 24 dominating dominating NOUN ejpam-4877 94 25 sequence sequence NOUN ejpam-4877 94 26 , , PUNCT ejpam-4877 94 27 then then ADV ejpam-4877 94 28 γth(g γth(g NOUN ejpam-4877 94 29 ) ) PUNCT ejpam-4877 94 30 = = PUNCT ejpam-4877 95 1 |ŝ| |ŝ| PROPN ejpam-4877 95 2 . . PUNCT ejpam-4877 95 3 proof proof NOUN ejpam-4877 95 4 . . PUNCT ejpam-4877 96 1 ( ( PUNCT ejpam-4877 96 2 i i NOUN ejpam-4877 96 3 ) ) PUNCT ejpam-4877 96 4 suppose suppose VERB ejpam-4877 96 5 that that SCONJ ejpam-4877 96 6 there there PRON ejpam-4877 96 7 exists exist VERB ejpam-4877 96 8 i i PRON ejpam-4877 96 9 ∈ ∈ PROPN ejpam-4877 96 10 { { PUNCT ejpam-4877 96 11 2 2 NUM ejpam-4877 96 12 , , PUNCT ejpam-4877 96 13 3 3 NUM ejpam-4877 96 14 , , PUNCT ejpam-4877 96 15 . . PUNCT ejpam-4877 96 16 . . PUNCT ejpam-4877 97 1 . . PUNCT ejpam-4877 98 1 , , PUNCT ejpam-4877 98 2 t t X ejpam-4877 98 3 } } PUNCT ejpam-4877 98 4 such such ADJ ejpam-4877 98 5 that that DET ejpam-4877 98 6 n2 n2 ADJ ejpam-4877 98 7 g(ui)\∪ g(ui)\∪ PROPN ejpam-4877 98 8 i−1 i−1 PROPN ejpam-4877 98 9 j=1n j=1n NOUN ejpam-4877 98 10 2 2 NUM ejpam-4877 98 11 g(uj g(uj NOUN ejpam-4877 98 12 ) ) PUNCT ejpam-4877 98 13 = = PUNCT ejpam-4877 98 14 ∅. ∅. VERB ejpam-4877 98 15 then then ADV ejpam-4877 98 16 n2 n2 PROPN ejpam-4877 98 17 g(ui g(ui PROPN ejpam-4877 98 18 ) ) PUNCT ejpam-4877 98 19 ⊆ ⊆ NUM ejpam-4877 98 20 ∪i−1 ∪i−1 PROPN ejpam-4877 98 21 j=1n j=1n PROPN ejpam-4877 98 22 2 2 NUM ejpam-4877 98 23 g(uj g(uj NOUN ejpam-4877 98 24 ) ) PUNCT ejpam-4877 98 25 . . PUNCT ejpam-4877 99 1 this this PRON ejpam-4877 99 2 means mean VERB ejpam-4877 99 3 that that SCONJ ejpam-4877 99 4 d\{ui d\{ui PROPN ejpam-4877 99 5 } } PUNCT ejpam-4877 99 6 is be AUX ejpam-4877 99 7 a a DET ejpam-4877 99 8 total total ADJ ejpam-4877 99 9 hop hop NOUN ejpam-4877 99 10 dominating dominating NOUN ejpam-4877 99 11 set set NOUN ejpam-4877 99 12 of of ADP ejpam-4877 99 13 g g PROPN ejpam-4877 99 14 , , PUNCT ejpam-4877 99 15 which which PRON ejpam-4877 99 16 is be AUX ejpam-4877 99 17 a a DET ejpam-4877 99 18 contradiction contradiction NOUN ejpam-4877 99 19 to to ADP ejpam-4877 99 20 the the DET ejpam-4877 99 21 minimality minimality NOUN ejpam-4877 99 22 of of ADP ejpam-4877 99 23 d. d. PROPN ejpam-4877 99 24 hence hence ADV ejpam-4877 99 25 , , PUNCT ejpam-4877 99 26 n2 n2 PROPN ejpam-4877 99 27 g(ui g(ui PROPN ejpam-4877 99 28 ) ) PUNCT ejpam-4877 99 29 \ \ NOUN ejpam-4877 99 30 ∪ ∪ ADP ejpam-4877 99 31 i−1 i−1 PROPN ejpam-4877 99 32 j=1n j=1n PROPN ejpam-4877 99 33 2 2 NUM ejpam-4877 99 34 g(uj g(uj NOUN ejpam-4877 99 35 ) ) PUNCT ejpam-4877 99 36 ̸= ̸= NOUN ejpam-4877 99 37 ∅ ∅ NOUN ejpam-4877 99 38 for for ADP ejpam-4877 99 39 each each DET ejpam-4877 99 40 i i PRON ejpam-4877 99 41 ∈ ∈ PROPN ejpam-4877 99 42 { { PUNCT ejpam-4877 99 43 2 2 NUM ejpam-4877 99 44 , , PUNCT ejpam-4877 99 45 3 3 NUM ejpam-4877 99 46 , , PUNCT ejpam-4877 99 47 . . PUNCT ejpam-4877 99 48 . . PUNCT ejpam-4877 100 1 . . PUNCT ejpam-4877 101 1 , , PUNCT ejpam-4877 101 2 t t PROPN ejpam-4877 101 3 } } PUNCT ejpam-4877 101 4 , , PUNCT ejpam-4877 101 5 and and CCONJ ejpam-4877 101 6 so so ADV ejpam-4877 101 7 s s VERB ejpam-4877 101 8 is be AUX ejpam-4877 101 9 grundy grundy PROPN ejpam-4877 101 10 total total ADJ ejpam-4877 101 11 hop hop NOUN ejpam-4877 101 12 dominating dominating NOUN ejpam-4877 101 13 sequence sequence NOUN ejpam-4877 101 14 . . PUNCT ejpam-4877 102 1 consequently consequently ADV ejpam-4877 102 2 , , PUNCT ejpam-4877 102 3 γth(g γth(g NOUN ejpam-4877 102 4 ) ) PUNCT ejpam-4877 102 5 ≤ ≤ NOUN ejpam-4877 102 6 γthgr(g γthgr(g PROPN ejpam-4877 102 7 ) ) PUNCT ejpam-4877 102 8 . . PUNCT ejpam-4877 103 1 ( ( PUNCT ejpam-4877 103 2 ii ii NOUN ejpam-4877 103 3 ) ) PUNCT ejpam-4877 103 4 from from ADP ejpam-4877 103 5 ( ( PUNCT ejpam-4877 103 6 i i NOUN ejpam-4877 103 7 ) ) PUNCT ejpam-4877 103 8 , , PUNCT ejpam-4877 103 9 every every DET ejpam-4877 103 10 γth γth NOUN ejpam-4877 103 11 - - PUNCT ejpam-4877 103 12 set set NOUN ejpam-4877 103 13 of of ADP ejpam-4877 103 14 g g PROPN ejpam-4877 103 15 forms form VERB ejpam-4877 103 16 a a DET ejpam-4877 103 17 grundy grundy PROPN ejpam-4877 103 18 total total NOUN ejpam-4877 103 19 hop hop NOUN ejpam-4877 103 20 dominating dominating NOUN ejpam-4877 103 21 sequence sequence NOUN ejpam-4877 103 22 . . PUNCT ejpam-4877 104 1 since since SCONJ ejpam-4877 104 2 s s PROPN ejpam-4877 104 3 is be AUX ejpam-4877 104 4 a a DET ejpam-4877 104 5 minimum minimum ADJ ejpam-4877 104 6 grundy grundy PROPN ejpam-4877 104 7 total total NOUN ejpam-4877 104 8 hop hop NOUN ejpam-4877 104 9 dominating dominating NOUN ejpam-4877 104 10 sequence sequence NOUN ejpam-4877 104 11 , , PUNCT ejpam-4877 104 12 it it PRON ejpam-4877 104 13 follows follow VERB ejpam-4877 104 14 that that SCONJ ejpam-4877 104 15 |ŝ| |ŝ| PROPN ejpam-4877 104 16 ≤ ≤ NUM ejpam-4877 104 17 γth(g γth(g NOUN ejpam-4877 104 18 ) ) PUNCT ejpam-4877 104 19 . . PUNCT ejpam-4877 105 1 on on ADP ejpam-4877 105 2 the the DET ejpam-4877 105 3 other other ADJ ejpam-4877 105 4 hand hand NOUN ejpam-4877 105 5 , , PUNCT ejpam-4877 105 6 since since SCONJ ejpam-4877 105 7 every every DET ejpam-4877 105 8 grundy grundy PROPN ejpam-4877 105 9 total total NOUN ejpam-4877 105 10 hop hop NOUN ejpam-4877 105 11 dominating dominating NOUN ejpam-4877 105 12 sequence sequence NOUN ejpam-4877 105 13 forms form VERB ejpam-4877 105 14 a a DET ejpam-4877 105 15 total total ADJ ejpam-4877 105 16 hop hop NOUN ejpam-4877 105 17 dominating dominating NOUN ejpam-4877 105 18 set set VERB ejpam-4877 105 19 by by ADP ejpam-4877 105 20 definition definition NOUN ejpam-4877 105 21 , , PUNCT ejpam-4877 105 22 it it PRON ejpam-4877 105 23 follows follow VERB ejpam-4877 105 24 that that SCONJ ejpam-4877 105 25 γth(g γth(g NOUN ejpam-4877 105 26 ) ) PUNCT ejpam-4877 105 27 ≤ ≤ NOUN ejpam-4877 105 28 |ŝ| |ŝ| PROPN ejpam-4877 105 29 . . PUNCT ejpam-4877 106 1 consequently consequently ADV ejpam-4877 106 2 , , PUNCT ejpam-4877 106 3 γth(g γth(g NOUN ejpam-4877 106 4 ) ) PUNCT ejpam-4877 106 5 = = SYM ejpam-4877 106 6 |ŝ| |ŝ| PROPN ejpam-4877 106 7 . . PUNCT ejpam-4877 106 8 theorem theorem NOUN ejpam-4877 106 9 2 2 NUM ejpam-4877 106 10 . . PUNCT ejpam-4877 107 1 let let VERB ejpam-4877 107 2 g g PRON ejpam-4877 107 3 be be AUX ejpam-4877 107 4 a a DET ejpam-4877 107 5 graph graph NOUN ejpam-4877 107 6 of of ADP ejpam-4877 107 7 order order NOUN ejpam-4877 107 8 n n PRON ejpam-4877 107 9 with with ADP ejpam-4877 107 10 γ(c γ(c PROPN ejpam-4877 107 11 ) ) PUNCT ejpam-4877 107 12 ̸= ̸= PROPN ejpam-4877 107 13 1 1 NUM ejpam-4877 107 14 for for ADP ejpam-4877 107 15 each each DET ejpam-4877 107 16 component component NOUN ejpam-4877 107 17 c c PROPN ejpam-4877 107 18 of of ADP ejpam-4877 107 19 g. g. PROPN ejpam-4877 107 20 then then ADV ejpam-4877 107 21 s s VERB ejpam-4877 107 22 = = PUNCT ejpam-4877 107 23 ( ( PUNCT ejpam-4877 107 24 u1 u1 PROPN ejpam-4877 107 25 , , PUNCT ejpam-4877 107 26 u2 u2 PROPN ejpam-4877 107 27 , , PUNCT ejpam-4877 107 28 · · PUNCT ejpam-4877 107 29 · · PUNCT ejpam-4877 107 30 · · PUNCT ejpam-4877 107 31 , , PUNCT ejpam-4877 107 32 ul ul INTJ ejpam-4877 107 33 ) ) PUNCT ejpam-4877 107 34 is be AUX ejpam-4877 107 35 a a DET ejpam-4877 107 36 maximum maximum ADJ ejpam-4877 107 37 legal legal ADJ ejpam-4877 107 38 open open ADJ ejpam-4877 107 39 hop hop NOUN ejpam-4877 107 40 neighborhood neighborhood NOUN ejpam-4877 107 41 sequence sequence NOUN ejpam-4877 107 42 of of ADP ejpam-4877 107 43 g g PROPN ejpam-4877 108 1 if if SCONJ ejpam-4877 108 2 and and CCONJ ejpam-4877 108 3 only only ADV ejpam-4877 108 4 if if SCONJ ejpam-4877 108 5 s s NOUN ejpam-4877 108 6 is be AUX ejpam-4877 108 7 a a DET ejpam-4877 108 8 grundy grundy PROPN ejpam-4877 108 9 total total NOUN ejpam-4877 108 10 hop hop NOUN ejpam-4877 108 11 dominating dominating NOUN ejpam-4877 108 12 sequence sequence NOUN ejpam-4877 108 13 of of ADP ejpam-4877 108 14 g g PROPN ejpam-4877 108 15 and and CCONJ ejpam-4877 108 16 γthgr(g γthgr(g NOUN ejpam-4877 108 17 ) ) PUNCT ejpam-4877 109 1 = = PUNCT ejpam-4877 109 2 l. l. NOUN ejpam-4877 109 3 proof proof NOUN ejpam-4877 109 4 . . PUNCT ejpam-4877 110 1 let let VERB ejpam-4877 110 2 s s PRON ejpam-4877 110 3 = = PUNCT ejpam-4877 110 4 ( ( PUNCT ejpam-4877 110 5 u1 u1 PROPN ejpam-4877 110 6 , , PUNCT ejpam-4877 110 7 · · PUNCT ejpam-4877 110 8 · · PUNCT ejpam-4877 110 9 · · PUNCT ejpam-4877 110 10 , , PUNCT ejpam-4877 110 11 ul ul INTJ ejpam-4877 110 12 ) ) PUNCT ejpam-4877 110 13 be be AUX ejpam-4877 110 14 a a DET ejpam-4877 110 15 maximum maximum ADJ ejpam-4877 110 16 legal legal ADJ ejpam-4877 110 17 open open ADJ ejpam-4877 110 18 hop hop NOUN ejpam-4877 110 19 neighborhood neighborhood NOUN ejpam-4877 110 20 sequence sequence NOUN ejpam-4877 110 21 of of ADP ejpam-4877 110 22 g. g. PROPN ejpam-4877 110 23 suppose suppose VERB ejpam-4877 110 24 on on ADP ejpam-4877 110 25 the the DET ejpam-4877 110 26 contrary contrary NOUN ejpam-4877 110 27 that that PRON ejpam-4877 110 28 ŝ ŝ VERB ejpam-4877 110 29 is be AUX ejpam-4877 110 30 not not PART ejpam-4877 110 31 a a DET ejpam-4877 110 32 total total ADJ ejpam-4877 110 33 hop hop NOUN ejpam-4877 110 34 dominating dominating NOUN ejpam-4877 110 35 set set NOUN ejpam-4877 110 36 of of ADP ejpam-4877 110 37 g. g. PROPN ejpam-4877 110 38 then then ADV ejpam-4877 110 39 there there PRON ejpam-4877 110 40 exists exist VERB ejpam-4877 110 41 u u PROPN ejpam-4877 110 42 ∈ ∈ PROPN ejpam-4877 110 43 v v ADP ejpam-4877 110 44 ( ( PUNCT ejpam-4877 110 45 g g NOUN ejpam-4877 110 46 ) ) PUNCT ejpam-4877 110 47 such such ADJ ejpam-4877 110 48 that that DET ejpam-4877 110 49 u u PROPN ejpam-4877 110 50 /∈ /∈ PROPN ejpam-4877 110 51 n2 n2 PROPN ejpam-4877 110 52 g(ŝ g(ŝ PROPN ejpam-4877 110 53 ) ) PUNCT ejpam-4877 110 54 . . PUNCT ejpam-4877 111 1 this this PRON ejpam-4877 111 2 means mean VERB ejpam-4877 111 3 that that SCONJ ejpam-4877 111 4 u u PROPN ejpam-4877 111 5 /∈ /∈ PROPN ejpam-4877 111 6 n2 n2 ADJ ejpam-4877 111 7 g(v g(v PROPN ejpam-4877 111 8 ) ) PUNCT ejpam-4877 111 9 for for ADP ejpam-4877 111 10 every every DET ejpam-4877 111 11 v v NUM ejpam-4877 111 12 ∈ ∈ PROPN ejpam-4877 111 13 ŝ. ŝ. NOUN ejpam-4877 111 14 since since SCONJ ejpam-4877 111 15 u u NOUN ejpam-4877 111 16 is be AUX ejpam-4877 111 17 not not PART ejpam-4877 111 18 hop hop ADV ejpam-4877 111 19 dominated dominate VERB ejpam-4877 111 20 by by ADP ejpam-4877 111 21 any any DET ejpam-4877 111 22 v v NOUN ejpam-4877 111 23 ∈ ∈ PROPN ejpam-4877 111 24 ŝ ŝ NOUN ejpam-4877 111 25 , , PUNCT ejpam-4877 111 26 u u PROPN ejpam-4877 111 27 ∈ ∈ PROPN ejpam-4877 111 28 n2 n2 PROPN ejpam-4877 111 29 g(t g(t PROPN ejpam-4877 111 30 ) ) PUNCT ejpam-4877 111 31 for for ADP ejpam-4877 111 32 some some DET ejpam-4877 111 33 t t NOUN ejpam-4877 111 34 ∈ ∈ PROPN ejpam-4877 111 35 v v ADP ejpam-4877 111 36 ( ( PUNCT ejpam-4877 111 37 g g NOUN ejpam-4877 111 38 ) ) PUNCT ejpam-4877 111 39 \ \ NOUN ejpam-4877 111 40 ŝ. ŝ. NOUN ejpam-4877 111 41 this this PRON ejpam-4877 111 42 means mean VERB ejpam-4877 111 43 that that SCONJ ejpam-4877 111 44 n2 n2 PROPN ejpam-4877 111 45 g(t g(t PROPN ejpam-4877 111 46 ) ) PUNCT ejpam-4877 111 47 \ \ PROPN ejpam-4877 111 48 ⋃k ⋃k NOUN ejpam-4877 111 49 i=1n i=1n PROPN ejpam-4877 111 50 2 2 NUM ejpam-4877 111 51 g(ui g(ui PROPN ejpam-4877 111 52 ) ) PUNCT ejpam-4877 111 53 ̸= ̸= PROPN ejpam-4877 111 54 ∅. ∅. ADP ejpam-4877 111 55 thus thus ADV ejpam-4877 111 56 , , PUNCT ejpam-4877 111 57 s′ s′ ADJ ejpam-4877 111 58 = = PUNCT ejpam-4877 111 59 ( ( PUNCT ejpam-4877 111 60 u1 u1 PROPN ejpam-4877 111 61 , , PUNCT ejpam-4877 111 62 · · PUNCT ejpam-4877 111 63 · · PUNCT ejpam-4877 111 64 · · PUNCT ejpam-4877 111 65 , , PUNCT ejpam-4877 111 66 ul ul INTJ ejpam-4877 111 67 , , PUNCT ejpam-4877 111 68 t t PROPN ejpam-4877 111 69 ) ) PUNCT ejpam-4877 111 70 is be AUX ejpam-4877 111 71 a a DET ejpam-4877 111 72 legal legal ADJ ejpam-4877 111 73 open open ADJ ejpam-4877 111 74 hop hop NOUN ejpam-4877 111 75 neighborhood neighborhood NOUN ejpam-4877 111 76 sequence sequence NOUN ejpam-4877 111 77 of of ADP ejpam-4877 111 78 g g NOUN ejpam-4877 111 79 , , PUNCT ejpam-4877 111 80 a a DET ejpam-4877 111 81 contradiction contradiction NOUN ejpam-4877 111 82 to to ADP ejpam-4877 111 83 the the DET ejpam-4877 111 84 maximality maximality NOUN ejpam-4877 111 85 of of ADP ejpam-4877 111 86 s. s. PROPN ejpam-4877 111 87 therefore therefore ADV ejpam-4877 111 88 , , PUNCT ejpam-4877 111 89 ŝ ŝ X ejpam-4877 111 90 is be AUX ejpam-4877 111 91 a a DET ejpam-4877 111 92 total total ADJ ejpam-4877 111 93 hop hop NOUN ejpam-4877 111 94 dominating dominating NOUN ejpam-4877 111 95 set set NOUN ejpam-4877 111 96 of of ADP ejpam-4877 111 97 g. g. PROPN ejpam-4877 111 98 consequently consequently ADV ejpam-4877 111 99 , , PUNCT ejpam-4877 111 100 s s VERB ejpam-4877 111 101 is be AUX ejpam-4877 111 102 a a DET ejpam-4877 111 103 grundy grundy PROPN ejpam-4877 111 104 total total NOUN ejpam-4877 111 105 hop hop NOUN ejpam-4877 111 106 dominating dominating NOUN ejpam-4877 111 107 sequence sequence NOUN ejpam-4877 111 108 of of ADP ejpam-4877 111 109 g g PROPN ejpam-4877 111 110 and and CCONJ ejpam-4877 111 111 γthgr(g γthgr(g NOUN ejpam-4877 111 112 ) ) PUNCT ejpam-4877 112 1 = = VERB ejpam-4877 112 2 l. l. PROPN ejpam-4877 112 3 the the DET ejpam-4877 112 4 converse converse NOUN ejpam-4877 112 5 is be AUX ejpam-4877 112 6 clear clear ADJ ejpam-4877 112 7 . . PUNCT ejpam-4877 113 1 the the DET ejpam-4877 113 2 next next ADJ ejpam-4877 113 3 result result NOUN ejpam-4877 113 4 follows follow VERB ejpam-4877 113 5 from from ADP ejpam-4877 113 6 theorem theorem ADJ ejpam-4877 113 7 2 2 NUM ejpam-4877 113 8 . . PUNCT ejpam-4877 113 9 corollary corollary ADJ ejpam-4877 113 10 1 1 NUM ejpam-4877 113 11 . . PUNCT ejpam-4877 114 1 let let VERB ejpam-4877 114 2 g g PRON ejpam-4877 114 3 be be AUX ejpam-4877 114 4 a a DET ejpam-4877 114 5 graph graph NOUN ejpam-4877 114 6 of of ADP ejpam-4877 114 7 order order NOUN ejpam-4877 114 8 n n PRON ejpam-4877 114 9 with with ADP ejpam-4877 114 10 γ(c γ(c PROPN ejpam-4877 114 11 ) ) PUNCT ejpam-4877 114 12 ̸= ̸= PROPN ejpam-4877 114 13 1 1 NUM ejpam-4877 114 14 for for ADP ejpam-4877 114 15 each each DET ejpam-4877 114 16 component component NOUN ejpam-4877 114 17 c c NOUN ejpam-4877 114 18 of of ADP ejpam-4877 114 19 g g PROPN ejpam-4877 114 20 and and CCONJ ejpam-4877 114 21 let let VERB ejpam-4877 114 22 t t NOUN ejpam-4877 114 23 = = SYM ejpam-4877 114 24 ( ( PUNCT ejpam-4877 114 25 x1 x1 PROPN ejpam-4877 114 26 , , PUNCT ejpam-4877 114 27 · · PUNCT ejpam-4877 114 28 · · PUNCT ejpam-4877 114 29 · · PUNCT ejpam-4877 114 30 , , PUNCT ejpam-4877 114 31 xj xj NOUN ejpam-4877 114 32 ) ) PUNCT ejpam-4877 114 33 be be VERB ejpam-4877 114 34 a a DET ejpam-4877 114 35 legal legal ADJ ejpam-4877 114 36 open open ADJ ejpam-4877 114 37 hop hop NOUN ejpam-4877 114 38 neighborhood neighborhood NOUN ejpam-4877 114 39 sequence sequence NOUN ejpam-4877 114 40 of of ADP ejpam-4877 114 41 g. g. PROPN ejpam-4877 114 42 then then ADV ejpam-4877 114 43 |t̂ |t̂ VERB ejpam-4877 114 44 | | ADV ejpam-4877 114 45 = = SYM ejpam-4877 114 46 j j PROPN ejpam-4877 114 47 ≤ ≤ PROPN ejpam-4877 114 48 γthgr(g γthgr(g PROPN ejpam-4877 114 49 ) ) PUNCT ejpam-4877 114 50 . . PUNCT ejpam-4877 115 1 theorem theorem NOUN ejpam-4877 115 2 3 3 X ejpam-4877 115 3 . . PUNCT ejpam-4877 116 1 let let VERB ejpam-4877 116 2 g g PRON ejpam-4877 116 3 be be AUX ejpam-4877 116 4 a a DET ejpam-4877 116 5 graph graph NOUN ejpam-4877 116 6 of of ADP ejpam-4877 116 7 order order NOUN ejpam-4877 116 8 n n PRON ejpam-4877 116 9 with with ADP ejpam-4877 116 10 γ(c γ(c PROPN ejpam-4877 116 11 ) ) PUNCT ejpam-4877 116 12 ̸= ̸= PROPN ejpam-4877 116 13 1 1 NUM ejpam-4877 116 14 for for ADP ejpam-4877 116 15 each each DET ejpam-4877 116 16 component component NOUN ejpam-4877 116 17 c c PROPN ejpam-4877 116 18 of of ADP ejpam-4877 116 19 g. g. PROPN ejpam-4877 116 20 then then ADV ejpam-4877 116 21 4 4 NUM ejpam-4877 116 22 ≤ ≤ NUM ejpam-4877 116 23 γthgr(g γthgr(g PROPN ejpam-4877 116 24 ) ) PUNCT ejpam-4877 116 25 ≤ ≤ NOUN ejpam-4877 117 1 n n CCONJ ejpam-4877 117 2 and and CCONJ ejpam-4877 117 3 these these DET ejpam-4877 117 4 bounds bound NOUN ejpam-4877 117 5 can can AUX ejpam-4877 117 6 not not PART ejpam-4877 117 7 be be AUX ejpam-4877 117 8 improved improve VERB ejpam-4877 117 9 . . PUNCT ejpam-4877 118 1 j.a j.a PROPN ejpam-4877 118 2 . . PROPN ejpam-4877 118 3 hassan hassan PROPN ejpam-4877 118 4 , , PUNCT ejpam-4877 118 5 s. s. PROPN ejpam-4877 118 6 canoy canoy PROPN ejpam-4877 118 7 / / SYM ejpam-4877 118 8 eur eur PROPN ejpam-4877 118 9 . . PUNCT ejpam-4877 119 1 j. j. PROPN ejpam-4877 119 2 pure pure PROPN ejpam-4877 119 3 appl appl PROPN ejpam-4877 119 4 . . PROPN ejpam-4877 119 5 math math PROPN ejpam-4877 119 6 , , PUNCT ejpam-4877 119 7 16 16 NUM ejpam-4877 119 8 ( ( PUNCT ejpam-4877 119 9 4 4 NUM ejpam-4877 119 10 ) ) PUNCT ejpam-4877 119 11 ( ( PUNCT ejpam-4877 119 12 2023 2023 NUM ejpam-4877 119 13 ) ) PUNCT ejpam-4877 119 14 , , PUNCT ejpam-4877 119 15 2597 2597 NUM ejpam-4877 119 16 - - SYM ejpam-4877 119 17 2612 2612 NUM ejpam-4877 119 18 2601 2601 NUM ejpam-4877 119 19 proof proof NOUN ejpam-4877 119 20 . . PUNCT ejpam-4877 120 1 clearly clearly ADV ejpam-4877 120 2 γthgr(g γthgr(g NUM ejpam-4877 120 3 ) ) PUNCT ejpam-4877 120 4 = = SYM ejpam-4877 121 1 1 1 NUM ejpam-4877 121 2 is be AUX ejpam-4877 121 3 not not PART ejpam-4877 121 4 possible possible ADJ ejpam-4877 121 5 . . PUNCT ejpam-4877 122 1 suppose suppose VERB ejpam-4877 122 2 γthgr(g γthgr(g NOUN ejpam-4877 122 3 ) ) PUNCT ejpam-4877 122 4 = = SYM ejpam-4877 122 5 2 2 X ejpam-4877 122 6 , , PUNCT ejpam-4877 122 7 say say VERB ejpam-4877 122 8 , , PUNCT ejpam-4877 122 9 s s PART ejpam-4877 122 10 = = PUNCT ejpam-4877 122 11 ( ( PUNCT ejpam-4877 122 12 v1 v1 PROPN ejpam-4877 122 13 , , PUNCT ejpam-4877 122 14 v2 v2 PROPN ejpam-4877 122 15 ) ) PUNCT ejpam-4877 122 16 is be AUX ejpam-4877 122 17 a a DET ejpam-4877 122 18 grundy grundy PROPN ejpam-4877 122 19 total total NOUN ejpam-4877 122 20 hop hop NOUN ejpam-4877 122 21 dominating dominating NOUN ejpam-4877 122 22 sequence sequence NOUN ejpam-4877 122 23 . . PUNCT ejpam-4877 123 1 since since SCONJ ejpam-4877 123 2 ŝ ŝ NUM ejpam-4877 123 3 is be AUX ejpam-4877 123 4 a a DET ejpam-4877 123 5 total total ADJ ejpam-4877 123 6 hop hop NOUN ejpam-4877 123 7 dominating dominating NOUN ejpam-4877 123 8 set set NOUN ejpam-4877 123 9 , , PUNCT ejpam-4877 123 10 v1 v1 PROPN ejpam-4877 123 11 ∈ ∈ PROPN ejpam-4877 123 12 n2 n2 NOUN ejpam-4877 123 13 g(v2 g(v2 NOUN ejpam-4877 123 14 ) ) PUNCT ejpam-4877 123 15 . . PUNCT ejpam-4877 124 1 let let VERB ejpam-4877 124 2 v v NUM ejpam-4877 124 3 ∈ ∈ NOUN ejpam-4877 124 4 ng(v1 ng(v1 NOUN ejpam-4877 124 5 ) ) PUNCT ejpam-4877 124 6 ∩ ∩ NOUN ejpam-4877 124 7 ng(v2 ng(v2 NOUN ejpam-4877 124 8 ) ) PUNCT ejpam-4877 124 9 . . PUNCT ejpam-4877 125 1 then then ADV ejpam-4877 125 2 v v X ejpam-4877 125 3 /∈ /∈ PUNCT ejpam-4877 125 4 n2 n2 ADJ ejpam-4877 125 5 g(v1 g(v1 PROPN ejpam-4877 125 6 ) ) PUNCT ejpam-4877 125 7 ∪ ∪ ADP ejpam-4877 125 8 n2 n2 ADJ ejpam-4877 125 9 g(v2 g(v2 NOUN ejpam-4877 125 10 ) ) PUNCT ejpam-4877 125 11 , , PUNCT ejpam-4877 125 12 a a DET ejpam-4877 125 13 contradiction contradiction NOUN ejpam-4877 125 14 . . PUNCT ejpam-4877 126 1 next next ADV ejpam-4877 126 2 , , PUNCT ejpam-4877 126 3 suppose suppose VERB ejpam-4877 126 4 γthgr(g γthgr(g NOUN ejpam-4877 126 5 ) ) PUNCT ejpam-4877 126 6 = = SYM ejpam-4877 126 7 3 3 X ejpam-4877 126 8 , , PUNCT ejpam-4877 126 9 say say VERB ejpam-4877 126 10 s s X ejpam-4877 126 11 = = PUNCT ejpam-4877 126 12 ( ( PUNCT ejpam-4877 126 13 v1 v1 PROPN ejpam-4877 126 14 , , PUNCT ejpam-4877 126 15 v2 v2 PROPN ejpam-4877 126 16 , , PUNCT ejpam-4877 126 17 v3 v3 PROPN ejpam-4877 126 18 ) ) PUNCT ejpam-4877 126 19 is be AUX ejpam-4877 126 20 a a DET ejpam-4877 126 21 grundy grundy PROPN ejpam-4877 126 22 total total NOUN ejpam-4877 126 23 hop hop NOUN ejpam-4877 126 24 dominating dominating NOUN ejpam-4877 126 25 sequence sequence NOUN ejpam-4877 126 26 . . PUNCT ejpam-4877 127 1 since since SCONJ ejpam-4877 127 2 ŝ ŝ NUM ejpam-4877 127 3 is be AUX ejpam-4877 127 4 a a DET ejpam-4877 127 5 total total ADJ ejpam-4877 127 6 hop hop NOUN ejpam-4877 127 7 dominating dominating NOUN ejpam-4877 127 8 set set NOUN ejpam-4877 127 9 , , PUNCT ejpam-4877 127 10 v1 v1 PROPN ejpam-4877 127 11 is be AUX ejpam-4877 127 12 hop hop ADV ejpam-4877 127 13 dominated dominate VERB ejpam-4877 127 14 by by ADP ejpam-4877 127 15 v2 v2 PROPN ejpam-4877 127 16 or or CCONJ ejpam-4877 127 17 v3 v3 PROPN ejpam-4877 127 18 . . PUNCT ejpam-4877 128 1 suppose suppose VERB ejpam-4877 128 2 dg(v1 dg(v1 NOUN ejpam-4877 128 3 , , PUNCT ejpam-4877 128 4 v2 v2 NOUN ejpam-4877 128 5 ) ) PUNCT ejpam-4877 128 6 = = SYM ejpam-4877 128 7 2 2 NUM ejpam-4877 128 8 and and CCONJ ejpam-4877 128 9 let let VERB ejpam-4877 128 10 p p PRON ejpam-4877 128 11 ∈ ∈ NOUN ejpam-4877 128 12 ng(v1 ng(v1 NOUN ejpam-4877 128 13 ) ) PUNCT ejpam-4877 128 14 ∩ ∩ NOUN ejpam-4877 128 15 ng(v2 ng(v2 NOUN ejpam-4877 128 16 ) ) PUNCT ejpam-4877 128 17 . . PUNCT ejpam-4877 129 1 then then ADV ejpam-4877 129 2 p p PROPN ejpam-4877 129 3 ∈ ∈ PROPN ejpam-4877 129 4 n2 n2 PROPN ejpam-4877 129 5 g(v3 g(v3 PROPN ejpam-4877 129 6 ) ) PUNCT ejpam-4877 129 7 and and CCONJ ejpam-4877 129 8 v3 v3 PROPN ejpam-4877 129 9 ∈ ∈ PROPN ejpam-4877 129 10 n2 n2 PROPN ejpam-4877 129 11 g(v1 g(v1 PROPN ejpam-4877 129 12 ) ) PUNCT ejpam-4877 129 13 ∪ ∪ ADP ejpam-4877 129 14 n2 n2 ADJ ejpam-4877 129 15 g(v2 g(v2 NOUN ejpam-4877 129 16 ) ) PUNCT ejpam-4877 129 17 . . PUNCT ejpam-4877 130 1 let let VERB ejpam-4877 130 2 s∗ s∗ PROPN ejpam-4877 130 3 = = SYM ejpam-4877 130 4 ( ( PUNCT ejpam-4877 130 5 p p X ejpam-4877 130 6 , , PUNCT ejpam-4877 130 7 v1 v1 NOUN ejpam-4877 130 8 , , PUNCT ejpam-4877 130 9 v2 v2 PROPN ejpam-4877 130 10 , , PUNCT ejpam-4877 130 11 v3 v3 PROPN ejpam-4877 130 12 ) ) PUNCT ejpam-4877 130 13 . . PUNCT ejpam-4877 131 1 then then ADV ejpam-4877 131 2 ŝ∗ ŝ∗ NOUN ejpam-4877 131 3 is be AUX ejpam-4877 131 4 a a DET ejpam-4877 131 5 total total ADJ ejpam-4877 131 6 hop hop NOUN ejpam-4877 131 7 dominating dominating NOUN ejpam-4877 131 8 set set NOUN ejpam-4877 131 9 . . PUNCT ejpam-4877 132 1 moreover moreover ADV ejpam-4877 132 2 , , PUNCT ejpam-4877 132 3 observe observe VERB ejpam-4877 132 4 that that SCONJ ejpam-4877 132 5 v3 v3 PROPN ejpam-4877 132 6 ∈ ∈ PROPN ejpam-4877 132 7 n2 n2 NOUN ejpam-4877 132 8 g(p g(p PROPN ejpam-4877 132 9 ) ) PUNCT ejpam-4877 132 10 , , PUNCT ejpam-4877 132 11 v2 v2 PROPN ejpam-4877 132 12 ∈ ∈ PROPN ejpam-4877 132 13 n2 n2 NOUN ejpam-4877 132 14 g(v1 g(v1 NOUN ejpam-4877 132 15 ) ) PUNCT ejpam-4877 132 16 \ \ PROPN ejpam-4877 132 17 n2 n2 ADJ ejpam-4877 132 18 g(p g(p PROPN ejpam-4877 132 19 ) ) PUNCT ejpam-4877 132 20 , , PUNCT ejpam-4877 132 21 v1 v1 PROPN ejpam-4877 132 22 ∈ ∈ PROPN ejpam-4877 133 1 [ [ X ejpam-4877 133 2 n2 n2 ADJ ejpam-4877 133 3 g(v2 g(v2 NOUN ejpam-4877 133 4 ) ) PUNCT ejpam-4877 133 5 \ \ PUNCT ejpam-4877 133 6 ( ( PUNCT ejpam-4877 133 7 n2 n2 PROPN ejpam-4877 133 8 g(v1 g(v1 NOUN ejpam-4877 133 9 ) ) PUNCT ejpam-4877 133 10 ∪ ∪ ADP ejpam-4877 133 11 n2 n2 ADJ ejpam-4877 133 12 g(p g(p PROPN ejpam-4877 133 13 ) ) PUNCT ejpam-4877 133 14 ) ) PUNCT ejpam-4877 133 15 ] ] PUNCT ejpam-4877 133 16 and and CCONJ ejpam-4877 133 17 p p NOUN ejpam-4877 133 18 ∈ ∈ PROPN ejpam-4877 134 1 [ [ X ejpam-4877 134 2 n2 n2 NOUN ejpam-4877 134 3 g(v3 g(v3 PROPN ejpam-4877 134 4 ) ) PUNCT ejpam-4877 134 5 \ \ PUNCT ejpam-4877 134 6 ( ( PUNCT ejpam-4877 134 7 n2 n2 ADJ ejpam-4877 134 8 g(v2 g(v2 PROPN ejpam-4877 134 9 ) ) PUNCT ejpam-4877 134 10 ∪ ∪ ADP ejpam-4877 134 11 n2 n2 PROPN ejpam-4877 134 12 g(v1 g(v1 NOUN ejpam-4877 134 13 ) ) PUNCT ejpam-4877 134 14 ∪ ∪ ADP ejpam-4877 134 15 n2 n2 ADJ ejpam-4877 134 16 g(p g(p PROPN ejpam-4877 134 17 ) ) PUNCT ejpam-4877 134 18 ) ) PUNCT ejpam-4877 134 19 ] ] PUNCT ejpam-4877 134 20 . . PUNCT ejpam-4877 135 1 hence hence ADV ejpam-4877 135 2 , , PUNCT ejpam-4877 135 3 s∗ s∗ PROPN ejpam-4877 135 4 is be AUX ejpam-4877 135 5 a a DET ejpam-4877 135 6 legal legal ADJ ejpam-4877 135 7 open open ADJ ejpam-4877 135 8 hop hop NOUN ejpam-4877 135 9 neighborhood neighborhood NOUN ejpam-4877 135 10 sequence sequence NOUN ejpam-4877 135 11 , , PUNCT ejpam-4877 135 12 and and CCONJ ejpam-4877 135 13 so so ADV ejpam-4877 135 14 s∗ s∗ PROPN ejpam-4877 135 15 is be AUX ejpam-4877 135 16 a a DET ejpam-4877 135 17 grundy grundy PROPN ejpam-4877 135 18 total total NOUN ejpam-4877 135 19 hop hop NOUN ejpam-4877 135 20 dominating dominating NOUN ejpam-4877 135 21 sequence sequence NOUN ejpam-4877 135 22 of of ADP ejpam-4877 135 23 g g NOUN ejpam-4877 135 24 , , PUNCT ejpam-4877 135 25 contrary contrary ADJ ejpam-4877 135 26 to to ADP ejpam-4877 135 27 our our PRON ejpam-4877 135 28 assumption assumption NOUN ejpam-4877 135 29 that that SCONJ ejpam-4877 135 30 γthgr(g γthgr(g NOUN ejpam-4877 135 31 ) ) PUNCT ejpam-4877 136 1 = = SYM ejpam-4877 137 1 3 3 X ejpam-4877 137 2 . . X ejpam-4877 137 3 therefore therefore ADV ejpam-4877 137 4 , , PUNCT ejpam-4877 137 5 γthgr(g γthgr(g NOUN ejpam-4877 137 6 ) ) PUNCT ejpam-4877 137 7 ≥ ≥ NOUN ejpam-4877 137 8 4 4 NUM ejpam-4877 137 9 . . X ejpam-4877 138 1 for for ADP ejpam-4877 138 2 tightness tightness NOUN ejpam-4877 138 3 of of ADP ejpam-4877 138 4 the the DET ejpam-4877 138 5 bounds bound NOUN ejpam-4877 138 6 , , PUNCT ejpam-4877 138 7 consider consider VERB ejpam-4877 138 8 g g NOUN ejpam-4877 138 9 = = SYM ejpam-4877 138 10 c4 c4 NOUN ejpam-4877 138 11 and and CCONJ ejpam-4877 138 12 h h NOUN ejpam-4877 138 13 = = PROPN ejpam-4877 138 14 p8 p8 PROPN ejpam-4877 138 15 . . PUNCT ejpam-4877 139 1 then then ADV ejpam-4877 139 2 γthgr(g γthgr(g NUM ejpam-4877 139 3 ) ) PUNCT ejpam-4877 139 4 = = SYM ejpam-4877 139 5 4 4 NUM ejpam-4877 139 6 and and CCONJ ejpam-4877 139 7 γthgr(h γthgr(h NOUN ejpam-4877 139 8 ) ) PUNCT ejpam-4877 139 9 = = SYM ejpam-4877 139 10 8 8 X ejpam-4877 139 11 . . PUNCT ejpam-4877 139 12 theorem theorem NOUN ejpam-4877 139 13 4 4 NUM ejpam-4877 139 14 . . PUNCT ejpam-4877 140 1 let let VERB ejpam-4877 140 2 g g PRON ejpam-4877 140 3 be be AUX ejpam-4877 140 4 a a DET ejpam-4877 140 5 connected connected ADJ ejpam-4877 140 6 graph graph NOUN ejpam-4877 140 7 of of ADP ejpam-4877 140 8 order order NOUN ejpam-4877 140 9 n n PRON ejpam-4877 140 10 such such ADJ ejpam-4877 140 11 that that PRON ejpam-4877 140 12 γ(g γ(g PROPN ejpam-4877 140 13 ) ) PUNCT ejpam-4877 140 14 ̸= ̸= PROPN ejpam-4877 140 15 1 1 NUM ejpam-4877 140 16 . . PUNCT ejpam-4877 141 1 if if SCONJ ejpam-4877 141 2 n n NOUN ejpam-4877 141 3 = = NOUN ejpam-4877 141 4 2m 2m NUM ejpam-4877 141 5 , , PUNCT ejpam-4877 141 6 m m VERB ejpam-4877 141 7 ≥ ≥ NOUN ejpam-4877 141 8 2 2 NUM ejpam-4877 141 9 and and CCONJ ejpam-4877 141 10 the the DET ejpam-4877 141 11 vertices vertex NOUN ejpam-4877 141 12 of of ADP ejpam-4877 141 13 g g NOUN ejpam-4877 141 14 can can AUX ejpam-4877 141 15 be be AUX ejpam-4877 141 16 labeled label VERB ejpam-4877 141 17 as as ADP ejpam-4877 141 18 u1 u1 NOUN ejpam-4877 141 19 , , PUNCT ejpam-4877 141 20 . . PUNCT ejpam-4877 141 21 . . PUNCT ejpam-4877 142 1 . . PUNCT ejpam-4877 143 1 , , PUNCT ejpam-4877 143 2 um um INTJ ejpam-4877 143 3 , , PUNCT ejpam-4877 143 4 v1 v1 PROPN ejpam-4877 143 5 , , PUNCT ejpam-4877 143 6 . . PUNCT ejpam-4877 143 7 . . PUNCT ejpam-4877 144 1 . . PUNCT ejpam-4877 145 1 , , PUNCT ejpam-4877 145 2 vm vm PROPN ejpam-4877 145 3 in in ADP ejpam-4877 145 4 such such DET ejpam-4877 145 5 a a DET ejpam-4877 145 6 way way NOUN ejpam-4877 145 7 that that SCONJ ejpam-4877 145 8 ( ( PUNCT ejpam-4877 145 9 i i NOUN ejpam-4877 145 10 ) ) PUNCT ejpam-4877 145 11 dg(ui dg(ui PROPN ejpam-4877 145 12 , , PUNCT ejpam-4877 145 13 vi vi NOUN ejpam-4877 145 14 ) ) PUNCT ejpam-4877 145 15 = = SYM ejpam-4877 145 16 2 2 NUM ejpam-4877 145 17 for for ADP ejpam-4877 145 18 each each DET ejpam-4877 145 19 i i PRON ejpam-4877 145 20 , , PUNCT ejpam-4877 145 21 ( ( PUNCT ejpam-4877 145 22 ii ii NOUN ejpam-4877 145 23 ) ) PUNCT ejpam-4877 145 24 { { PUNCT ejpam-4877 145 25 u1 u1 NOUN ejpam-4877 145 26 , , PUNCT ejpam-4877 145 27 . . PUNCT ejpam-4877 145 28 . . PUNCT ejpam-4877 146 1 . . PUNCT ejpam-4877 147 1 , , PUNCT ejpam-4877 147 2 um um INTJ ejpam-4877 147 3 } } PUNCT ejpam-4877 147 4 is be AUX ejpam-4877 147 5 a a DET ejpam-4877 147 6 hop hop NOUN ejpam-4877 147 7 independent independent ADJ ejpam-4877 147 8 set set NOUN ejpam-4877 147 9 of of ADP ejpam-4877 147 10 g g NOUN ejpam-4877 147 11 , , PUNCT ejpam-4877 147 12 and and CCONJ ejpam-4877 147 13 ( ( PUNCT ejpam-4877 147 14 iii iii NOUN ejpam-4877 147 15 ) ) PUNCT ejpam-4877 147 16 dg(ui dg(ui NOUN ejpam-4877 147 17 , , PUNCT ejpam-4877 147 18 vj vj PROPN ejpam-4877 147 19 ) ) PUNCT ejpam-4877 147 20 = = SYM ejpam-4877 147 21 2 2 NUM ejpam-4877 147 22 implies imply VERB ejpam-4877 147 23 that that SCONJ ejpam-4877 147 24 i i PRON ejpam-4877 147 25 ≥ ≥ VERB ejpam-4877 147 26 j j NOUN ejpam-4877 147 27 , , PUNCT ejpam-4877 147 28 then then ADV ejpam-4877 147 29 γthgr(g γthgr(g NOUN ejpam-4877 147 30 ) ) PUNCT ejpam-4877 148 1 = = SYM ejpam-4877 148 2 n. n. NOUN ejpam-4877 148 3 proof proof NOUN ejpam-4877 148 4 . . PUNCT ejpam-4877 149 1 suppose suppose VERB ejpam-4877 149 2 the the DET ejpam-4877 149 3 vertices vertex NOUN ejpam-4877 149 4 of of ADP ejpam-4877 149 5 g g NOUN ejpam-4877 149 6 can can AUX ejpam-4877 149 7 be be AUX ejpam-4877 149 8 labeled label VERB ejpam-4877 149 9 as as ADP ejpam-4877 149 10 described describe VERB ejpam-4877 149 11 . . PUNCT ejpam-4877 150 1 clearly clearly ADV ejpam-4877 150 2 , , PUNCT ejpam-4877 150 3 ŝ ŝ X ejpam-4877 150 4 = = SYM ejpam-4877 150 5 { { PUNCT ejpam-4877 150 6 u1 u1 NOUN ejpam-4877 150 7 , , PUNCT ejpam-4877 150 8 · · PUNCT ejpam-4877 150 9 · · PUNCT ejpam-4877 150 10 · · PUNCT ejpam-4877 150 11 , , PUNCT ejpam-4877 150 12 um um INTJ ejpam-4877 150 13 , , PUNCT ejpam-4877 150 14 vm vm PROPN ejpam-4877 150 15 , , PUNCT ejpam-4877 150 16 · · PUNCT ejpam-4877 150 17 · · PUNCT ejpam-4877 150 18 · · PUNCT ejpam-4877 150 19 , , PUNCT ejpam-4877 150 20 v1 v1 NOUN ejpam-4877 150 21 } } PUNCT ejpam-4877 150 22 is be AUX ejpam-4877 150 23 a a DET ejpam-4877 150 24 total total ADJ ejpam-4877 150 25 hop hop NOUN ejpam-4877 150 26 dominating dominating NOUN ejpam-4877 150 27 set set NOUN ejpam-4877 150 28 of of ADP ejpam-4877 150 29 g. g. PROPN ejpam-4877 151 1 observe observe VERB ejpam-4877 151 2 that that SCONJ ejpam-4877 151 3 vi vi PROPN ejpam-4877 151 4 ∈ ∈ PROPN ejpam-4877 151 5 n2 n2 PROPN ejpam-4877 151 6 g(ui g(ui PROPN ejpam-4877 151 7 ) ) PUNCT ejpam-4877 151 8 \ \ NOUN ejpam-4877 151 9 ⋃i−1 ⋃i−1 NOUN ejpam-4877 151 10 j=1n j=1n PROPN ejpam-4877 151 11 2 2 NUM ejpam-4877 151 12 g(uj g(uj PROPN ejpam-4877 151 13 ) ) PUNCT ejpam-4877 151 14 for for ADP ejpam-4877 151 15 each each DET ejpam-4877 151 16 i i PRON ejpam-4877 151 17 ∈ ∈ PROPN ejpam-4877 151 18 { { PUNCT ejpam-4877 151 19 2 2 NUM ejpam-4877 151 20 , , PUNCT ejpam-4877 151 21 . . PUNCT ejpam-4877 151 22 . . PUNCT ejpam-4877 152 1 . . PUNCT ejpam-4877 153 1 , , PUNCT ejpam-4877 153 2 m m VERB ejpam-4877 153 3 } } PUNCT ejpam-4877 153 4 by by ADP ejpam-4877 153 5 ( ( PUNCT ejpam-4877 153 6 i i NOUN ejpam-4877 153 7 ) ) PUNCT ejpam-4877 153 8 and and CCONJ ejpam-4877 153 9 ( ( PUNCT ejpam-4877 153 10 iii iii NOUN ejpam-4877 153 11 ) ) PUNCT ejpam-4877 153 12 and and CCONJ ejpam-4877 153 13 um um INTJ ejpam-4877 153 14 ∈ ∈ PROPN ejpam-4877 153 15 n2 n2 PROPN ejpam-4877 153 16 g(vm g(vm PROPN ejpam-4877 153 17 ) ) PUNCT ejpam-4877 153 18 \⋃m \⋃m PROPN ejpam-4877 153 19 j=1n j=1n VERB ejpam-4877 153 20 2 2 NUM ejpam-4877 153 21 g(uj g(uj NOUN ejpam-4877 153 22 ) ) PUNCT ejpam-4877 153 23 by by ADP ejpam-4877 153 24 ( ( PUNCT ejpam-4877 153 25 i i NOUN ejpam-4877 153 26 ) ) PUNCT ejpam-4877 153 27 and and CCONJ ejpam-4877 153 28 ( ( PUNCT ejpam-4877 153 29 ii ii NOUN ejpam-4877 153 30 ) ) PUNCT ejpam-4877 153 31 . . PUNCT ejpam-4877 154 1 now now ADV ejpam-4877 154 2 , , PUNCT ejpam-4877 154 3 for for ADP ejpam-4877 154 4 any any DET ejpam-4877 154 5 k k PROPN ejpam-4877 154 6 < < X ejpam-4877 154 7 m m PROPN ejpam-4877 154 8 , , PUNCT ejpam-4877 154 9 uk uk PROPN ejpam-4877 154 10 ∈ ∈ PROPN ejpam-4877 154 11 n2 n2 NOUN ejpam-4877 154 12 g(vk g(vk PROPN ejpam-4877 154 13 ) ) PUNCT ejpam-4877 154 14 \ \ PUNCT ejpam-4877 155 1   VERB ejpam-4877 155 2 m⋃ m⋃ NOUN ejpam-4877 155 3 j=1 j=1 PROPN ejpam-4877 155 4 n2 n2 PROPN ejpam-4877 155 5 g(uj g(uj PROPN ejpam-4877 155 6 ) ) PUNCT ejpam-4877 155 7   PROPN ejpam-4877 155 8 ∪ ∪ ADV ejpam-4877 155 9 ( ( PUNCT ejpam-4877 155 10 k+1⋃ k+1⋃ NOUN ejpam-4877 155 11 i i PROPN ejpam-4877 155 12 = = NOUN ejpam-4877 155 13 m m NOUN ejpam-4877 155 14 n2 n2 NOUN ejpam-4877 155 15 g(vi g(vi NUM ejpam-4877 155 16 ) ) PUNCT ejpam-4877 155 17 ) ) PUNCT ejpam-4877 155 18   NOUN ejpam-4877 155 19 by by ADP ejpam-4877 155 20 ( ( PUNCT ejpam-4877 155 21 i i NOUN ejpam-4877 155 22 ) ) PUNCT ejpam-4877 155 23 , , PUNCT ejpam-4877 155 24 ( ( PUNCT ejpam-4877 155 25 ii ii NOUN ejpam-4877 155 26 ) ) PUNCT ejpam-4877 155 27 , , PUNCT ejpam-4877 155 28 and and CCONJ ejpam-4877 155 29 ( ( PUNCT ejpam-4877 155 30 iii iii NOUN ejpam-4877 155 31 ) ) PUNCT ejpam-4877 155 32 . . PUNCT ejpam-4877 156 1 therefore therefore ADV ejpam-4877 156 2 , , PUNCT ejpam-4877 156 3 s s VERB ejpam-4877 156 4 is be AUX ejpam-4877 156 5 a a DET ejpam-4877 156 6 grundy grundy PROPN ejpam-4877 156 7 total total NOUN ejpam-4877 156 8 hop hop NOUN ejpam-4877 156 9 dominating dominating NOUN ejpam-4877 156 10 sequence sequence NOUN ejpam-4877 156 11 , , PUNCT ejpam-4877 156 12 showing show VERB ejpam-4877 156 13 that that SCONJ ejpam-4877 156 14 γthgr(g γthgr(g NOUN ejpam-4877 156 15 ) ) PUNCT ejpam-4877 156 16 = = PUNCT ejpam-4877 156 17 n. n. NOUN ejpam-4877 156 18 proposition proposition NOUN ejpam-4877 156 19 1 1 X ejpam-4877 156 20 . . PUNCT ejpam-4877 157 1 let let VERB ejpam-4877 157 2 n n PRON ejpam-4877 157 3 and and CCONJ ejpam-4877 157 4 m m AUX ejpam-4877 157 5 be be AUX ejpam-4877 157 6 any any DET ejpam-4877 157 7 positive positive ADJ ejpam-4877 157 8 integers integer NOUN ejpam-4877 158 1 such such ADJ ejpam-4877 158 2 that that SCONJ ejpam-4877 158 3 n n NUM ejpam-4877 158 4 ≥ ≥ NOUN ejpam-4877 158 5 4 4 NUM ejpam-4877 158 6 . . PUNCT ejpam-4877 158 7 then then ADV ejpam-4877 158 8 γthgr(pn γthgr(pn ADJ ejpam-4877 158 9 ) ) PUNCT ejpam-4877 158 10 = = PUNCT ejpam-4877 159 1   PRON ejpam-4877 159 2 n− n− NOUN ejpam-4877 159 3 2 2 NUM ejpam-4877 159 4 if if SCONJ ejpam-4877 159 5 n n NOUN ejpam-4877 159 6 = = SYM ejpam-4877 159 7 4m+ 4m+ NUM ejpam-4877 159 8 2 2 NUM ejpam-4877 159 9 n− n− NOUN ejpam-4877 159 10 1 1 NUM ejpam-4877 159 11 if if SCONJ ejpam-4877 159 12 n n PRON ejpam-4877 159 13 ≥ ≥ NOUN ejpam-4877 159 14 5 5 NUM ejpam-4877 159 15 and and CCONJ ejpam-4877 159 16 odd odd ADJ ejpam-4877 159 17 n n NOUN ejpam-4877 159 18 if if SCONJ ejpam-4877 159 19 n n NOUN ejpam-4877 159 20 = = SYM ejpam-4877 159 21 4 4 NUM ejpam-4877 159 22 m m NOUN ejpam-4877 159 23 proof proof NOUN ejpam-4877 159 24 . . PUNCT ejpam-4877 160 1 let let VERB ejpam-4877 160 2 pn pn VERB ejpam-4877 160 3 = = PUNCT ejpam-4877 161 1 [ [ X ejpam-4877 161 2 v1 v1 NOUN ejpam-4877 161 3 , , PUNCT ejpam-4877 161 4 v2 v2 NOUN ejpam-4877 161 5 , , PUNCT ejpam-4877 161 6 . . PUNCT ejpam-4877 161 7 . . PUNCT ejpam-4877 161 8 . . PUNCT ejpam-4877 162 1 , , PUNCT ejpam-4877 162 2 vn vn X ejpam-4877 162 3 ] ] PUNCT ejpam-4877 162 4 . . PUNCT ejpam-4877 163 1 clearly clearly ADV ejpam-4877 163 2 , , PUNCT ejpam-4877 163 3 γthgr(p6 γthgr(p6 ADJ ejpam-4877 163 4 ) ) PUNCT ejpam-4877 163 5 = = SYM ejpam-4877 164 1 4 4 X ejpam-4877 164 2 . . X ejpam-4877 164 3 for for ADP ejpam-4877 164 4 n n NOUN ejpam-4877 164 5 = = SYM ejpam-4877 164 6 4 4 NUM ejpam-4877 164 7 m m NOUN ejpam-4877 164 8 + + NUM ejpam-4877 164 9 2 2 NUM ejpam-4877 164 10 ≥ ≥ NOUN ejpam-4877 164 11 10 10 NUM ejpam-4877 164 12 , , PUNCT ejpam-4877 164 13 let let VERB ejpam-4877 164 14 s s AUX ejpam-4877 164 15 = = PUNCT ejpam-4877 164 16 ( ( PUNCT ejpam-4877 164 17 v1 v1 PROPN ejpam-4877 164 18 , , PUNCT ejpam-4877 164 19 v2 v2 NOUN ejpam-4877 164 20 , , PUNCT ejpam-4877 164 21 . . PUNCT ejpam-4877 164 22 . . PUNCT ejpam-4877 165 1 . . PUNCT ejpam-4877 166 1 , , PUNCT ejpam-4877 166 2 vn−2 vn−2 PROPN ejpam-4877 166 3 ) ) PUNCT ejpam-4877 166 4 . . PUNCT ejpam-4877 167 1 clearly clearly ADV ejpam-4877 167 2 , , PUNCT ejpam-4877 167 3 s s VERB ejpam-4877 167 4 is be AUX ejpam-4877 167 5 a a DET ejpam-4877 167 6 grundy grundy PROPN ejpam-4877 167 7 total total NOUN ejpam-4877 167 8 hop hop NOUN ejpam-4877 167 9 dominating dominating NOUN ejpam-4877 167 10 sequence sequence NOUN ejpam-4877 167 11 of of ADP ejpam-4877 167 12 pn pn PROPN ejpam-4877 167 13 . . PUNCT ejpam-4877 168 1 thus thus ADV ejpam-4877 168 2 , , PUNCT ejpam-4877 168 3 j.a j.a PROPN ejpam-4877 168 4 . . PROPN ejpam-4877 168 5 hassan hassan PROPN ejpam-4877 168 6 , , PUNCT ejpam-4877 168 7 s. s. PROPN ejpam-4877 168 8 canoy canoy PROPN ejpam-4877 168 9 / / SYM ejpam-4877 168 10 eur eur PROPN ejpam-4877 168 11 . . PUNCT ejpam-4877 169 1 j. j. PROPN ejpam-4877 169 2 pure pure PROPN ejpam-4877 169 3 appl appl PROPN ejpam-4877 169 4 . . PROPN ejpam-4877 169 5 math math PROPN ejpam-4877 169 6 , , PUNCT ejpam-4877 169 7 16 16 NUM ejpam-4877 169 8 ( ( PUNCT ejpam-4877 169 9 4 4 NUM ejpam-4877 169 10 ) ) PUNCT ejpam-4877 169 11 ( ( PUNCT ejpam-4877 169 12 2023 2023 NUM ejpam-4877 169 13 ) ) PUNCT ejpam-4877 169 14 , , PUNCT ejpam-4877 169 15 2597 2597 NUM ejpam-4877 169 16 - - SYM ejpam-4877 169 17 2612 2612 NUM ejpam-4877 169 18 2602 2602 NUM ejpam-4877 169 19 γthgr(pn γthgr(pn PROPN ejpam-4877 169 20 ) ) PUNCT ejpam-4877 169 21 ≥ ≥ NOUN ejpam-4877 169 22 n n CCONJ ejpam-4877 170 1 − − NOUN ejpam-4877 170 2 2 2 NUM ejpam-4877 170 3 . . PUNCT ejpam-4877 171 1 on on ADP ejpam-4877 171 2 the the DET ejpam-4877 171 3 other other ADJ ejpam-4877 171 4 hand hand NOUN ejpam-4877 171 5 , , PUNCT ejpam-4877 171 6 let let VERB ejpam-4877 171 7 s′ s′ ADJ ejpam-4877 171 8 = = SYM ejpam-4877 171 9 ( ( PUNCT ejpam-4877 171 10 w1 w1 NOUN ejpam-4877 171 11 , , PUNCT ejpam-4877 171 12 w2 w2 NOUN ejpam-4877 171 13 , , PUNCT ejpam-4877 171 14 . . PUNCT ejpam-4877 171 15 . . PUNCT ejpam-4877 172 1 . . PUNCT ejpam-4877 173 1 , , PUNCT ejpam-4877 173 2 wk wk AUX ejpam-4877 173 3 ) ) PUNCT ejpam-4877 173 4 be be AUX ejpam-4877 173 5 a a DET ejpam-4877 173 6 grundy grundy PROPN ejpam-4877 173 7 total total NOUN ejpam-4877 173 8 hop hop NOUN ejpam-4877 173 9 dominating dominating NOUN ejpam-4877 173 10 sequence sequence NOUN ejpam-4877 173 11 of of ADP ejpam-4877 173 12 pn pn PROPN ejpam-4877 173 13 . . PROPN ejpam-4877 174 1 notice notice VERB ejpam-4877 174 2 that that SCONJ ejpam-4877 174 3 one one NUM ejpam-4877 174 4 of of ADP ejpam-4877 174 5 the the DET ejpam-4877 174 6 vertices vertex NOUN ejpam-4877 174 7 v1 v1 NOUN ejpam-4877 174 8 , , PUNCT ejpam-4877 174 9 v5 v5 NOUN ejpam-4877 174 10 , , PUNCT ejpam-4877 174 11 v9 v9 NOUN ejpam-4877 174 12 , , PUNCT ejpam-4877 174 13 . . PUNCT ejpam-4877 174 14 . . PUNCT ejpam-4877 175 1 . . PUNCT ejpam-4877 176 1 , , PUNCT ejpam-4877 176 2 vn−5 vn−5 PROPN ejpam-4877 176 3 , , PUNCT ejpam-4877 176 4 vn−1 vn−1 PROPN ejpam-4877 176 5 is be AUX ejpam-4877 176 6 not not PART ejpam-4877 176 7 in in ADP ejpam-4877 176 8 ŝ′. ŝ′. ADJ ejpam-4877 176 9 suppose suppose VERB ejpam-4877 176 10 all all DET ejpam-4877 176 11 the the DET ejpam-4877 176 12 vertices vertex NOUN ejpam-4877 176 13 v1 v1 NOUN ejpam-4877 176 14 , , PUNCT ejpam-4877 176 15 v5 v5 NOUN ejpam-4877 176 16 , , PUNCT ejpam-4877 176 17 v9 v9 NOUN ejpam-4877 176 18 , , PUNCT ejpam-4877 176 19 . . PUNCT ejpam-4877 176 20 . . PUNCT ejpam-4877 177 1 . . PUNCT ejpam-4877 178 1 , , PUNCT ejpam-4877 178 2 vn−5 vn−5 PROPN ejpam-4877 178 3 , , PUNCT ejpam-4877 178 4 vn−1 vn−1 PROPN ejpam-4877 178 5 are be AUX ejpam-4877 178 6 in in ADP ejpam-4877 178 7 ŝ. ŝ. NOUN ejpam-4877 178 8 if if SCONJ ejpam-4877 178 9 v5 v5 PROPN ejpam-4877 178 10 comes come VERB ejpam-4877 178 11 before before ADP ejpam-4877 178 12 v1 v1 NOUN ejpam-4877 178 13 , , PUNCT ejpam-4877 178 14 then then ADV ejpam-4877 178 15 n2 n2 PROPN ejpam-4877 178 16 pn pn PROPN ejpam-4877 178 17 ( ( PUNCT ejpam-4877 178 18 v1 v1 PROPN ejpam-4877 178 19 ) ) PUNCT ejpam-4877 178 20 ⊆ ⊆ NUM ejpam-4877 178 21 n2 n2 ADJ ejpam-4877 178 22 pn pn PROPN ejpam-4877 178 23 ( ( PUNCT ejpam-4877 178 24 v5 v5 PROPN ejpam-4877 178 25 ) ) PUNCT ejpam-4877 178 26 , , PUNCT ejpam-4877 178 27 a a DET ejpam-4877 178 28 contradiction contradiction NOUN ejpam-4877 178 29 . . PUNCT ejpam-4877 179 1 so so ADV ejpam-4877 179 2 , , PUNCT ejpam-4877 179 3 v5 v5 PROPN ejpam-4877 179 4 comes come VERB ejpam-4877 179 5 after after ADP ejpam-4877 179 6 v1 v1 NOUN ejpam-4877 179 7 . . PUNCT ejpam-4877 180 1 next next ADV ejpam-4877 180 2 , , PUNCT ejpam-4877 180 3 suppose suppose VERB ejpam-4877 180 4 v9 v9 PROPN ejpam-4877 180 5 comes come VERB ejpam-4877 180 6 before before ADP ejpam-4877 180 7 v5 v5 PROPN ejpam-4877 180 8 , , PUNCT ejpam-4877 180 9 then then ADV ejpam-4877 180 10 n2 n2 PROPN ejpam-4877 180 11 pn pn PROPN ejpam-4877 180 12 ( ( PUNCT ejpam-4877 180 13 v5 v5 PROPN ejpam-4877 180 14 ) ) PUNCT ejpam-4877 180 15 ⊆ ⊆ NUM ejpam-4877 180 16 n2 n2 ADJ ejpam-4877 180 17 pn pn PROPN ejpam-4877 180 18 ( ( PUNCT ejpam-4877 180 19 v1 v1 PROPN ejpam-4877 180 20 ) ) PUNCT ejpam-4877 180 21 ∪ ∪ NOUN ejpam-4877 180 22 n2 n2 PROPN ejpam-4877 180 23 pn pn PROPN ejpam-4877 180 24 ( ( PUNCT ejpam-4877 180 25 v9 v9 PROPN ejpam-4877 180 26 ) ) PUNCT ejpam-4877 180 27 , , PUNCT ejpam-4877 180 28 a a DET ejpam-4877 180 29 contradiction contradiction NOUN ejpam-4877 180 30 . . PUNCT ejpam-4877 181 1 thus thus ADV ejpam-4877 181 2 , , PUNCT ejpam-4877 181 3 v9 v9 PROPN ejpam-4877 181 4 comes come VERB ejpam-4877 181 5 after after ADP ejpam-4877 181 6 v5 v5 PROPN ejpam-4877 181 7 . . PUNCT ejpam-4877 182 1 continuing continue VERB ejpam-4877 182 2 in in ADP ejpam-4877 182 3 this this DET ejpam-4877 182 4 manner manner NOUN ejpam-4877 182 5 , , PUNCT ejpam-4877 182 6 we we PRON ejpam-4877 182 7 find find VERB ejpam-4877 182 8 that that SCONJ ejpam-4877 182 9 the the DET ejpam-4877 182 10 following follow VERB ejpam-4877 182 11 order order NOUN ejpam-4877 182 12 of of ADP ejpam-4877 182 13 appearance appearance NOUN ejpam-4877 182 14 ( ( PUNCT ejpam-4877 182 15 not not PART ejpam-4877 182 16 necessarily necessarily ADV ejpam-4877 182 17 consecutive consecutive ADJ ejpam-4877 182 18 ) ) PUNCT ejpam-4877 182 19 of of ADP ejpam-4877 182 20 the the DET ejpam-4877 182 21 given give VERB ejpam-4877 182 22 vertices vertex NOUN ejpam-4877 182 23 in in ADP ejpam-4877 182 24 the the DET ejpam-4877 182 25 grundy grundy PROPN ejpam-4877 182 26 total total NOUN ejpam-4877 182 27 hop hop NOUN ejpam-4877 182 28 dominating dominating NOUN ejpam-4877 182 29 sequence sequence NOUN ejpam-4877 182 30 s′ s′ NOUN ejpam-4877 182 31 : : PUNCT ejpam-4877 182 32 v1 v1 NOUN ejpam-4877 182 33 , , PUNCT ejpam-4877 182 34 v5 v5 NOUN ejpam-4877 182 35 , , PUNCT ejpam-4877 182 36 v9 v9 NOUN ejpam-4877 182 37 , , PUNCT ejpam-4877 182 38 . . PUNCT ejpam-4877 182 39 . . PUNCT ejpam-4877 183 1 . . PUNCT ejpam-4877 184 1 , , PUNCT ejpam-4877 184 2 vn−5 vn−5 PROPN ejpam-4877 184 3 , , PUNCT ejpam-4877 184 4 vn−1 vn−1 PROPN ejpam-4877 184 5 . . PUNCT ejpam-4877 185 1 however however ADV ejpam-4877 185 2 , , PUNCT ejpam-4877 185 3 n2 n2 PROPN ejpam-4877 185 4 pn pn PROPN ejpam-4877 185 5 ( ( PUNCT ejpam-4877 185 6 vn−1 vn−1 PROPN ejpam-4877 185 7 ) ) PUNCT ejpam-4877 185 8 ⊆ ⊆ NUM ejpam-4877 185 9 npn(vn−5 npn(vn−5 PROPN ejpam-4877 185 10 ) ) PUNCT ejpam-4877 185 11 . . PUNCT ejpam-4877 186 1 hence hence ADV ejpam-4877 186 2 , , PUNCT ejpam-4877 186 3 vn−1 vn−1 PROPN ejpam-4877 186 4 /∈ /∈ PUNCT ejpam-4877 186 5 ŝ′ ŝ′ PROPN ejpam-4877 186 6 , , PUNCT ejpam-4877 186 7 a a DET ejpam-4877 186 8 contradiction contradiction NOUN ejpam-4877 186 9 . . PUNCT ejpam-4877 187 1 similarly similarly ADV ejpam-4877 187 2 , , PUNCT ejpam-4877 187 3 one one NUM ejpam-4877 187 4 of of ADP ejpam-4877 187 5 the the DET ejpam-4877 187 6 vertices vertex NOUN ejpam-4877 187 7 v2 v2 PROPN ejpam-4877 187 8 , , PUNCT ejpam-4877 187 9 v6 v6 NOUN ejpam-4877 187 10 , , PUNCT ejpam-4877 187 11 v10 v10 PROPN ejpam-4877 187 12 , , PUNCT ejpam-4877 187 13 . . PUNCT ejpam-4877 187 14 . . PUNCT ejpam-4877 188 1 . . PUNCT ejpam-4877 189 1 , , PUNCT ejpam-4877 189 2 vn−4 vn−4 NOUN ejpam-4877 189 3 , , PUNCT ejpam-4877 189 4 vn vn PROPN ejpam-4877 189 5 is be AUX ejpam-4877 189 6 not not PART ejpam-4877 189 7 in in ADP ejpam-4877 189 8 ŝ′. ŝ′. ADJ ejpam-4877 189 9 therefore therefore ADV ejpam-4877 189 10 , , PUNCT ejpam-4877 189 11 γthgr(pn γthgr(pn ADJ ejpam-4877 189 12 ) ) PUNCT ejpam-4877 189 13 = = SYM ejpam-4877 190 1 k k X ejpam-4877 190 2 ≤ ≤ ADJ ejpam-4877 190 3 n− n− NOUN ejpam-4877 190 4 2 2 NUM ejpam-4877 190 5 . . PUNCT ejpam-4877 190 6 consequently consequently ADV ejpam-4877 190 7 , , PUNCT ejpam-4877 190 8 γthgr(pn γthgr(pn ADJ ejpam-4877 190 9 ) ) PUNCT ejpam-4877 190 10 = = PUNCT ejpam-4877 190 11 n− n− NOUN ejpam-4877 190 12 2 2 NUM ejpam-4877 190 13 for for ADP ejpam-4877 190 14 all all DET ejpam-4877 190 15 n n NOUN ejpam-4877 190 16 = = SYM ejpam-4877 190 17 4m+ 4m+ NUM ejpam-4877 190 18 2 2 NUM ejpam-4877 190 19 . . PUNCT ejpam-4877 191 1 next next ADV ejpam-4877 191 2 , , PUNCT ejpam-4877 191 3 let let VERB ejpam-4877 191 4 n n PRON ejpam-4877 191 5 ≥ ≥ X ejpam-4877 191 6 5 5 NUM ejpam-4877 191 7 and and CCONJ ejpam-4877 191 8 odd odd ADJ ejpam-4877 191 9 . . PUNCT ejpam-4877 192 1 clearly clearly ADV ejpam-4877 192 2 , , PUNCT ejpam-4877 192 3 γthgr(p5 γthgr(p5 ADJ ejpam-4877 192 4 ) ) PUNCT ejpam-4877 192 5 = = SYM ejpam-4877 192 6 4 4 X ejpam-4877 192 7 . . PUNCT ejpam-4877 192 8 suppose suppose VERB ejpam-4877 192 9 n n PRON ejpam-4877 192 10 ≥ ≥ X ejpam-4877 192 11 7 7 NUM ejpam-4877 192 12 and and CCONJ ejpam-4877 192 13 odd odd ADJ ejpam-4877 192 14 . . PUNCT ejpam-4877 193 1 for for ADP ejpam-4877 193 2 n n PRON ejpam-4877 193 3 ∈ ∈ PROPN ejpam-4877 193 4 { { PUNCT ejpam-4877 193 5 7 7 NUM ejpam-4877 193 6 , , PUNCT ejpam-4877 193 7 11 11 NUM ejpam-4877 193 8 , , PUNCT ejpam-4877 193 9 15 15 NUM ejpam-4877 193 10 , , PUNCT ejpam-4877 193 11 . . PUNCT ejpam-4877 193 12 . . PUNCT ejpam-4877 193 13 . . PUNCT ejpam-4877 194 1 } } PUNCT ejpam-4877 194 2 , , PUNCT ejpam-4877 194 3 let let VERB ejpam-4877 194 4 s1 s1 PROPN ejpam-4877 194 5 = = SYM ejpam-4877 194 6 ( ( PUNCT ejpam-4877 194 7 v1 v1 PROPN ejpam-4877 194 8 , , PUNCT ejpam-4877 194 9 v2 v2 PROPN ejpam-4877 194 10 , , PUNCT ejpam-4877 194 11 v5 v5 PROPN ejpam-4877 194 12 , , PUNCT ejpam-4877 194 13 v6 v6 NOUN ejpam-4877 194 14 , , PUNCT ejpam-4877 194 15 · · PUNCT ejpam-4877 194 16 · · PUNCT ejpam-4877 194 17 · · PUNCT ejpam-4877 194 18 , , PUNCT ejpam-4877 194 19 vn−6 vn−6 PROPN ejpam-4877 194 20 , , PUNCT ejpam-4877 194 21 vn−5 vn−5 PROPN ejpam-4877 194 22 , , PUNCT ejpam-4877 194 23 vn−2 vn−2 PROPN ejpam-4877 194 24 , , PUNCT ejpam-4877 194 25 vn vn PROPN ejpam-4877 194 26 , , PUNCT ejpam-4877 194 27 vn−3 vn−3 PROPN ejpam-4877 194 28 , , PUNCT ejpam-4877 194 29 vn−4 vn−4 NOUN ejpam-4877 194 30 , , PUNCT ejpam-4877 194 31 vn−7 vn−7 PROPN ejpam-4877 194 32 , , PUNCT ejpam-4877 194 33 vn−8 vn−8 PROPN ejpam-4877 194 34 , , PUNCT ejpam-4877 194 35 · · PUNCT ejpam-4877 194 36 · · PUNCT ejpam-4877 194 37 · · PUNCT ejpam-4877 194 38 , , PUNCT ejpam-4877 194 39 v4 v4 PROPN ejpam-4877 194 40 , , PUNCT ejpam-4877 194 41 v3 v3 PROPN ejpam-4877 194 42 ) ) PUNCT ejpam-4877 194 43 . . PUNCT ejpam-4877 195 1 then then ADV ejpam-4877 195 2 s1 s1 PROPN ejpam-4877 195 3 is be AUX ejpam-4877 195 4 a a DET ejpam-4877 195 5 grundy grundy PROPN ejpam-4877 195 6 total total NOUN ejpam-4877 195 7 hop hop NOUN ejpam-4877 195 8 dominating dominating NOUN ejpam-4877 195 9 sequence sequence NOUN ejpam-4877 195 10 of of ADP ejpam-4877 195 11 pn pn PROPN ejpam-4877 195 12 . . PROPN ejpam-4877 196 1 hence hence ADV ejpam-4877 196 2 , , PUNCT ejpam-4877 196 3 γthgr(pn γthgr(pn ADJ ejpam-4877 196 4 ) ) PUNCT ejpam-4877 196 5 ≥ ≥ NOUN ejpam-4877 196 6 n− n− NOUN ejpam-4877 196 7 1 1 NUM ejpam-4877 196 8 . . PUNCT ejpam-4877 197 1 next next ADV ejpam-4877 197 2 , , PUNCT ejpam-4877 197 3 for for ADP ejpam-4877 197 4 n n PRON ejpam-4877 197 5 ∈ ∈ NOUN ejpam-4877 197 6 { { PUNCT ejpam-4877 197 7 9 9 NUM ejpam-4877 197 8 , , PUNCT ejpam-4877 197 9 13 13 NUM ejpam-4877 197 10 , , PUNCT ejpam-4877 197 11 17 17 NUM ejpam-4877 197 12 , , PUNCT ejpam-4877 197 13 . . PUNCT ejpam-4877 197 14 . . PUNCT ejpam-4877 197 15 . . PUNCT ejpam-4877 198 1 } } PUNCT ejpam-4877 198 2 , , PUNCT ejpam-4877 198 3 let let VERB ejpam-4877 198 4 s2 s2 VERB ejpam-4877 198 5 = = SYM ejpam-4877 198 6 ( ( PUNCT ejpam-4877 198 7 v1 v1 PROPN ejpam-4877 198 8 , , PUNCT ejpam-4877 198 9 v2 v2 PROPN ejpam-4877 198 10 , , PUNCT ejpam-4877 198 11 v5 v5 PROPN ejpam-4877 198 12 , , PUNCT ejpam-4877 198 13 v6 v6 NOUN ejpam-4877 198 14 , , PUNCT ejpam-4877 198 15 · · PUNCT ejpam-4877 198 16 · · PUNCT ejpam-4877 198 17 · · PUNCT ejpam-4877 198 18 , , PUNCT ejpam-4877 198 19 vn−4 vn−4 NOUN ejpam-4877 198 20 , , PUNCT ejpam-4877 198 21 vn−3 vn−3 PROPN ejpam-4877 198 22 , , PUNCT ejpam-4877 198 23 vn−2 vn−2 PROPN ejpam-4877 198 24 , , PUNCT ejpam-4877 198 25 vn−1 vn−1 PROPN ejpam-4877 198 26 , , PUNCT ejpam-4877 198 27 vn−6 vn−6 PROPN ejpam-4877 198 28 , , PUNCT ejpam-4877 198 29 vn−5 vn−5 PROPN ejpam-4877 198 30 , , PUNCT ejpam-4877 198 31 · · PUNCT ejpam-4877 198 32 · · PUNCT ejpam-4877 198 33 · · PUNCT ejpam-4877 198 34 , , PUNCT ejpam-4877 198 35 v3 v3 PROPN ejpam-4877 198 36 , , PUNCT ejpam-4877 198 37 v4 v4 PROPN ejpam-4877 198 38 ) ) PUNCT ejpam-4877 198 39 . . PUNCT ejpam-4877 199 1 observe observe VERB ejpam-4877 199 2 that that SCONJ ejpam-4877 199 3 s2 s2 PROPN ejpam-4877 199 4 is be AUX ejpam-4877 199 5 a a DET ejpam-4877 199 6 grundy grundy PROPN ejpam-4877 199 7 total total NOUN ejpam-4877 199 8 hop hop NOUN ejpam-4877 199 9 dominating dominating NOUN ejpam-4877 199 10 sequence sequence NOUN ejpam-4877 199 11 of of ADP ejpam-4877 199 12 pn pn PROPN ejpam-4877 199 13 . . PROPN ejpam-4877 200 1 hence hence ADV ejpam-4877 200 2 , , PUNCT ejpam-4877 200 3 γthgr(pn γthgr(pn ADJ ejpam-4877 200 4 ) ) PUNCT ejpam-4877 200 5 ≥ ≥ NOUN ejpam-4877 200 6 n− n− NOUN ejpam-4877 200 7 1 1 NUM ejpam-4877 200 8 . . PUNCT ejpam-4877 200 9 suppose suppose VERB ejpam-4877 200 10 γthgr(pn γthgr(pn ADJ ejpam-4877 200 11 ) ) PUNCT ejpam-4877 200 12 = = SYM ejpam-4877 200 13 n n CCONJ ejpam-4877 200 14 , , PUNCT ejpam-4877 200 15 say say VERB ejpam-4877 200 16 s0 s0 PROPN ejpam-4877 200 17 = = SYM ejpam-4877 200 18 ( ( PUNCT ejpam-4877 200 19 w1 w1 NOUN ejpam-4877 200 20 , , PUNCT ejpam-4877 200 21 w2 w2 NOUN ejpam-4877 200 22 , , PUNCT ejpam-4877 200 23 · · PUNCT ejpam-4877 200 24 · · PUNCT ejpam-4877 200 25 · · PUNCT ejpam-4877 200 26 , , PUNCT ejpam-4877 200 27 wn wn PROPN ejpam-4877 200 28 ) ) PUNCT ejpam-4877 200 29 is be AUX ejpam-4877 200 30 a a DET ejpam-4877 200 31 grundy grundy PROPN ejpam-4877 200 32 total total NOUN ejpam-4877 200 33 hop hop NOUN ejpam-4877 200 34 dominating dominating NOUN ejpam-4877 200 35 sequence sequence NOUN ejpam-4877 200 36 of of ADP ejpam-4877 200 37 pn pn PROPN ejpam-4877 200 38 . . PROPN ejpam-4877 200 39 observe observe VERB ejpam-4877 200 40 that that SCONJ ejpam-4877 200 41 for for ADP ejpam-4877 200 42 n n PRON ejpam-4877 200 43 ∈ ∈ NOUN ejpam-4877 200 44 { { PUNCT ejpam-4877 200 45 7 7 NUM ejpam-4877 200 46 , , PUNCT ejpam-4877 200 47 11 11 NUM ejpam-4877 200 48 , , PUNCT ejpam-4877 200 49 15 15 NUM ejpam-4877 200 50 , , PUNCT ejpam-4877 200 51 . . PUNCT ejpam-4877 200 52 . . PUNCT ejpam-4877 201 1 . . PUNCT ejpam-4877 202 1 } } PUNCT ejpam-4877 202 2 , , PUNCT ejpam-4877 202 3 one one NUM ejpam-4877 202 4 of of ADP ejpam-4877 202 5 the the DET ejpam-4877 202 6 vertices vertex NOUN ejpam-4877 202 7 v2 v2 PROPN ejpam-4877 202 8 , , PUNCT ejpam-4877 202 9 v6 v6 NOUN ejpam-4877 202 10 , , PUNCT ejpam-4877 202 11 v10 v10 PROPN ejpam-4877 202 12 , , PUNCT ejpam-4877 202 13 . . PUNCT ejpam-4877 202 14 . . PUNCT ejpam-4877 202 15 . . PUNCT ejpam-4877 203 1 , , PUNCT ejpam-4877 203 2 vn−4 vn−4 NOUN ejpam-4877 203 3 , , PUNCT ejpam-4877 203 4 vn vn PROPN ejpam-4877 203 5 is be AUX ejpam-4877 203 6 not not PART ejpam-4877 203 7 in in ADP ejpam-4877 203 8 ŝ0 ŝ0 PROPN ejpam-4877 203 9 . . PUNCT ejpam-4877 204 1 suppose suppose VERB ejpam-4877 204 2 all all DET ejpam-4877 204 3 vertices vertice VERB ejpam-4877 204 4 v2 v2 PROPN ejpam-4877 204 5 , , PUNCT ejpam-4877 204 6 v6 v6 NOUN ejpam-4877 204 7 , , PUNCT ejpam-4877 204 8 v10 v10 PROPN ejpam-4877 204 9 , , PUNCT ejpam-4877 204 10 . . PUNCT ejpam-4877 204 11 . . PUNCT ejpam-4877 205 1 . . PUNCT ejpam-4877 206 1 , , PUNCT ejpam-4877 206 2 vn−4 vn−4 NOUN ejpam-4877 206 3 , , PUNCT ejpam-4877 206 4 vn vn PROPN ejpam-4877 206 5 are be AUX ejpam-4877 206 6 in in ADP ejpam-4877 206 7 ŝ0 ŝ0 PROPN ejpam-4877 206 8 . . PUNCT ejpam-4877 207 1 if if SCONJ ejpam-4877 207 2 v6 v6 PROPN ejpam-4877 207 3 comes come VERB ejpam-4877 207 4 before before ADP ejpam-4877 207 5 v2 v2 PROPN ejpam-4877 207 6 , , PUNCT ejpam-4877 207 7 then then ADV ejpam-4877 207 8 n2 n2 PROPN ejpam-4877 207 9 pn pn PROPN ejpam-4877 207 10 ( ( PUNCT ejpam-4877 207 11 v2 v2 PROPN ejpam-4877 207 12 ) ) PUNCT ejpam-4877 207 13 ⊆ ⊆ NUM ejpam-4877 207 14 n2 n2 PROPN ejpam-4877 207 15 cn cn PROPN ejpam-4877 207 16 ( ( PUNCT ejpam-4877 207 17 v6 v6 PROPN ejpam-4877 207 18 ) ) PUNCT ejpam-4877 207 19 , , PUNCT ejpam-4877 207 20 a a DET ejpam-4877 207 21 contradiction contradiction NOUN ejpam-4877 207 22 . . PUNCT ejpam-4877 208 1 so so ADV ejpam-4877 208 2 , , PUNCT ejpam-4877 208 3 v6 v6 PROPN ejpam-4877 208 4 comes come VERB ejpam-4877 208 5 after after ADP ejpam-4877 208 6 v2 v2 PROPN ejpam-4877 208 7 . . PUNCT ejpam-4877 209 1 next next ADV ejpam-4877 209 2 , , PUNCT ejpam-4877 209 3 suppose suppose VERB ejpam-4877 209 4 v10 v10 NOUN ejpam-4877 209 5 comes come VERB ejpam-4877 209 6 before before ADP ejpam-4877 209 7 v6 v6 PROPN ejpam-4877 209 8 , , PUNCT ejpam-4877 209 9 then then ADV ejpam-4877 209 10 n2 n2 PROPN ejpam-4877 209 11 pn pn PROPN ejpam-4877 209 12 ( ( PUNCT ejpam-4877 209 13 v6 v6 PROPN ejpam-4877 209 14 ) ) PUNCT ejpam-4877 209 15 ⊆ ⊆ NUM ejpam-4877 209 16 n2 n2 NOUN ejpam-4877 209 17 pn pn PROPN ejpam-4877 209 18 ( ( PUNCT ejpam-4877 209 19 v2 v2 PROPN ejpam-4877 209 20 ) ) PUNCT ejpam-4877 209 21 ∪n2 ∪n2 PROPN ejpam-4877 209 22 pn pn PROPN ejpam-4877 209 23 ( ( PUNCT ejpam-4877 209 24 v10 v10 PROPN ejpam-4877 209 25 ) ) PUNCT ejpam-4877 209 26 , , PUNCT ejpam-4877 209 27 a a DET ejpam-4877 209 28 contradiction contradiction NOUN ejpam-4877 209 29 . . PUNCT ejpam-4877 210 1 thus thus ADV ejpam-4877 210 2 , , PUNCT ejpam-4877 210 3 v10 v10 PROPN ejpam-4877 210 4 comes come VERB ejpam-4877 210 5 after after ADP ejpam-4877 210 6 v6 v6 NOUN ejpam-4877 210 7 . . PUNCT ejpam-4877 211 1 continuing continue VERB ejpam-4877 211 2 in in ADP ejpam-4877 211 3 this this DET ejpam-4877 211 4 manner manner NOUN ejpam-4877 211 5 , , PUNCT ejpam-4877 211 6 we we PRON ejpam-4877 211 7 find find VERB ejpam-4877 211 8 that that SCONJ ejpam-4877 211 9 the the DET ejpam-4877 211 10 following follow VERB ejpam-4877 211 11 order order NOUN ejpam-4877 211 12 of of ADP ejpam-4877 211 13 appearance appearance NOUN ejpam-4877 211 14 ( ( PUNCT ejpam-4877 211 15 not not PART ejpam-4877 211 16 necessarily necessarily ADV ejpam-4877 211 17 consecutive consecutive ADJ ejpam-4877 211 18 ) ) PUNCT ejpam-4877 211 19 of of ADP ejpam-4877 211 20 the the DET ejpam-4877 211 21 given give VERB ejpam-4877 211 22 vertices vertex NOUN ejpam-4877 211 23 in in ADP ejpam-4877 211 24 the the DET ejpam-4877 211 25 grundy grundy PROPN ejpam-4877 211 26 total total NOUN ejpam-4877 211 27 hop hop NOUN ejpam-4877 211 28 dominating dominating NOUN ejpam-4877 211 29 sequence sequence NOUN ejpam-4877 211 30 s0 s0 NOUN ejpam-4877 211 31 : : PUNCT ejpam-4877 211 32 v2 v2 PROPN ejpam-4877 211 33 , , PUNCT ejpam-4877 211 34 v6 v6 NOUN ejpam-4877 211 35 , , PUNCT ejpam-4877 211 36 v10 v10 PROPN ejpam-4877 211 37 , , PUNCT ejpam-4877 211 38 . . PUNCT ejpam-4877 211 39 . . PUNCT ejpam-4877 212 1 . . PUNCT ejpam-4877 213 1 , , PUNCT ejpam-4877 213 2 vn−4 vn−4 NOUN ejpam-4877 213 3 , , PUNCT ejpam-4877 213 4 vn vn NOUN ejpam-4877 213 5 . . PUNCT ejpam-4877 214 1 however however ADV ejpam-4877 214 2 , , PUNCT ejpam-4877 214 3 n n PROPN ejpam-4877 214 4 2 2 NUM ejpam-4877 214 5 pn pn PROPN ejpam-4877 214 6 ( ( PUNCT ejpam-4877 214 7 vn vn PROPN ejpam-4877 214 8 ) ) PUNCT ejpam-4877 214 9 ⊆ ⊆ NUM ejpam-4877 214 10 npn(vn−4 npn(vn−4 NOUN ejpam-4877 214 11 ) ) PUNCT ejpam-4877 214 12 . . PUNCT ejpam-4877 215 1 hence hence ADV ejpam-4877 215 2 , , PUNCT ejpam-4877 215 3 vn vn PROPN ejpam-4877 215 4 /∈ /∈ PUNCT ejpam-4877 216 1 ŝ0 ŝ0 PROPN ejpam-4877 216 2 , , PUNCT ejpam-4877 216 3 a a DET ejpam-4877 216 4 contradiction contradiction NOUN ejpam-4877 216 5 . . PUNCT ejpam-4877 217 1 similarly similarly ADV ejpam-4877 217 2 , , PUNCT ejpam-4877 217 3 for for ADP ejpam-4877 217 4 n n PRON ejpam-4877 217 5 ∈ ∈ NOUN ejpam-4877 217 6 { { PUNCT ejpam-4877 217 7 9 9 NUM ejpam-4877 217 8 , , PUNCT ejpam-4877 217 9 13 13 NUM ejpam-4877 217 10 , , PUNCT ejpam-4877 217 11 17 17 NUM ejpam-4877 217 12 , , PUNCT ejpam-4877 217 13 . . PUNCT ejpam-4877 217 14 . . PUNCT ejpam-4877 217 15 . . PUNCT ejpam-4877 218 1 } } PUNCT ejpam-4877 218 2 , , PUNCT ejpam-4877 218 3 one one NUM ejpam-4877 218 4 of of ADP ejpam-4877 218 5 the the DET ejpam-4877 218 6 vertices vertex NOUN ejpam-4877 218 7 v1 v1 NOUN ejpam-4877 218 8 , , PUNCT ejpam-4877 218 9 v5 v5 NOUN ejpam-4877 218 10 , , PUNCT ejpam-4877 218 11 v9 v9 NOUN ejpam-4877 218 12 , , PUNCT ejpam-4877 218 13 . . PUNCT ejpam-4877 218 14 . . PUNCT ejpam-4877 219 1 . . PUNCT ejpam-4877 220 1 , , PUNCT ejpam-4877 220 2 vn−5 vn−5 PROPN ejpam-4877 220 3 , , PUNCT ejpam-4877 220 4 vn−1 vn−1 PROPN ejpam-4877 220 5 is be AUX ejpam-4877 220 6 not not PART ejpam-4877 220 7 in in ADP ejpam-4877 220 8 the the DET ejpam-4877 220 9 grundy grundy PROPN ejpam-4877 220 10 total total NOUN ejpam-4877 220 11 hop hop NOUN ejpam-4877 220 12 dominating dominating NOUN ejpam-4877 220 13 sequence sequence NOUN ejpam-4877 220 14 , , PUNCT ejpam-4877 220 15 say say VERB ejpam-4877 220 16 s′′. s′′. NOUN ejpam-4877 220 17 thus thus ADV ejpam-4877 220 18 , , PUNCT ejpam-4877 220 19 γthgr(pn γthgr(pn ADJ ejpam-4877 220 20 ) ) PUNCT ejpam-4877 220 21 ≤ ≤ NOUN ejpam-4877 220 22 n n CCONJ ejpam-4877 220 23 − − PROPN ejpam-4877 220 24 1 1 NUM ejpam-4877 220 25 . . PUNCT ejpam-4877 221 1 consequently consequently ADV ejpam-4877 221 2 , , PUNCT ejpam-4877 221 3 γthgr(pn γthgr(pn ADJ ejpam-4877 221 4 ) ) PUNCT ejpam-4877 221 5 = = PUNCT ejpam-4877 221 6 n− n− NOUN ejpam-4877 221 7 1 1 NUM ejpam-4877 221 8 for for ADP ejpam-4877 221 9 all all DET ejpam-4877 221 10 n n PRON ejpam-4877 221 11 ≥ ≥ NOUN ejpam-4877 221 12 5 5 NUM ejpam-4877 221 13 and and CCONJ ejpam-4877 221 14 odd odd ADJ ejpam-4877 221 15 . . PUNCT ejpam-4877 222 1 lastly lastly ADV ejpam-4877 222 2 , , PUNCT ejpam-4877 222 3 assume assume VERB ejpam-4877 222 4 that that SCONJ ejpam-4877 222 5 n n NOUN ejpam-4877 222 6 = = SYM ejpam-4877 222 7 4 4 NUM ejpam-4877 222 8 m. m. NOUN ejpam-4877 222 9 clearly clearly ADV ejpam-4877 222 10 , , PUNCT ejpam-4877 222 11 γthgr(p4 γthgr(p4 NOUN ejpam-4877 222 12 ) ) PUNCT ejpam-4877 222 13 = = PUNCT ejpam-4877 223 1 4 4 X ejpam-4877 223 2 . . X ejpam-4877 223 3 for for ADP ejpam-4877 223 4 n n NOUN ejpam-4877 223 5 = = SYM ejpam-4877 223 6 4 4 NUM ejpam-4877 223 7 m m NOUN ejpam-4877 223 8 ≥ ≥ NOUN ejpam-4877 223 9 8 8 NUM ejpam-4877 223 10 , , PUNCT ejpam-4877 223 11 let let VERB ejpam-4877 223 12 c c NOUN ejpam-4877 223 13 = = SYM ejpam-4877 223 14 ( ( PUNCT ejpam-4877 223 15 v1 v1 PROPN ejpam-4877 223 16 , , PUNCT ejpam-4877 223 17 v2 v2 PROPN ejpam-4877 223 18 , , PUNCT ejpam-4877 223 19 v5 v5 PROPN ejpam-4877 223 20 , , PUNCT ejpam-4877 223 21 v6 v6 NOUN ejpam-4877 223 22 , , PUNCT ejpam-4877 223 23 · · PUNCT ejpam-4877 223 24 · · PUNCT ejpam-4877 223 25 · · PUNCT ejpam-4877 223 26 , , PUNCT ejpam-4877 223 27 vn−3 vn−3 PROPN ejpam-4877 223 28 , , PUNCT ejpam-4877 223 29 vn−2 vn−2 PROPN ejpam-4877 223 30 , , PUNCT ejpam-4877 223 31 vn−1 vn−1 PROPN ejpam-4877 223 32 , , PUNCT ejpam-4877 223 33 vn vn NOUN ejpam-4877 223 34 , , PUNCT ejpam-4877 223 35 vn−5 vn−5 PROPN ejpam-4877 223 36 , , PUNCT ejpam-4877 223 37 vn−4 vn−4 NOUN ejpam-4877 223 38 , , PUNCT ejpam-4877 223 39 · · PUNCT ejpam-4877 223 40 · · PUNCT ejpam-4877 223 41 · · PUNCT ejpam-4877 223 42 , , PUNCT ejpam-4877 223 43 v3 v3 PROPN ejpam-4877 223 44 , , PUNCT ejpam-4877 223 45 v4 v4 PROPN ejpam-4877 223 46 ) ) PUNCT ejpam-4877 223 47 . . PUNCT ejpam-4877 224 1 then then ADV ejpam-4877 224 2 c c PROPN ejpam-4877 224 3 is be AUX ejpam-4877 224 4 a a DET ejpam-4877 224 5 grundy grundy PROPN ejpam-4877 224 6 total total NOUN ejpam-4877 224 7 hop hop NOUN ejpam-4877 224 8 dominating dominating NOUN ejpam-4877 224 9 sequence sequence NOUN ejpam-4877 224 10 of of ADP ejpam-4877 224 11 pn pn PROPN ejpam-4877 224 12 . . PUNCT ejpam-4877 225 1 thus thus ADV ejpam-4877 225 2 , , PUNCT ejpam-4877 225 3 γthgr(pn γthgr(pn ADJ ejpam-4877 225 4 ) ) PUNCT ejpam-4877 225 5 = = SYM ejpam-4877 225 6 n n PROPN ejpam-4877 225 7 for for ADP ejpam-4877 225 8 all all DET ejpam-4877 225 9 n n PRON ejpam-4877 225 10 = = SYM ejpam-4877 225 11 4 4 NUM ejpam-4877 225 12 m. m. NOUN ejpam-4877 225 13 j.a j.a PROPN ejpam-4877 225 14 . . PROPN ejpam-4877 226 1 hassan hassan PROPN ejpam-4877 226 2 , , PUNCT ejpam-4877 226 3 s. s. PROPN ejpam-4877 226 4 canoy canoy PROPN ejpam-4877 226 5 / / SYM ejpam-4877 226 6 eur eur PROPN ejpam-4877 226 7 . . PUNCT ejpam-4877 227 1 j. j. PROPN ejpam-4877 227 2 pure pure PROPN ejpam-4877 227 3 appl appl PROPN ejpam-4877 227 4 . . PROPN ejpam-4877 227 5 math math PROPN ejpam-4877 227 6 , , PUNCT ejpam-4877 227 7 16 16 NUM ejpam-4877 227 8 ( ( PUNCT ejpam-4877 227 9 4 4 NUM ejpam-4877 227 10 ) ) PUNCT ejpam-4877 227 11 ( ( PUNCT ejpam-4877 227 12 2023 2023 NUM ejpam-4877 227 13 ) ) PUNCT ejpam-4877 227 14 , , PUNCT ejpam-4877 227 15 2597 2597 NUM ejpam-4877 227 16 - - SYM ejpam-4877 227 17 2612 2612 NUM ejpam-4877 227 18 2603 2603 NUM ejpam-4877 227 19 proposition proposition NOUN ejpam-4877 227 20 2 2 NUM ejpam-4877 227 21 . . PUNCT ejpam-4877 228 1 let let VERB ejpam-4877 228 2 g g PRON ejpam-4877 228 3 be be AUX ejpam-4877 228 4 a a DET ejpam-4877 228 5 graph graph NOUN ejpam-4877 228 6 of of ADP ejpam-4877 228 7 order order NOUN ejpam-4877 228 8 n n PRON ejpam-4877 228 9 with with ADP ejpam-4877 228 10 γ(c γ(c PROPN ejpam-4877 228 11 ) ) PUNCT ejpam-4877 228 12 ̸= ̸= PROPN ejpam-4877 228 13 1 1 NUM ejpam-4877 228 14 for for ADP ejpam-4877 228 15 each each DET ejpam-4877 228 16 component component NOUN ejpam-4877 228 17 c c PROPN ejpam-4877 228 18 of of ADP ejpam-4877 228 19 g. g. PROPN ejpam-4877 228 20 if if SCONJ ejpam-4877 228 21 |n2 |n2 PROPN ejpam-4877 228 22 g(v)| g(v)| VERB ejpam-4877 228 23 ≥ ≥ NOUN ejpam-4877 228 24 m m VERB ejpam-4877 228 25 for for ADP ejpam-4877 228 26 every every DET ejpam-4877 228 27 v v NUM ejpam-4877 228 28 ∈ ∈ PROPN ejpam-4877 228 29 v v NOUN ejpam-4877 228 30 ( ( PUNCT ejpam-4877 228 31 g g NOUN ejpam-4877 228 32 ) ) PUNCT ejpam-4877 228 33 , , PUNCT ejpam-4877 228 34 then then ADV ejpam-4877 228 35 γthgr(g γthgr(g NOUN ejpam-4877 228 36 ) ) PUNCT ejpam-4877 228 37 ≤ ≤ NUM ejpam-4877 228 38 n− n− NOUN ejpam-4877 228 39 ( ( PUNCT ejpam-4877 228 40 m− m− PROPN ejpam-4877 228 41 1 1 NUM ejpam-4877 228 42 ) ) PUNCT ejpam-4877 228 43 . . PUNCT ejpam-4877 229 1 proof proof NOUN ejpam-4877 229 2 . . PUNCT ejpam-4877 230 1 suppose suppose VERB ejpam-4877 230 2 γthgr(g γthgr(g NOUN ejpam-4877 230 3 ) ) PUNCT ejpam-4877 230 4 = = SYM ejpam-4877 231 1 k k NOUN ejpam-4877 231 2 , , PUNCT ejpam-4877 231 3 say say VERB ejpam-4877 231 4 s s X ejpam-4877 231 5 = = PUNCT ejpam-4877 231 6 ( ( PUNCT ejpam-4877 231 7 w1 w1 NOUN ejpam-4877 231 8 , , PUNCT ejpam-4877 231 9 w2 w2 NOUN ejpam-4877 231 10 , , PUNCT ejpam-4877 231 11 · · PUNCT ejpam-4877 231 12 · · PUNCT ejpam-4877 231 13 · · PUNCT ejpam-4877 231 14 , , PUNCT ejpam-4877 231 15 wk wk X ejpam-4877 231 16 ) ) PUNCT ejpam-4877 231 17 is be AUX ejpam-4877 231 18 a a DET ejpam-4877 231 19 grundy grundy PROPN ejpam-4877 231 20 total total NOUN ejpam-4877 231 21 hop hop NOUN ejpam-4877 231 22 dominating dominating NOUN ejpam-4877 231 23 sequence sequence NOUN ejpam-4877 231 24 of of ADP ejpam-4877 231 25 g. g. PROPN ejpam-4877 231 26 assume assume VERB ejpam-4877 231 27 w1 w1 PROPN ejpam-4877 231 28 = = SYM ejpam-4877 231 29 vi vi PROPN ejpam-4877 231 30 for for ADP ejpam-4877 231 31 some some DET ejpam-4877 231 32 i i PRON ejpam-4877 231 33 ∈ ∈ PROPN ejpam-4877 231 34 { { PUNCT ejpam-4877 231 35 1 1 NUM ejpam-4877 231 36 , , PUNCT ejpam-4877 231 37 . . PUNCT ejpam-4877 231 38 . . PUNCT ejpam-4877 231 39 . . PUNCT ejpam-4877 231 40 , , PUNCT ejpam-4877 231 41 n n CCONJ ejpam-4877 231 42 } } PUNCT ejpam-4877 231 43 . . PUNCT ejpam-4877 232 1 then then ADV ejpam-4877 232 2 |n2 |n2 PROPN ejpam-4877 232 3 g(w1)| g(w1)| PROPN ejpam-4877 232 4 = = PUNCT ejpam-4877 232 5 |n2 |n2 PROPN ejpam-4877 232 6 g(vi)| g(vi)| X ejpam-4877 232 7 ≥ ≥ PROPN ejpam-4877 232 8 m m VERB ejpam-4877 232 9 for for ADP ejpam-4877 232 10 some some DET ejpam-4877 232 11 i i PRON ejpam-4877 232 12 ∈ ∈ PROPN ejpam-4877 232 13 { { PUNCT ejpam-4877 232 14 1 1 NUM ejpam-4877 232 15 , , PUNCT ejpam-4877 232 16 . . PUNCT ejpam-4877 232 17 . . PUNCT ejpam-4877 233 1 . . PUNCT ejpam-4877 233 2 , , PUNCT ejpam-4877 234 1 n n CCONJ ejpam-4877 234 2 } } PUNCT ejpam-4877 234 3 . . PUNCT ejpam-4877 235 1 it it PRON ejpam-4877 235 2 follows follow VERB ejpam-4877 235 3 that that SCONJ ejpam-4877 235 4 there there PRON ejpam-4877 235 5 are be VERB ejpam-4877 235 6 at at ADP ejpam-4877 235 7 most most ADJ ejpam-4877 235 8 n n PRON ejpam-4877 235 9 −m −m ADJ ejpam-4877 235 10 remaining remain VERB ejpam-4877 235 11 vertices vertex NOUN ejpam-4877 235 12 of of ADP ejpam-4877 235 13 g g NOUN ejpam-4877 235 14 that that PRON ejpam-4877 235 15 could could AUX ejpam-4877 235 16 be be AUX ejpam-4877 235 17 hop hop NOUN ejpam-4877 235 18 footprinted footprinte VERB ejpam-4877 235 19 by by ADP ejpam-4877 235 20 the the DET ejpam-4877 235 21 next next ADJ ejpam-4877 235 22 terms term NOUN ejpam-4877 235 23 of of ADP ejpam-4877 235 24 s. s. PROPN ejpam-4877 235 25 therefore therefore ADV ejpam-4877 235 26 , , PUNCT ejpam-4877 235 27 γthgr(g γthgr(g NOUN ejpam-4877 235 28 ) ) PUNCT ejpam-4877 235 29 = = SYM ejpam-4877 236 1 k k PROPN ejpam-4877 236 2 ≤ ≤ PROPN ejpam-4877 236 3 n−m+ n−m+ PROPN ejpam-4877 236 4 |{vi}| |{vi}| PROPN ejpam-4877 236 5 = = PUNCT ejpam-4877 236 6 n−m+ n−m+ PROPN ejpam-4877 236 7 1 1 NUM ejpam-4877 236 8 = = SYM ejpam-4877 236 9 n− n− PROPN ejpam-4877 236 10 ( ( PUNCT ejpam-4877 236 11 m− m− PROPN ejpam-4877 236 12 1 1 NUM ejpam-4877 236 13 ) ) PUNCT ejpam-4877 236 14 . . PUNCT ejpam-4877 237 1 the the DET ejpam-4877 237 2 next next ADJ ejpam-4877 237 3 result result NOUN ejpam-4877 237 4 follows follow VERB ejpam-4877 237 5 from from ADP ejpam-4877 237 6 proposition proposition NOUN ejpam-4877 237 7 2 2 NUM ejpam-4877 237 8 . . PUNCT ejpam-4877 237 9 corollary corollary ADJ ejpam-4877 237 10 2 2 NUM ejpam-4877 237 11 . . PUNCT ejpam-4877 238 1 let let VERB ejpam-4877 238 2 g g PRON ejpam-4877 238 3 be be AUX ejpam-4877 238 4 a a DET ejpam-4877 238 5 graph graph NOUN ejpam-4877 238 6 of of ADP ejpam-4877 238 7 order order NOUN ejpam-4877 238 8 n n PRON ejpam-4877 238 9 with with ADP ejpam-4877 238 10 γ(c γ(c PROPN ejpam-4877 238 11 ) ) PUNCT ejpam-4877 238 12 ̸= ̸= PROPN ejpam-4877 238 13 1 1 NUM ejpam-4877 238 14 for for ADP ejpam-4877 238 15 each each DET ejpam-4877 238 16 component component NOUN ejpam-4877 238 17 c c PROPN ejpam-4877 238 18 of of ADP ejpam-4877 238 19 g. g. PROPN ejpam-4877 238 20 if if SCONJ ejpam-4877 238 21 |n2 |n2 PROPN ejpam-4877 238 22 g(u)| g(u)| PROPN ejpam-4877 238 23 ≥ ≥ NUM ejpam-4877 238 24 2 2 NUM ejpam-4877 238 25 for for ADP ejpam-4877 238 26 every every PRON ejpam-4877 238 27 u u PROPN ejpam-4877 238 28 ∈ ∈ PROPN ejpam-4877 238 29 v v NOUN ejpam-4877 238 30 ( ( PUNCT ejpam-4877 238 31 g g NOUN ejpam-4877 238 32 ) ) PUNCT ejpam-4877 238 33 , , PUNCT ejpam-4877 238 34 then then ADV ejpam-4877 238 35 γthgr(g γthgr(g NOUN ejpam-4877 238 36 ) ) PUNCT ejpam-4877 238 37 ≤ ≤ NUM ejpam-4877 238 38 n− n− NOUN ejpam-4877 238 39 1 1 NUM ejpam-4877 238 40 . . PUNCT ejpam-4877 239 1 proposition proposition NOUN ejpam-4877 239 2 3 3 X ejpam-4877 239 3 . . PUNCT ejpam-4877 240 1 let let VERB ejpam-4877 240 2 n n PRON ejpam-4877 240 3 and and CCONJ ejpam-4877 240 4 m m AUX ejpam-4877 240 5 be be AUX ejpam-4877 240 6 any any DET ejpam-4877 240 7 positive positive ADJ ejpam-4877 240 8 integers integer NOUN ejpam-4877 240 9 such such ADJ ejpam-4877 240 10 that that SCONJ ejpam-4877 240 11 n n NUM ejpam-4877 240 12 ≥ ≥ NOUN ejpam-4877 240 13 4 4 NUM ejpam-4877 240 14 . . PUNCT ejpam-4877 240 15 then then ADV ejpam-4877 240 16 γthgr(cn γthgr(cn VERB ejpam-4877 240 17 ) ) PUNCT ejpam-4877 240 18 = = PUNCT ejpam-4877 241 1   NUM ejpam-4877 241 2 4 4 NUM ejpam-4877 241 3 if if SCONJ ejpam-4877 241 4 n n X ejpam-4877 241 5 = = SYM ejpam-4877 241 6 4 4 NUM ejpam-4877 241 7 , , PUNCT ejpam-4877 241 8 5 5 NUM ejpam-4877 241 9 , , PUNCT ejpam-4877 241 10 6 6 NUM ejpam-4877 241 11 6 6 NUM ejpam-4877 241 12 if if SCONJ ejpam-4877 241 13 n n NOUN ejpam-4877 241 14 = = SYM ejpam-4877 241 15 8 8 NUM ejpam-4877 241 16 n− n− NOUN ejpam-4877 241 17 4 4 NUM ejpam-4877 241 18 if if SCONJ ejpam-4877 241 19 n n NOUN ejpam-4877 241 20 = = SYM ejpam-4877 241 21 4 4 NUM ejpam-4877 241 22 m m NOUN ejpam-4877 241 23 ≥ ≥ NOUN ejpam-4877 241 24 12 12 NUM ejpam-4877 241 25 n− n− NOUN ejpam-4877 241 26 2 2 NUM ejpam-4877 241 27 if if SCONJ ejpam-4877 241 28 n n NOUN ejpam-4877 241 29 = = SYM ejpam-4877 241 30 4m+ 4m+ NUM ejpam-4877 241 31 2 2 NUM ejpam-4877 241 32 ≥ ≥ NOUN ejpam-4877 241 33 10 10 NUM ejpam-4877 241 34 n− n− NOUN ejpam-4877 241 35 1 1 NUM ejpam-4877 241 36 if if SCONJ ejpam-4877 241 37 n n PRON ejpam-4877 241 38 ≥ ≥ VERB ejpam-4877 241 39 7 7 NUM ejpam-4877 241 40 and and CCONJ ejpam-4877 241 41 odd odd ADJ ejpam-4877 241 42 proof proof NOUN ejpam-4877 241 43 . . PUNCT ejpam-4877 242 1 clearly clearly ADV ejpam-4877 242 2 for for ADP ejpam-4877 242 3 n n NOUN ejpam-4877 242 4 = = SYM ejpam-4877 242 5 4 4 NUM ejpam-4877 242 6 , , PUNCT ejpam-4877 242 7 5 5 NUM ejpam-4877 242 8 , , PUNCT ejpam-4877 242 9 6 6 NUM ejpam-4877 242 10 and and CCONJ ejpam-4877 242 11 n n CCONJ ejpam-4877 242 12 = = SYM ejpam-4877 242 13 8 8 NUM ejpam-4877 242 14 , , PUNCT ejpam-4877 242 15 γthgr(cn γthgr(cn NOUN ejpam-4877 242 16 ) ) PUNCT ejpam-4877 242 17 = = SYM ejpam-4877 242 18 4 4 NUM ejpam-4877 242 19 and and CCONJ ejpam-4877 242 20 γthgr(cn γthgr(cn NOUN ejpam-4877 242 21 ) ) PUNCT ejpam-4877 242 22 = = SYM ejpam-4877 242 23 6 6 NUM ejpam-4877 242 24 , , PUNCT ejpam-4877 242 25 respectively respectively ADV ejpam-4877 242 26 . . PUNCT ejpam-4877 243 1 for for ADP ejpam-4877 243 2 n n NOUN ejpam-4877 243 3 = = SYM ejpam-4877 243 4 4 4 NUM ejpam-4877 243 5 m m NOUN ejpam-4877 243 6 ≥ ≥ NOUN ejpam-4877 243 7 12 12 NUM ejpam-4877 243 8 , , PUNCT ejpam-4877 243 9 let let VERB ejpam-4877 243 10 v v X ejpam-4877 243 11 ( ( PUNCT ejpam-4877 243 12 cn cn PROPN ejpam-4877 243 13 ) ) PUNCT ejpam-4877 243 14 = = SYM ejpam-4877 243 15 { { PUNCT ejpam-4877 243 16 v1 v1 PROPN ejpam-4877 243 17 , , PUNCT ejpam-4877 243 18 v2 v2 PROPN ejpam-4877 243 19 , , PUNCT ejpam-4877 243 20 . . PUNCT ejpam-4877 243 21 . . PUNCT ejpam-4877 244 1 . . PUNCT ejpam-4877 245 1 , , PUNCT ejpam-4877 245 2 vn vn PROPN ejpam-4877 245 3 } } PUNCT ejpam-4877 245 4 . . PUNCT ejpam-4877 246 1 observe observe VERB ejpam-4877 246 2 that that SCONJ ejpam-4877 246 3 s s VERB ejpam-4877 246 4 = = PUNCT ejpam-4877 246 5 ( ( PUNCT ejpam-4877 246 6 v1 v1 PROPN ejpam-4877 246 7 , , PUNCT ejpam-4877 246 8 v2 v2 PROPN ejpam-4877 246 9 , , PUNCT ejpam-4877 246 10 · · PUNCT ejpam-4877 246 11 · · PUNCT ejpam-4877 246 12 · · PUNCT ejpam-4877 246 13 , , PUNCT ejpam-4877 246 14 vn−4 vn−4 NOUN ejpam-4877 246 15 ) ) PUNCT ejpam-4877 246 16 is be AUX ejpam-4877 246 17 a a DET ejpam-4877 246 18 grundy grundy PROPN ejpam-4877 246 19 total total NOUN ejpam-4877 246 20 hop hop NOUN ejpam-4877 246 21 dominating dominating NOUN ejpam-4877 246 22 sequence sequence NOUN ejpam-4877 246 23 of of ADP ejpam-4877 246 24 cn cn PROPN ejpam-4877 246 25 . . PUNCT ejpam-4877 247 1 thus thus ADV ejpam-4877 247 2 , , PUNCT ejpam-4877 247 3 γthgr(cn γthgr(cn ADJ ejpam-4877 247 4 ) ) PUNCT ejpam-4877 247 5 ≥ ≥ NOUN ejpam-4877 247 6 n n CCONJ ejpam-4877 247 7 − − NOUN ejpam-4877 247 8 4 4 NUM ejpam-4877 247 9 . . PUNCT ejpam-4877 248 1 on on ADP ejpam-4877 248 2 the the DET ejpam-4877 248 3 other other ADJ ejpam-4877 248 4 hand hand NOUN ejpam-4877 248 5 , , PUNCT ejpam-4877 248 6 let let VERB ejpam-4877 248 7 s′ s′ ADJ ejpam-4877 248 8 = = SYM ejpam-4877 248 9 ( ( PUNCT ejpam-4877 248 10 w1 w1 NOUN ejpam-4877 248 11 , , PUNCT ejpam-4877 248 12 w2 w2 NOUN ejpam-4877 248 13 , , PUNCT ejpam-4877 248 14 . . PUNCT ejpam-4877 248 15 . . PUNCT ejpam-4877 249 1 . . PUNCT ejpam-4877 250 1 , , PUNCT ejpam-4877 250 2 wk wk AUX ejpam-4877 250 3 ) ) PUNCT ejpam-4877 250 4 be be AUX ejpam-4877 250 5 a a DET ejpam-4877 250 6 grundy grundy PROPN ejpam-4877 250 7 total total NOUN ejpam-4877 250 8 hop hop NOUN ejpam-4877 250 9 dominating dominating NOUN ejpam-4877 250 10 sequence sequence NOUN ejpam-4877 250 11 of of ADP ejpam-4877 250 12 cn cn PROPN ejpam-4877 250 13 . . PROPN ejpam-4877 250 14 notice notice VERB ejpam-4877 250 15 that that SCONJ ejpam-4877 250 16 one one NUM ejpam-4877 250 17 of of ADP ejpam-4877 250 18 the the DET ejpam-4877 250 19 v1 v1 NOUN ejpam-4877 250 20 , , PUNCT ejpam-4877 250 21 v5 v5 NOUN ejpam-4877 250 22 , , PUNCT ejpam-4877 250 23 v9 v9 NOUN ejpam-4877 250 24 , , PUNCT ejpam-4877 250 25 . . PUNCT ejpam-4877 250 26 . . PUNCT ejpam-4877 251 1 . . PUNCT ejpam-4877 252 1 , , PUNCT ejpam-4877 252 2 vn−7 vn−7 PROPN ejpam-4877 252 3 , , PUNCT ejpam-4877 252 4 vn−3 vn−3 PROPN ejpam-4877 252 5 is be AUX ejpam-4877 252 6 not not PART ejpam-4877 252 7 in in ADP ejpam-4877 252 8 ŝ′. ŝ′. ADJ ejpam-4877 252 9 suppose suppose VERB ejpam-4877 252 10 all all DET ejpam-4877 252 11 the the DET ejpam-4877 252 12 vertices vertex NOUN ejpam-4877 252 13 v1 v1 NOUN ejpam-4877 252 14 , , PUNCT ejpam-4877 252 15 v5 v5 NOUN ejpam-4877 252 16 , , PUNCT ejpam-4877 252 17 v9 v9 NOUN ejpam-4877 252 18 , , PUNCT ejpam-4877 252 19 . . PUNCT ejpam-4877 252 20 . . PUNCT ejpam-4877 253 1 . . PUNCT ejpam-4877 254 1 , , PUNCT ejpam-4877 254 2 vn−7 vn−7 PROPN ejpam-4877 254 3 , , PUNCT ejpam-4877 254 4 vn−3 vn−3 PROPN ejpam-4877 254 5 are be AUX ejpam-4877 254 6 in in ADP ejpam-4877 254 7 ŝ′. ŝ′. ADJ ejpam-4877 254 8 wlog wlog NOUN ejpam-4877 254 9 , , PUNCT ejpam-4877 254 10 assume assume VERB ejpam-4877 254 11 that that SCONJ ejpam-4877 254 12 w1 w1 NOUN ejpam-4877 254 13 = = SYM ejpam-4877 254 14 v1 v1 NOUN ejpam-4877 254 15 . . PUNCT ejpam-4877 255 1 if if SCONJ ejpam-4877 255 2 v9 v9 PROPN ejpam-4877 255 3 comes come VERB ejpam-4877 255 4 before before ADP ejpam-4877 255 5 v5 v5 PROPN ejpam-4877 255 6 , , PUNCT ejpam-4877 255 7 then then ADV ejpam-4877 255 8 n2 n2 PROPN ejpam-4877 255 9 cn cn PROPN ejpam-4877 255 10 ( ( PUNCT ejpam-4877 255 11 v5 v5 PROPN ejpam-4877 255 12 ) ) PUNCT ejpam-4877 255 13 ⊆ ⊆ NUM ejpam-4877 255 14 n2 n2 PROPN ejpam-4877 255 15 cn cn PROPN ejpam-4877 255 16 ( ( PUNCT ejpam-4877 255 17 v1)∪n2 v1)∪n2 PROPN ejpam-4877 255 18 cn cn PROPN ejpam-4877 255 19 ( ( PUNCT ejpam-4877 255 20 v9 v9 PROPN ejpam-4877 255 21 ) ) PUNCT ejpam-4877 255 22 , , PUNCT ejpam-4877 255 23 a a DET ejpam-4877 255 24 contradiction contradiction NOUN ejpam-4877 255 25 . . PUNCT ejpam-4877 256 1 so so ADV ejpam-4877 256 2 , , PUNCT ejpam-4877 256 3 v9 v9 PROPN ejpam-4877 256 4 comes come VERB ejpam-4877 256 5 after after ADP ejpam-4877 256 6 v5 v5 PROPN ejpam-4877 256 7 . . PUNCT ejpam-4877 257 1 next next ADV ejpam-4877 257 2 , , PUNCT ejpam-4877 257 3 suppose suppose VERB ejpam-4877 257 4 v13 v13 NOUN ejpam-4877 257 5 comes come VERB ejpam-4877 257 6 before before ADP ejpam-4877 257 7 v9 v9 PROPN ejpam-4877 257 8 , , PUNCT ejpam-4877 257 9 then then ADV ejpam-4877 257 10 n2 n2 PROPN ejpam-4877 257 11 cn cn PROPN ejpam-4877 257 12 ( ( PUNCT ejpam-4877 257 13 v9 v9 PROPN ejpam-4877 257 14 ) ) PUNCT ejpam-4877 257 15 ⊆ ⊆ NUM ejpam-4877 257 16 n2 n2 PROPN ejpam-4877 257 17 cn cn PROPN ejpam-4877 257 18 ( ( PUNCT ejpam-4877 257 19 v1)∪n2 v1)∪n2 PROPN ejpam-4877 257 20 cn cn PROPN ejpam-4877 257 21 ( ( PUNCT ejpam-4877 257 22 v5)∪n2 v5)∪n2 PROPN ejpam-4877 257 23 cn cn PROPN ejpam-4877 257 24 ( ( PUNCT ejpam-4877 257 25 v13 v13 PROPN ejpam-4877 257 26 ) ) PUNCT ejpam-4877 257 27 , , PUNCT ejpam-4877 257 28 a a DET ejpam-4877 257 29 contradiction contradiction NOUN ejpam-4877 257 30 . . PUNCT ejpam-4877 258 1 thus thus ADV ejpam-4877 258 2 , , PUNCT ejpam-4877 258 3 v13 v13 PROPN ejpam-4877 258 4 comes come VERB ejpam-4877 258 5 after after ADP ejpam-4877 258 6 v9 v9 NOUN ejpam-4877 258 7 . . PUNCT ejpam-4877 259 1 continuing continue VERB ejpam-4877 259 2 in in ADP ejpam-4877 259 3 this this DET ejpam-4877 259 4 manner manner NOUN ejpam-4877 259 5 , , PUNCT ejpam-4877 259 6 we we PRON ejpam-4877 259 7 find find VERB ejpam-4877 259 8 that that SCONJ ejpam-4877 259 9 the the DET ejpam-4877 259 10 following follow VERB ejpam-4877 259 11 order order NOUN ejpam-4877 259 12 of of ADP ejpam-4877 259 13 appearance appearance NOUN ejpam-4877 259 14 ( ( PUNCT ejpam-4877 259 15 not not PART ejpam-4877 259 16 necessarily necessarily ADV ejpam-4877 259 17 consecutive consecutive ADJ ejpam-4877 259 18 ) ) PUNCT ejpam-4877 259 19 of of ADP ejpam-4877 259 20 the the DET ejpam-4877 259 21 given give VERB ejpam-4877 259 22 vertices vertex NOUN ejpam-4877 259 23 in in ADP ejpam-4877 259 24 the the DET ejpam-4877 259 25 grundy grundy PROPN ejpam-4877 259 26 total total NOUN ejpam-4877 259 27 hop hop NOUN ejpam-4877 259 28 dominating dominating NOUN ejpam-4877 259 29 sequence sequence NOUN ejpam-4877 259 30 s′ s′ NOUN ejpam-4877 259 31 : : PUNCT ejpam-4877 259 32 v1 v1 NOUN ejpam-4877 259 33 , , PUNCT ejpam-4877 259 34 v5 v5 NOUN ejpam-4877 259 35 , , PUNCT ejpam-4877 259 36 v9 v9 NOUN ejpam-4877 259 37 , , PUNCT ejpam-4877 259 38 . . PUNCT ejpam-4877 259 39 . . PUNCT ejpam-4877 260 1 . . PUNCT ejpam-4877 261 1 , , PUNCT ejpam-4877 261 2 vn−7 vn−7 PROPN ejpam-4877 261 3 , , PUNCT ejpam-4877 261 4 vn−3 vn−3 PROPN ejpam-4877 261 5 . . PUNCT ejpam-4877 262 1 however however ADV ejpam-4877 262 2 , , PUNCT ejpam-4877 262 3 n2 n2 PROPN ejpam-4877 262 4 cn cn PROPN ejpam-4877 262 5 ( ( PUNCT ejpam-4877 262 6 vn−3 vn−3 PROPN ejpam-4877 262 7 ) ) PUNCT ejpam-4877 262 8 ⊆ ⊆ NUM ejpam-4877 262 9 n2 n2 PROPN ejpam-4877 262 10 cn cn PROPN ejpam-4877 262 11 ( ( PUNCT ejpam-4877 262 12 v1 v1 PROPN ejpam-4877 262 13 ) ) PUNCT ejpam-4877 262 14 ∪ ∪ ADP ejpam-4877 262 15 ncn(vn−7 ncn(vn−7 PROPN ejpam-4877 262 16 ) ) PUNCT ejpam-4877 262 17 . . PUNCT ejpam-4877 263 1 hence hence ADV ejpam-4877 263 2 , , PUNCT ejpam-4877 263 3 vn−3 vn−3 PROPN ejpam-4877 263 4 /∈ /∈ PUNCT ejpam-4877 263 5 ŝ′ ŝ′ PROPN ejpam-4877 263 6 , , PUNCT ejpam-4877 263 7 a a DET ejpam-4877 263 8 contradiction contradiction NOUN ejpam-4877 263 9 . . PUNCT ejpam-4877 264 1 similarly similarly ADV ejpam-4877 264 2 , , PUNCT ejpam-4877 264 3 one one NUM ejpam-4877 264 4 of of ADP ejpam-4877 264 5 the the DET ejpam-4877 264 6 vertices vertex NOUN ejpam-4877 264 7 v2 v2 PROPN ejpam-4877 264 8 , , PUNCT ejpam-4877 264 9 v6 v6 NOUN ejpam-4877 264 10 , , PUNCT ejpam-4877 264 11 v10 v10 PROPN ejpam-4877 264 12 , , PUNCT ejpam-4877 264 13 . . PUNCT ejpam-4877 264 14 . . PUNCT ejpam-4877 264 15 . . PUNCT ejpam-4877 265 1 , , PUNCT ejpam-4877 265 2 vn−6 vn−6 PROPN ejpam-4877 265 3 , , PUNCT ejpam-4877 265 4 vn−2 vn−2 PROPN ejpam-4877 265 5 , , PUNCT ejpam-4877 265 6 v3 v3 PROPN ejpam-4877 265 7 , , PUNCT ejpam-4877 265 8 v7 v7 VERB ejpam-4877 265 9 , , PUNCT ejpam-4877 265 10 v11 v11 NOUN ejpam-4877 265 11 , , PUNCT ejpam-4877 265 12 . . PUNCT ejpam-4877 265 13 . . PUNCT ejpam-4877 266 1 . . PUNCT ejpam-4877 267 1 , , PUNCT ejpam-4877 267 2 vn−5 vn−5 PROPN ejpam-4877 267 3 , , PUNCT ejpam-4877 267 4 vn−1 vn−1 ADJ ejpam-4877 267 5 , , PUNCT ejpam-4877 267 6 and and CCONJ ejpam-4877 267 7 v4 v4 NOUN ejpam-4877 267 8 , , PUNCT ejpam-4877 267 9 v8 v8 PROPN ejpam-4877 267 10 , , PUNCT ejpam-4877 267 11 v12 v12 VERB ejpam-4877 267 12 , , PUNCT ejpam-4877 267 13 . . PUNCT ejpam-4877 267 14 . . PUNCT ejpam-4877 268 1 . . PUNCT ejpam-4877 269 1 , , PUNCT ejpam-4877 269 2 vn−4 vn−4 NOUN ejpam-4877 269 3 , , PUNCT ejpam-4877 269 4 vn vn NOUN ejpam-4877 269 5 , , PUNCT ejpam-4877 269 6 respectively respectively ADV ejpam-4877 269 7 , , PUNCT ejpam-4877 269 8 is be AUX ejpam-4877 269 9 not not PART ejpam-4877 269 10 in in ADP ejpam-4877 269 11 ŝ′. ŝ′. ADJ ejpam-4877 269 12 therefore therefore ADV ejpam-4877 269 13 , , PUNCT ejpam-4877 269 14 γthgr(pn γthgr(pn ADJ ejpam-4877 269 15 ) ) PUNCT ejpam-4877 269 16 = = SYM ejpam-4877 270 1 k k X ejpam-4877 270 2 ≤ ≤ NUM ejpam-4877 270 3 n− n− NOUN ejpam-4877 270 4 4 4 NUM ejpam-4877 270 5 . . PUNCT ejpam-4877 271 1 consequently consequently ADV ejpam-4877 271 2 , , PUNCT ejpam-4877 271 3 γthgr(pn γthgr(pn ADJ ejpam-4877 271 4 ) ) PUNCT ejpam-4877 271 5 = = PUNCT ejpam-4877 271 6 n− n− NOUN ejpam-4877 271 7 4 4 NUM ejpam-4877 271 8 for for ADP ejpam-4877 271 9 all all DET ejpam-4877 271 10 n n PRON ejpam-4877 271 11 = = SYM ejpam-4877 271 12 4 4 NUM ejpam-4877 271 13 m. m. NOUN ejpam-4877 271 14 next next ADV ejpam-4877 271 15 , , PUNCT ejpam-4877 271 16 for for ADP ejpam-4877 271 17 n n NOUN ejpam-4877 271 18 = = SYM ejpam-4877 271 19 4m+ 4m+ NUM ejpam-4877 271 20 2 2 NUM ejpam-4877 271 21 ≥ ≥ NOUN ejpam-4877 271 22 10 10 NUM ejpam-4877 271 23 , , PUNCT ejpam-4877 271 24 let let VERB ejpam-4877 271 25 s1 s1 PROPN ejpam-4877 271 26 = = SYM ejpam-4877 271 27 ( ( PUNCT ejpam-4877 271 28 v1 v1 PROPN ejpam-4877 271 29 , , PUNCT ejpam-4877 271 30 v5 v5 NOUN ejpam-4877 271 31 , , PUNCT ejpam-4877 271 32 . . PUNCT ejpam-4877 271 33 . . PUNCT ejpam-4877 272 1 . . PUNCT ejpam-4877 273 1 , , PUNCT ejpam-4877 273 2 vn−5 vn−5 PROPN ejpam-4877 273 3 , , PUNCT ejpam-4877 273 4 vn−1 vn−1 PROPN ejpam-4877 273 5 , , PUNCT ejpam-4877 273 6 v3 v3 PROPN ejpam-4877 273 7 , , PUNCT ejpam-4877 273 8 v7 v7 VERB ejpam-4877 273 9 , , PUNCT ejpam-4877 273 10 . . PUNCT ejpam-4877 273 11 . . PUNCT ejpam-4877 274 1 . . PUNCT ejpam-4877 275 1 , , PUNCT ejpam-4877 275 2 vn−7 vn−7 PROPN ejpam-4877 275 3 , , PUNCT ejpam-4877 275 4 v2 v2 PROPN ejpam-4877 275 5 , , PUNCT ejpam-4877 275 6 v6 v6 NOUN ejpam-4877 275 7 , , PUNCT ejpam-4877 275 8 . . PUNCT ejpam-4877 275 9 . . PUNCT ejpam-4877 276 1 . . PUNCT ejpam-4877 277 1 , , PUNCT ejpam-4877 277 2 vn−4 vn−4 NOUN ejpam-4877 277 3 , , PUNCT ejpam-4877 277 4 vn vn NOUN ejpam-4877 277 5 , , PUNCT ejpam-4877 277 6 v4 v4 PROPN ejpam-4877 277 7 , , PUNCT ejpam-4877 277 8 v8 v8 PROPN ejpam-4877 277 9 , , PUNCT ejpam-4877 277 10 . . PUNCT ejpam-4877 277 11 . . PUNCT ejpam-4877 278 1 . . PUNCT ejpam-4877 279 1 , , PUNCT ejpam-4877 279 2 vn−6 vn−6 PROPN ejpam-4877 279 3 ) ) PUNCT ejpam-4877 279 4 . . PUNCT ejpam-4877 280 1 then then ADV ejpam-4877 280 2 s1 s1 PROPN ejpam-4877 280 3 is be AUX ejpam-4877 280 4 a a DET ejpam-4877 280 5 grundy grundy PROPN ejpam-4877 280 6 total total NOUN ejpam-4877 280 7 hop hop NOUN ejpam-4877 280 8 dominating dominating NOUN ejpam-4877 280 9 sequence sequence NOUN ejpam-4877 280 10 of of ADP ejpam-4877 280 11 cn cn PROPN ejpam-4877 280 12 . . PUNCT ejpam-4877 281 1 hence hence ADV ejpam-4877 281 2 , , PUNCT ejpam-4877 281 3 γthgr(cn γthgr(cn ADJ ejpam-4877 281 4 ) ) PUNCT ejpam-4877 281 5 ≥ ≥ NOUN ejpam-4877 281 6 n− n− NOUN ejpam-4877 281 7 2 2 NUM ejpam-4877 281 8 . . PUNCT ejpam-4877 282 1 on on ADP ejpam-4877 282 2 the the DET ejpam-4877 282 3 other other ADJ ejpam-4877 282 4 hand hand NOUN ejpam-4877 282 5 , , PUNCT ejpam-4877 282 6 let let VERB ejpam-4877 282 7 s s PRON ejpam-4877 282 8 = = PUNCT ejpam-4877 282 9 ( ( PUNCT ejpam-4877 282 10 w1 w1 NOUN ejpam-4877 282 11 , , PUNCT ejpam-4877 282 12 w2 w2 NOUN ejpam-4877 282 13 , , PUNCT ejpam-4877 282 14 . . PUNCT ejpam-4877 282 15 . . PUNCT ejpam-4877 283 1 . . PUNCT ejpam-4877 284 1 , , PUNCT ejpam-4877 284 2 wk wk AUX ejpam-4877 284 3 ) ) PUNCT ejpam-4877 284 4 be be AUX ejpam-4877 284 5 a a DET ejpam-4877 284 6 grundy grundy PROPN ejpam-4877 284 7 total total NOUN ejpam-4877 284 8 hop hop NOUN ejpam-4877 284 9 dominating dominating NOUN ejpam-4877 284 10 sequence sequence NOUN ejpam-4877 284 11 of of ADP ejpam-4877 284 12 cn cn PROPN ejpam-4877 284 13 . . PUNCT ejpam-4877 285 1 then then ADV ejpam-4877 285 2 applying apply VERB ejpam-4877 285 3 the the DET ejpam-4877 285 4 same same ADJ ejpam-4877 285 5 arguments argument NOUN ejpam-4877 285 6 with with ADP ejpam-4877 285 7 the the DET ejpam-4877 285 8 first first ADJ ejpam-4877 285 9 part part NOUN ejpam-4877 285 10 , , PUNCT ejpam-4877 285 11 one one PRON ejpam-4877 285 12 can can AUX ejpam-4877 285 13 show show VERB ejpam-4877 285 14 that that SCONJ ejpam-4877 285 15 one one NUM ejpam-4877 285 16 of of ADP ejpam-4877 285 17 the the DET ejpam-4877 285 18 j.a j.a PROPN ejpam-4877 285 19 . . PUNCT ejpam-4877 286 1 hassan hassan PROPN ejpam-4877 286 2 , , PUNCT ejpam-4877 286 3 s. s. PROPN ejpam-4877 286 4 canoy canoy PROPN ejpam-4877 286 5 / / SYM ejpam-4877 286 6 eur eur PROPN ejpam-4877 286 7 . . PUNCT ejpam-4877 287 1 j. j. PROPN ejpam-4877 287 2 pure pure PROPN ejpam-4877 287 3 appl appl PROPN ejpam-4877 287 4 . . PROPN ejpam-4877 287 5 math math PROPN ejpam-4877 287 6 , , PUNCT ejpam-4877 287 7 16 16 NUM ejpam-4877 287 8 ( ( PUNCT ejpam-4877 287 9 4 4 NUM ejpam-4877 287 10 ) ) PUNCT ejpam-4877 287 11 ( ( PUNCT ejpam-4877 287 12 2023 2023 NUM ejpam-4877 287 13 ) ) PUNCT ejpam-4877 287 14 , , PUNCT ejpam-4877 287 15 2597 2597 NUM ejpam-4877 287 16 - - SYM ejpam-4877 287 17 2612 2612 NUM ejpam-4877 287 18 2604 2604 NUM ejpam-4877 287 19 v1 v1 NOUN ejpam-4877 287 20 , , PUNCT ejpam-4877 287 21 v5 v5 NOUN ejpam-4877 287 22 , , PUNCT ejpam-4877 287 23 v9 v9 NOUN ejpam-4877 287 24 , , PUNCT ejpam-4877 287 25 . . PUNCT ejpam-4877 287 26 . . PUNCT ejpam-4877 288 1 . . PUNCT ejpam-4877 289 1 , , PUNCT ejpam-4877 289 2 vn−5 vn−5 PROPN ejpam-4877 289 3 , , PUNCT ejpam-4877 289 4 vn−1 vn−1 PROPN ejpam-4877 289 5 , , PUNCT ejpam-4877 289 6 v3 v3 PROPN ejpam-4877 289 7 , , PUNCT ejpam-4877 289 8 v7 v7 VERB ejpam-4877 289 9 , , PUNCT ejpam-4877 289 10 . . PUNCT ejpam-4877 289 11 . . PUNCT ejpam-4877 290 1 . . PUNCT ejpam-4877 291 1 , , PUNCT ejpam-4877 291 2 vn−7 vn−7 PROPN ejpam-4877 291 3 , , PUNCT ejpam-4877 291 4 vn−3 vn−3 PROPN ejpam-4877 291 5 and and CCONJ ejpam-4877 291 6 v2 v2 PROPN ejpam-4877 291 7 , , PUNCT ejpam-4877 291 8 v6 v6 NOUN ejpam-4877 291 9 , , PUNCT ejpam-4877 291 10 v10 v10 PROPN ejpam-4877 291 11 , , PUNCT ejpam-4877 291 12 . . PUNCT ejpam-4877 291 13 . . PUNCT ejpam-4877 292 1 . . PUNCT ejpam-4877 293 1 , , PUNCT ejpam-4877 293 2 vn−4 vn−4 NOUN ejpam-4877 293 3 , , PUNCT ejpam-4877 293 4 vn vn NOUN ejpam-4877 293 5 , , PUNCT ejpam-4877 293 6 v4 v4 PROPN ejpam-4877 293 7 , , PUNCT ejpam-4877 293 8 v8 v8 PROPN ejpam-4877 293 9 , , PUNCT ejpam-4877 293 10 . . PUNCT ejpam-4877 293 11 . . PUNCT ejpam-4877 294 1 . . PUNCT ejpam-4877 295 1 , , PUNCT ejpam-4877 295 2 vn−6 vn−6 PROPN ejpam-4877 295 3 , , PUNCT ejpam-4877 295 4 vn−2 vn−2 PROPN ejpam-4877 295 5 is be AUX ejpam-4877 295 6 not not PART ejpam-4877 295 7 in in ADP ejpam-4877 295 8 ŝ1 ŝ1 PROPN ejpam-4877 295 9 , , PUNCT ejpam-4877 295 10 respectively respectively ADV ejpam-4877 295 11 . . PUNCT ejpam-4877 296 1 hence hence ADV ejpam-4877 296 2 , , PUNCT ejpam-4877 296 3 γ γ X ejpam-4877 296 4 th th X ejpam-4877 296 5 gr(cn gr(cn NOUN ejpam-4877 296 6 ) ) PUNCT ejpam-4877 296 7 = = SYM ejpam-4877 297 1 k k PROPN ejpam-4877 297 2 ≤ ≤ PROPN ejpam-4877 297 3 n−2 n−2 PROPN ejpam-4877 297 4 . . PUNCT ejpam-4877 298 1 consequently consequently ADV ejpam-4877 298 2 , , PUNCT ejpam-4877 298 3 γthgr(cn γthgr(cn NOUN ejpam-4877 298 4 ) ) PUNCT ejpam-4877 298 5 = = SYM ejpam-4877 298 6 n−2 n−2 PROPN ejpam-4877 298 7 . . PUNCT ejpam-4877 299 1 let let VERB ejpam-4877 299 2 n n PRON ejpam-4877 299 3 ≥ ≥ X ejpam-4877 299 4 7 7 NUM ejpam-4877 299 5 and and CCONJ ejpam-4877 299 6 odd odd ADJ ejpam-4877 299 7 . . PUNCT ejpam-4877 300 1 clearly clearly ADV ejpam-4877 300 2 , , PUNCT ejpam-4877 300 3 γthgr(c7 γthgr(c7 NOUN ejpam-4877 300 4 ) ) PUNCT ejpam-4877 300 5 = = SYM ejpam-4877 301 1 6 6 X ejpam-4877 301 2 . . PUNCT ejpam-4877 301 3 suppose suppose VERB ejpam-4877 301 4 n n PRON ejpam-4877 301 5 ≥ ≥ NOUN ejpam-4877 301 6 9 9 NUM ejpam-4877 301 7 and and CCONJ ejpam-4877 301 8 odd odd ADJ ejpam-4877 301 9 . . PUNCT ejpam-4877 302 1 for for ADP ejpam-4877 302 2 n n PRON ejpam-4877 302 3 ∈ ∈ NOUN ejpam-4877 302 4 { { PUNCT ejpam-4877 302 5 9 9 NUM ejpam-4877 302 6 , , PUNCT ejpam-4877 302 7 13 13 NUM ejpam-4877 302 8 , , PUNCT ejpam-4877 302 9 17 17 NUM ejpam-4877 302 10 , , PUNCT ejpam-4877 302 11 . . PUNCT ejpam-4877 302 12 . . PUNCT ejpam-4877 302 13 . . PUNCT ejpam-4877 303 1 } } PUNCT ejpam-4877 303 2 , , PUNCT ejpam-4877 303 3 let let VERB ejpam-4877 303 4 s2 s2 VERB ejpam-4877 303 5 = = SYM ejpam-4877 303 6 ( ( PUNCT ejpam-4877 303 7 v1 v1 PROPN ejpam-4877 303 8 , , PUNCT ejpam-4877 303 9 v5 v5 NOUN ejpam-4877 303 10 , , PUNCT ejpam-4877 303 11 . . PUNCT ejpam-4877 303 12 . . PUNCT ejpam-4877 304 1 . . PUNCT ejpam-4877 305 1 , , PUNCT ejpam-4877 305 2 vn vn PROPN ejpam-4877 305 3 , , PUNCT ejpam-4877 305 4 v4 v4 PROPN ejpam-4877 305 5 , , PUNCT ejpam-4877 305 6 . . PUNCT ejpam-4877 305 7 . . PUNCT ejpam-4877 305 8 . . PUNCT ejpam-4877 306 1 , , PUNCT ejpam-4877 306 2 vn−1 vn−1 PROPN ejpam-4877 306 3 , , PUNCT ejpam-4877 306 4 v3 v3 PROPN ejpam-4877 306 5 , , PUNCT ejpam-4877 306 6 . . PUNCT ejpam-4877 306 7 . . PUNCT ejpam-4877 307 1 . . PUNCT ejpam-4877 308 1 , , PUNCT ejpam-4877 308 2 vn−2 vn−2 PROPN ejpam-4877 308 3 , , PUNCT ejpam-4877 308 4 v2 v2 PROPN ejpam-4877 308 5 , , PUNCT ejpam-4877 308 6 . . PUNCT ejpam-4877 308 7 . . PUNCT ejpam-4877 309 1 . . PUNCT ejpam-4877 310 1 , , PUNCT ejpam-4877 310 2 vn−7 vn−7 PROPN ejpam-4877 310 3 ) ) PUNCT ejpam-4877 310 4 . . PUNCT ejpam-4877 311 1 then then ADV ejpam-4877 311 2 s2 s2 PROPN ejpam-4877 311 3 is be AUX ejpam-4877 311 4 a a DET ejpam-4877 311 5 grundy grundy PROPN ejpam-4877 311 6 total total NOUN ejpam-4877 311 7 hop hop NOUN ejpam-4877 311 8 dominating dominating NOUN ejpam-4877 311 9 sequence sequence NOUN ejpam-4877 311 10 of of ADP ejpam-4877 311 11 cn cn PROPN ejpam-4877 311 12 . . PUNCT ejpam-4877 312 1 hence hence ADV ejpam-4877 312 2 , , PUNCT ejpam-4877 312 3 γ γ X ejpam-4877 312 4 th th X ejpam-4877 312 5 gr(cn gr(cn PROPN ejpam-4877 312 6 ) ) PUNCT ejpam-4877 312 7 ≥ ≥ NOUN ejpam-4877 313 1 n−1 n−1 PROPN ejpam-4877 313 2 . . PROPN ejpam-4877 314 1 next next ADV ejpam-4877 314 2 , , PUNCT ejpam-4877 314 3 for for ADP ejpam-4877 314 4 n n PRON ejpam-4877 314 5 ∈ ∈ NOUN ejpam-4877 314 6 { { PUNCT ejpam-4877 314 7 11 11 NUM ejpam-4877 314 8 , , PUNCT ejpam-4877 314 9 15 15 NUM ejpam-4877 314 10 , , PUNCT ejpam-4877 314 11 19 19 NUM ejpam-4877 314 12 , , PUNCT ejpam-4877 314 13 . . PUNCT ejpam-4877 314 14 . . PUNCT ejpam-4877 314 15 . . PUNCT ejpam-4877 315 1 } } PUNCT ejpam-4877 315 2 , , PUNCT ejpam-4877 315 3 let let VERB ejpam-4877 315 4 s3 s3 PROPN ejpam-4877 315 5 = = SYM ejpam-4877 315 6 ( ( PUNCT ejpam-4877 315 7 v1 v1 PROPN ejpam-4877 315 8 , , PUNCT ejpam-4877 315 9 v5 v5 NOUN ejpam-4877 315 10 , , PUNCT ejpam-4877 315 11 . . PUNCT ejpam-4877 315 12 . . PUNCT ejpam-4877 316 1 . . PUNCT ejpam-4877 317 1 , , PUNCT ejpam-4877 317 2 vn−2 vn−2 PROPN ejpam-4877 317 3 , , PUNCT ejpam-4877 317 4 v2 v2 PROPN ejpam-4877 317 5 , , PUNCT ejpam-4877 317 6 . . PUNCT ejpam-4877 317 7 . . PUNCT ejpam-4877 318 1 . . PUNCT ejpam-4877 319 1 , , PUNCT ejpam-4877 319 2 vn−1 vn−1 PROPN ejpam-4877 319 3 , , PUNCT ejpam-4877 319 4 v3 v3 PROPN ejpam-4877 319 5 , , PUNCT ejpam-4877 319 6 . . PUNCT ejpam-4877 319 7 . . PUNCT ejpam-4877 319 8 . . PUNCT ejpam-4877 320 1 , , PUNCT ejpam-4877 320 2 vn vn PROPN ejpam-4877 320 3 , , PUNCT ejpam-4877 320 4 v4 v4 PROPN ejpam-4877 320 5 , , PUNCT ejpam-4877 320 6 . . PUNCT ejpam-4877 320 7 . . PUNCT ejpam-4877 320 8 . . PUNCT ejpam-4877 321 1 , , PUNCT ejpam-4877 321 2 vn−7 vn−7 PROPN ejpam-4877 321 3 ) ) PUNCT ejpam-4877 321 4 . . PUNCT ejpam-4877 322 1 then then ADV ejpam-4877 322 2 s3 s3 PROPN ejpam-4877 322 3 is be AUX ejpam-4877 322 4 a a DET ejpam-4877 322 5 grundy grundy PROPN ejpam-4877 322 6 total total NOUN ejpam-4877 322 7 hop hop NOUN ejpam-4877 322 8 dominating dominating NOUN ejpam-4877 322 9 sequence sequence NOUN ejpam-4877 322 10 of of ADP ejpam-4877 322 11 cn cn PROPN ejpam-4877 322 12 . . PUNCT ejpam-4877 323 1 hence hence ADV ejpam-4877 323 2 , , PUNCT ejpam-4877 323 3 γthgr(cn γthgr(cn ADJ ejpam-4877 323 4 ) ) PUNCT ejpam-4877 323 5 ≥ ≥ NOUN ejpam-4877 323 6 n− n− NOUN ejpam-4877 323 7 1 1 NUM ejpam-4877 323 8 . . PUNCT ejpam-4877 324 1 on on ADP ejpam-4877 324 2 the the DET ejpam-4877 324 3 other other ADJ ejpam-4877 324 4 hand hand NOUN ejpam-4877 324 5 , , PUNCT ejpam-4877 324 6 since since SCONJ ejpam-4877 324 7 |ncn(v)| |ncn(v)| ADJ ejpam-4877 324 8 = = SYM ejpam-4877 324 9 2 2 NUM ejpam-4877 324 10 for for ADP ejpam-4877 324 11 every every DET ejpam-4877 324 12 v v NUM ejpam-4877 324 13 ∈ ∈ NOUN ejpam-4877 324 14 v v NOUN ejpam-4877 324 15 ( ( PUNCT ejpam-4877 324 16 cn cn PROPN ejpam-4877 324 17 ) ) PUNCT ejpam-4877 324 18 , , PUNCT ejpam-4877 324 19 it it PRON ejpam-4877 324 20 follows follow VERB ejpam-4877 324 21 that that PRON ejpam-4877 324 22 γthgr(cn γthgr(cn NOUN ejpam-4877 324 23 ) ) PUNCT ejpam-4877 324 24 ≤ ≤ NOUN ejpam-4877 325 1 n n CCONJ ejpam-4877 325 2 − − PROPN ejpam-4877 325 3 1 1 NUM ejpam-4877 325 4 by by ADP ejpam-4877 325 5 corollary corollary ADJ ejpam-4877 325 6 2 2 NUM ejpam-4877 325 7 . . PUNCT ejpam-4877 326 1 therefore therefore ADV ejpam-4877 326 2 , , PUNCT ejpam-4877 326 3 γthgr(cn γthgr(cn NOUN ejpam-4877 326 4 ) ) PUNCT ejpam-4877 326 5 = = PUNCT ejpam-4877 326 6 n− n− NOUN ejpam-4877 326 7 1 1 NUM ejpam-4877 326 8 . . PUNCT ejpam-4877 326 9 theorem theorem NOUN ejpam-4877 326 10 5 5 NUM ejpam-4877 326 11 . . PUNCT ejpam-4877 327 1 let let VERB ejpam-4877 327 2 g g PRON ejpam-4877 327 3 be be AUX ejpam-4877 327 4 a a DET ejpam-4877 327 5 graph graph NOUN ejpam-4877 327 6 of of ADP ejpam-4877 327 7 order order NOUN ejpam-4877 327 8 n n PRON ejpam-4877 327 9 with with ADP ejpam-4877 327 10 γ(c γ(c PROPN ejpam-4877 327 11 ) ) PUNCT ejpam-4877 327 12 ̸= ̸= PROPN ejpam-4877 327 13 1 1 NUM ejpam-4877 327 14 for for ADP ejpam-4877 327 15 each each DET ejpam-4877 327 16 component component NOUN ejpam-4877 327 17 c c PROPN ejpam-4877 327 18 of of ADP ejpam-4877 327 19 g. g. PROPN ejpam-4877 327 20 then then ADV ejpam-4877 327 21 γthgr(g γthgr(g PROPN ejpam-4877 327 22 ) ) PUNCT ejpam-4877 327 23 ≤ ≤ NOUN ejpam-4877 327 24 2γhgr(g 2γhgr(g NUM ejpam-4877 327 25 ) ) PUNCT ejpam-4877 327 26 . . PUNCT ejpam-4877 328 1 proof proof NOUN ejpam-4877 328 2 . . PUNCT ejpam-4877 329 1 let let VERB ejpam-4877 329 2 s s PRON ejpam-4877 329 3 = = PUNCT ejpam-4877 329 4 ( ( PUNCT ejpam-4877 329 5 v1 v1 PROPN ejpam-4877 329 6 , , PUNCT ejpam-4877 329 7 v2 v2 PROPN ejpam-4877 329 8 , , PUNCT ejpam-4877 329 9 · · PUNCT ejpam-4877 329 10 · · PUNCT ejpam-4877 329 11 · · PUNCT ejpam-4877 329 12 , , PUNCT ejpam-4877 329 13 vk vk AUX ejpam-4877 329 14 ) ) PUNCT ejpam-4877 329 15 be be AUX ejpam-4877 329 16 a a DET ejpam-4877 329 17 grundy grundy PROPN ejpam-4877 329 18 total total NOUN ejpam-4877 329 19 hop hop NOUN ejpam-4877 329 20 dominating dominating NOUN ejpam-4877 329 21 sequence sequence NOUN ejpam-4877 329 22 of of ADP ejpam-4877 329 23 g g NOUN ejpam-4877 329 24 , , PUNCT ejpam-4877 329 25 where where SCONJ ejpam-4877 329 26 k k PROPN ejpam-4877 329 27 = = SYM ejpam-4877 329 28 γthgr(g γthgr(g PROPN ejpam-4877 329 29 ) ) PUNCT ejpam-4877 329 30 . . PUNCT ejpam-4877 330 1 we we PRON ejpam-4877 330 2 will will AUX ejpam-4877 330 3 prove prove VERB ejpam-4877 330 4 that that SCONJ ejpam-4877 330 5 at at ADP ejpam-4877 330 6 most most ADJ ejpam-4877 330 7 k/2 k/2 NOUN ejpam-4877 330 8 vertices vertex NOUN ejpam-4877 330 9 can can AUX ejpam-4877 330 10 be be AUX ejpam-4877 330 11 removed remove VERB ejpam-4877 330 12 from from ADP ejpam-4877 330 13 s s PRON ejpam-4877 330 14 in in ADP ejpam-4877 330 15 such such DET ejpam-4877 330 16 a a DET ejpam-4877 330 17 way way NOUN ejpam-4877 330 18 the the DET ejpam-4877 330 19 resulting result VERB ejpam-4877 330 20 sequence sequence NOUN ejpam-4877 330 21 s′ s′ VERB ejpam-4877 330 22 forms form VERB ejpam-4877 330 23 a a DET ejpam-4877 330 24 legal legal ADJ ejpam-4877 330 25 closed closed ADJ ejpam-4877 330 26 hop hop NOUN ejpam-4877 330 27 neighborhood neighborhood NOUN ejpam-4877 330 28 sequence sequence NOUN ejpam-4877 330 29 of of ADP ejpam-4877 330 30 g. g. PROPN ejpam-4877 330 31 notice notice VERB ejpam-4877 330 32 that that SCONJ ejpam-4877 330 33 a a DET ejpam-4877 330 34 vertex vertex NOUN ejpam-4877 330 35 vi vi PROPN ejpam-4877 330 36 ∈ ∈ PROPN ejpam-4877 330 37 ŝ ŝ X ejpam-4877 330 38 prevents prevent VERB ejpam-4877 330 39 s s NOUN ejpam-4877 330 40 from from ADP ejpam-4877 330 41 being be AUX ejpam-4877 330 42 a a DET ejpam-4877 330 43 legal legal ADJ ejpam-4877 330 44 closed close VERB ejpam-4877 330 45 hop hop NOUN ejpam-4877 330 46 neighborhood neighborhood NOUN ejpam-4877 330 47 sequence sequence NOUN ejpam-4877 330 48 only only ADV ejpam-4877 330 49 if if SCONJ ejpam-4877 330 50 n2 n2 ADJ ejpam-4877 330 51 g[vi g[vi PROPN ejpam-4877 330 52 ] ] PUNCT ejpam-4877 330 53 \ \ PROPN ejpam-4877 330 54 ⋃i−1 ⋃i−1 NOUN ejpam-4877 330 55 j=1n j=1n PROPN ejpam-4877 330 56 2 2 NUM ejpam-4877 330 57 g[vj g[vj PROPN ejpam-4877 330 58 ] ] PUNCT ejpam-4877 330 59 = = PUNCT ejpam-4877 330 60 ∅ ∅ NOUN ejpam-4877 330 61 for for ADP ejpam-4877 330 62 each each DET ejpam-4877 330 63 i i PRON ejpam-4877 330 64 ∈ ∈ PROPN ejpam-4877 330 65 { { PUNCT ejpam-4877 330 66 1 1 NUM ejpam-4877 330 67 , , PUNCT ejpam-4877 330 68 . . PUNCT ejpam-4877 330 69 . . PUNCT ejpam-4877 330 70 . . PUNCT ejpam-4877 331 1 , , PUNCT ejpam-4877 331 2 k k X ejpam-4877 331 3 } } PUNCT ejpam-4877 331 4 . . PUNCT ejpam-4877 332 1 since since SCONJ ejpam-4877 332 2 s s PROPN ejpam-4877 332 3 is be AUX ejpam-4877 332 4 a a DET ejpam-4877 332 5 grundy grundy PROPN ejpam-4877 332 6 total total NOUN ejpam-4877 332 7 hop hop NOUN ejpam-4877 332 8 dominating dominating NOUN ejpam-4877 332 9 sequence sequence NOUN ejpam-4877 332 10 , , PUNCT ejpam-4877 332 11 vi vi PROPN ejpam-4877 332 12 hop hop NOUN ejpam-4877 332 13 footprinted footprinte VERB ejpam-4877 332 14 only only ADV ejpam-4877 332 15 vertices vertex NOUN ejpam-4877 332 16 from from ADP ejpam-4877 332 17 s s PRON ejpam-4877 332 18 that that PRON ejpam-4877 332 19 precedes precede VERB ejpam-4877 332 20 vi vi PROPN ejpam-4877 332 21 . . PUNCT ejpam-4877 333 1 that that PRON ejpam-4877 333 2 is be AUX ejpam-4877 333 3 , , PUNCT ejpam-4877 333 4 h−1 h−1 PROPN ejpam-4877 333 5 s s X ejpam-4877 333 6 ( ( PUNCT ejpam-4877 333 7 vi vi NOUN ejpam-4877 333 8 ) ) PUNCT ejpam-4877 333 9 ⊆ ⊆ NUM ejpam-4877 333 10 { { PUNCT ejpam-4877 333 11 v1 v1 NOUN ejpam-4877 333 12 , , PUNCT ejpam-4877 333 13 . . PUNCT ejpam-4877 333 14 . . PUNCT ejpam-4877 334 1 . . PUNCT ejpam-4877 335 1 , , PUNCT ejpam-4877 335 2 vi−1 vi−1 PROPN ejpam-4877 335 3 } } PUNCT ejpam-4877 335 4 , , PUNCT ejpam-4877 335 5 where where SCONJ ejpam-4877 335 6 hs hs INTJ ejpam-4877 335 7 : : PUNCT ejpam-4877 335 8 v v X ejpam-4877 335 9 ( ( PUNCT ejpam-4877 335 10 g g NOUN ejpam-4877 335 11 ) ) PUNCT ejpam-4877 335 12 → → SYM ejpam-4877 335 13 ŝ ŝ X ejpam-4877 335 14 is be AUX ejpam-4877 335 15 a a DET ejpam-4877 335 16 hop hop NOUN ejpam-4877 335 17 footprinter footprinter NOUN ejpam-4877 335 18 function function NOUN ejpam-4877 335 19 , , PUNCT ejpam-4877 335 20 mapping map VERB ejpam-4877 335 21 each each DET ejpam-4877 335 22 vertex vertex NOUN ejpam-4877 335 23 to to ADP ejpam-4877 335 24 its its PRON ejpam-4877 335 25 hop hop NOUN ejpam-4877 335 26 footprinter footprinter NOUN ejpam-4877 335 27 . . PUNCT ejpam-4877 336 1 set set VERB ejpam-4877 336 2 t t PROPN ejpam-4877 337 1 = = SYM ejpam-4877 337 2 { { PUNCT ejpam-4877 337 3 vi vi NOUN ejpam-4877 337 4 ∈ ∈ PROPN ejpam-4877 337 5 ŝ ŝ NOUN ejpam-4877 337 6 : : PUNCT ejpam-4877 337 7 h−1 h−1 PROPN ejpam-4877 337 8 s s X ejpam-4877 337 9 ( ( PUNCT ejpam-4877 337 10 vi vi NOUN ejpam-4877 337 11 ) ) PUNCT ejpam-4877 337 12 ⊆ ⊆ NUM ejpam-4877 337 13 { { PUNCT ejpam-4877 337 14 v1 v1 NOUN ejpam-4877 337 15 , , PUNCT ejpam-4877 337 16 . . PUNCT ejpam-4877 337 17 . . PUNCT ejpam-4877 337 18 . . PUNCT ejpam-4877 338 1 , , PUNCT ejpam-4877 338 2 vi−1 vi−1 PROPN ejpam-4877 338 3 } } PUNCT ejpam-4877 338 4 } } PUNCT ejpam-4877 338 5 . . PUNCT ejpam-4877 339 1 since since SCONJ ejpam-4877 339 2 v1 v1 PROPN ejpam-4877 339 3 /∈ /∈ PUNCT ejpam-4877 340 1 t t PROPN ejpam-4877 340 2 , , PUNCT ejpam-4877 340 3 t t PROPN ejpam-4877 340 4 ̸= ̸= PROPN ejpam-4877 340 5 ŝ. ŝ. NOUN ejpam-4877 340 6 suppose suppose VERB ejpam-4877 340 7 that that SCONJ ejpam-4877 340 8 h−1 h−1 PROPN ejpam-4877 340 9 s s PART ejpam-4877 340 10 ( ( PUNCT ejpam-4877 340 11 vj)∩t vj)∩t X ejpam-4877 340 12 ̸= ̸= PROPN ejpam-4877 340 13 ∅ ∅ NOUN ejpam-4877 340 14 for for ADP ejpam-4877 340 15 some some DET ejpam-4877 340 16 vj vj PROPN ejpam-4877 340 17 ∈ ∈ PROPN ejpam-4877 340 18 t t PROPN ejpam-4877 340 19 . . PUNCT ejpam-4877 341 1 let let VERB ejpam-4877 341 2 vi vi NOUN ejpam-4877 341 3 ∈ ∈ PROPN ejpam-4877 342 1 h−1 h−1 PROPN ejpam-4877 342 2 s s PART ejpam-4877 342 3 ( ( PUNCT ejpam-4877 342 4 vj)∩t vj)∩t X ejpam-4877 342 5 . . PUNCT ejpam-4877 343 1 since since SCONJ ejpam-4877 343 2 vj vj PROPN ejpam-4877 343 3 ∈ ∈ PROPN ejpam-4877 343 4 t t PROPN ejpam-4877 343 5 , , PUNCT ejpam-4877 343 6 the the DET ejpam-4877 343 7 vertex vertex NOUN ejpam-4877 343 8 vi vi PROPN ejpam-4877 343 9 that that PRON ejpam-4877 343 10 is be AUX ejpam-4877 343 11 hop hop NOUN ejpam-4877 343 12 footprinted footprinte VERB ejpam-4877 343 13 by by ADP ejpam-4877 343 14 vj vj INTJ ejpam-4877 343 15 satisties satistie NOUN ejpam-4877 344 1 i i PRON ejpam-4877 344 2 < < X ejpam-4877 344 3 j. j. PROPN ejpam-4877 344 4 since since SCONJ ejpam-4877 344 5 vi vi PROPN ejpam-4877 344 6 ∈ ∈ PROPN ejpam-4877 344 7 t t PROPN ejpam-4877 344 8 , , PUNCT ejpam-4877 344 9 vi vi PROPN ejpam-4877 344 10 hop hop NOUN ejpam-4877 344 11 footprints footprint NOUN ejpam-4877 344 12 some some DET ejpam-4877 344 13 vertex vertex NOUN ejpam-4877 344 14 vt vt NOUN ejpam-4877 344 15 , , PUNCT ejpam-4877 344 16 where where SCONJ ejpam-4877 344 17 t t PROPN ejpam-4877 344 18 < < X ejpam-4877 344 19 i. i. PROPN ejpam-4877 345 1 this this PRON ejpam-4877 345 2 means mean VERB ejpam-4877 345 3 that that SCONJ ejpam-4877 345 4 hs(vt hs(vt NOUN ejpam-4877 345 5 ) ) PUNCT ejpam-4877 345 6 = = SYM ejpam-4877 345 7 vi vi PROPN ejpam-4877 345 8 , , PUNCT ejpam-4877 345 9 where where SCONJ ejpam-4877 345 10 1 1 NUM ejpam-4877 345 11 ≤ ≤ NOUN ejpam-4877 345 12 t t NOUN ejpam-4877 345 13 ≤ ≤ NOUN ejpam-4877 345 14 i− i− ADP ejpam-4877 345 15 1 1 NUM ejpam-4877 345 16 . . PUNCT ejpam-4877 346 1 it it PRON ejpam-4877 346 2 follows follow VERB ejpam-4877 346 3 that that PRON ejpam-4877 346 4 vi vi PROPN ejpam-4877 346 5 /∈ /∈ PUNCT ejpam-4877 346 6 n2 n2 ADJ ejpam-4877 346 7 g(vj g(vj PROPN ejpam-4877 346 8 ) ) PUNCT ejpam-4877 346 9 \∪ \∪ PUNCT ejpam-4877 347 1 j−1 j−1 PROPN ejpam-4877 347 2 k=1n k=1n VERB ejpam-4877 347 3 2 2 NUM ejpam-4877 347 4 g(vk g(vk PROPN ejpam-4877 347 5 ) ) PUNCT ejpam-4877 347 6 , , PUNCT ejpam-4877 347 7 that that ADV ejpam-4877 347 8 is is ADV ejpam-4877 347 9 , , PUNCT ejpam-4877 347 10 hs(vi hs(vi PROPN ejpam-4877 347 11 ) ) PUNCT ejpam-4877 347 12 ̸= ̸= PROPN ejpam-4877 347 13 vj vj INTJ ejpam-4877 347 14 , , PUNCT ejpam-4877 347 15 contrary contrary ADV ejpam-4877 347 16 to to ADP ejpam-4877 347 17 the the DET ejpam-4877 347 18 assumption assumption NOUN ejpam-4877 347 19 that that SCONJ ejpam-4877 347 20 vi vi PROPN ejpam-4877 347 21 ∈ ∈ PROPN ejpam-4877 347 22 h−1 h−1 PROPN ejpam-4877 347 23 s s PART ejpam-4877 347 24 ( ( PUNCT ejpam-4877 347 25 vj vj PROPN ejpam-4877 347 26 ) ) PUNCT ejpam-4877 347 27 . . PUNCT ejpam-4877 348 1 therefore therefore ADV ejpam-4877 348 2 , , PUNCT ejpam-4877 348 3 h h NOUN ejpam-4877 348 4 −1 −1 NOUN ejpam-4877 348 5 s s X ejpam-4877 348 6 ( ( PUNCT ejpam-4877 348 7 vj)∩t vj)∩t X ejpam-4877 348 8 = = SYM ejpam-4877 348 9 ∅ ∅ NOUN ejpam-4877 348 10 for for ADP ejpam-4877 348 11 every every DET ejpam-4877 348 12 vertex vertex NOUN ejpam-4877 348 13 vj vj X ejpam-4877 348 14 ∈ ∈ PROPN ejpam-4877 348 15 t t PROPN ejpam-4877 348 16 . . PUNCT ejpam-4877 349 1 now now ADV ejpam-4877 349 2 , , PUNCT ejpam-4877 349 3 suppose suppose VERB ejpam-4877 349 4 that that SCONJ ejpam-4877 349 5 vi vi PROPN ejpam-4877 349 6 , , PUNCT ejpam-4877 349 7 vj vj PROPN ejpam-4877 349 8 ∈ ∈ PROPN ejpam-4877 349 9 t t PROPN ejpam-4877 349 10 , , PUNCT ejpam-4877 349 11 where where SCONJ ejpam-4877 349 12 i i PRON ejpam-4877 349 13 < < X ejpam-4877 349 14 j. j. PROPN ejpam-4877 349 15 by by ADP ejpam-4877 349 16 definition definition NOUN ejpam-4877 349 17 , , PUNCT ejpam-4877 349 18 h−1 h−1 PROPN ejpam-4877 349 19 s s X ejpam-4877 349 20 ( ( PUNCT ejpam-4877 349 21 vi vi NOUN ejpam-4877 349 22 ) ) PUNCT ejpam-4877 349 23 ⊆ ⊆ NUM ejpam-4877 349 24 { { PUNCT ejpam-4877 349 25 v1 v1 NOUN ejpam-4877 349 26 , , PUNCT ejpam-4877 349 27 . . PUNCT ejpam-4877 349 28 . . PUNCT ejpam-4877 350 1 . . PUNCT ejpam-4877 351 1 , , PUNCT ejpam-4877 351 2 vi−1 vi−1 PROPN ejpam-4877 351 3 } } PUNCT ejpam-4877 351 4 and and CCONJ ejpam-4877 351 5 h−1 h−1 PROPN ejpam-4877 351 6 s s PART ejpam-4877 351 7 ( ( PUNCT ejpam-4877 351 8 vj vj INTJ ejpam-4877 351 9 ) ) PUNCT ejpam-4877 351 10 ⊆ ⊆ NUM ejpam-4877 351 11 { { PUNCT ejpam-4877 351 12 v1 v1 NOUN ejpam-4877 351 13 , , PUNCT ejpam-4877 351 14 . . PUNCT ejpam-4877 351 15 . . PUNCT ejpam-4877 352 1 . . PUNCT ejpam-4877 353 1 , , PUNCT ejpam-4877 353 2 vj−1 vj−1 PROPN ejpam-4877 353 3 } } PUNCT ejpam-4877 353 4 . . PUNCT ejpam-4877 354 1 since since SCONJ ejpam-4877 354 2 every every DET ejpam-4877 354 3 vertex vertex NOUN ejpam-4877 354 4 is be AUX ejpam-4877 354 5 hop hop NOUN ejpam-4877 354 6 footprinted footprinte VERB ejpam-4877 354 7 by by ADP ejpam-4877 354 8 a a DET ejpam-4877 354 9 unique unique ADJ ejpam-4877 354 10 vertex vertex NOUN ejpam-4877 354 11 in in ADP ejpam-4877 354 12 ŝ ŝ PROPN ejpam-4877 354 13 , , PUNCT ejpam-4877 354 14 it it PRON ejpam-4877 354 15 follows follow VERB ejpam-4877 354 16 that that SCONJ ejpam-4877 354 17 h−1 h−1 PROPN ejpam-4877 354 18 s s X ejpam-4877 354 19 ( ( PUNCT ejpam-4877 354 20 vi vi NOUN ejpam-4877 354 21 ) ) PUNCT ejpam-4877 354 22 ∩ ∩ NOUN ejpam-4877 355 1 h−1 h−1 PROPN ejpam-4877 355 2 s s X ejpam-4877 355 3 ( ( PUNCT ejpam-4877 355 4 vj vj INTJ ejpam-4877 355 5 ) ) PUNCT ejpam-4877 355 6 = = PUNCT ejpam-4877 355 7 ∅. ∅. NOUN ejpam-4877 355 8 since since SCONJ ejpam-4877 355 9 h−1 h−1 PROPN ejpam-4877 355 10 s s PART ejpam-4877 355 11 ( ( PUNCT ejpam-4877 355 12 vj vj ADJ ejpam-4877 355 13 ) ) PUNCT ejpam-4877 355 14 ∩ ∩ ADJ ejpam-4877 355 15 t t NOUN ejpam-4877 355 16 = = PUNCT ejpam-4877 355 17 ∅ ∅ NOUN ejpam-4877 355 18 for for ADP ejpam-4877 355 19 every every DET ejpam-4877 355 20 vertex vertex NOUN ejpam-4877 355 21 vj vj X ejpam-4877 355 22 ∈ ∈ PROPN ejpam-4877 355 23 t t PROPN ejpam-4877 355 24 , , PUNCT ejpam-4877 355 25 { { PUNCT ejpam-4877 355 26 h−1 h−1 PROPN ejpam-4877 355 27 s s X ejpam-4877 355 28 ( ( PUNCT ejpam-4877 355 29 vi vi NOUN ejpam-4877 355 30 ) ) PUNCT ejpam-4877 355 31 : : PUNCT ejpam-4877 355 32 vi vi PROPN ejpam-4877 355 33 ∈ ∈ PROPN ejpam-4877 355 34 t t PROPN ejpam-4877 355 35 } } PUNCT ejpam-4877 355 36 forms form VERB ejpam-4877 355 37 a a DET ejpam-4877 355 38 partition partition NOUN ejpam-4877 355 39 of of ADP ejpam-4877 355 40 a a DET ejpam-4877 355 41 subset subset NOUN ejpam-4877 355 42 of of ADP ejpam-4877 355 43 ŝ ŝ X ejpam-4877 355 44 \ \ PROPN ejpam-4877 355 45 t t PROPN ejpam-4877 355 46 . . PUNCT ejpam-4877 356 1 note note VERB ejpam-4877 356 2 that that SCONJ ejpam-4877 356 3 for for ADP ejpam-4877 356 4 each each DET ejpam-4877 356 5 vi vi PROPN ejpam-4877 356 6 ∈ ∈ PROPN ejpam-4877 356 7 t t NOUN ejpam-4877 356 8 , , PUNCT ejpam-4877 356 9 |h−1 |h−1 NOUN ejpam-4877 356 10 s s PART ejpam-4877 356 11 ( ( PUNCT ejpam-4877 356 12 vi)| vi)| NOUN ejpam-4877 356 13 ≥ ≥ NOUN ejpam-4877 356 14 1 1 NUM ejpam-4877 356 15 , , PUNCT ejpam-4877 356 16 and and CCONJ ejpam-4877 356 17 so so ADV ejpam-4877 356 18 |t |t VERB ejpam-4877 357 1 | | ADV ejpam-4877 357 2 ≤ ≤ PUNCT ejpam-4877 358 1 | | ADV ejpam-4877 358 2 ⋃ ⋃ NOUN ejpam-4877 358 3 vi∈t vi∈t VERB ejpam-4877 359 1 h−1 h−1 PROPN ejpam-4877 359 2 s s PART ejpam-4877 359 3 ( ( PUNCT ejpam-4877 359 4 vi)| vi)| NOUN ejpam-4877 359 5 ≤ ≤ NOUN ejpam-4877 359 6 |ŝ| |ŝ| PUNCT ejpam-4877 359 7 − − NUM ejpam-4877 359 8 |t |t NOUN ejpam-4877 359 9 | | ADV ejpam-4877 359 10 implying imply VERB ejpam-4877 359 11 that that SCONJ ejpam-4877 359 12 2|t 2|t PROPN ejpam-4877 359 13 | | ADV ejpam-4877 359 14 ≤ ≤ VERB ejpam-4877 359 15 |ŝ| |ŝ| PROPN ejpam-4877 359 16 = = SYM ejpam-4877 359 17 k. k. PROPN ejpam-4877 359 18 hence hence ADV ejpam-4877 359 19 , , PUNCT ejpam-4877 359 20 |t |t VERB ejpam-4877 360 1 | | ADV ejpam-4877 360 2 ≤ ≤ ADV ejpam-4877 360 3 k k NOUN ejpam-4877 360 4 2 2 X ejpam-4877 360 5 . . PUNCT ejpam-4877 361 1 let let VERB ejpam-4877 361 2 s′ s′ NOUN ejpam-4877 361 3 be be AUX ejpam-4877 361 4 a a DET ejpam-4877 361 5 sequence sequence NOUN ejpam-4877 361 6 obtained obtain VERB ejpam-4877 361 7 from from ADP ejpam-4877 361 8 s s PRON ejpam-4877 361 9 by by ADP ejpam-4877 361 10 deleting delete VERB ejpam-4877 361 11 vertices vertex NOUN ejpam-4877 361 12 from from ADP ejpam-4877 361 13 t t PROPN ejpam-4877 361 14 . . PUNCT ejpam-4877 362 1 then then ADV ejpam-4877 362 2 s′ s′ PROPN ejpam-4877 362 3 is be AUX ejpam-4877 362 4 a a DET ejpam-4877 362 5 legal legal ADJ ejpam-4877 362 6 closed close VERB ejpam-4877 362 7 hop hop NOUN ejpam-4877 362 8 neighborhood neighborhood NOUN ejpam-4877 362 9 sequence sequence NOUN ejpam-4877 362 10 of of ADP ejpam-4877 362 11 g. g. PROPN ejpam-4877 362 12 thus thus ADV ejpam-4877 362 13 , , PUNCT ejpam-4877 362 14 γhgr(g γhgr(g PROPN ejpam-4877 362 15 ) ) PUNCT ejpam-4877 362 16 ≥ ≥ NOUN ejpam-4877 362 17 |s′| |s′| NOUN ejpam-4877 362 18 = = PUNCT ejpam-4877 363 1 k k PROPN ejpam-4877 363 2 − − PROPN ejpam-4877 363 3 |t |t INTJ ejpam-4877 364 1 | | ADV ejpam-4877 364 2 ≥ ≥ NOUN ejpam-4877 365 1 k k NOUN ejpam-4877 365 2 − − PUNCT ejpam-4877 366 1 k k NOUN ejpam-4877 366 2 2 2 X ejpam-4877 366 3 = = SYM ejpam-4877 366 4 k k NOUN ejpam-4877 366 5 2 2 X ejpam-4877 366 6 = = SYM ejpam-4877 366 7 1 1 NUM ejpam-4877 366 8 2γ 2γ NUM ejpam-4877 366 9 th th X ejpam-4877 366 10 gr(g gr(g PROPN ejpam-4877 366 11 ) ) PUNCT ejpam-4877 366 12 . . PUNCT ejpam-4877 367 1 consequently consequently ADV ejpam-4877 367 2 , , PUNCT ejpam-4877 367 3 γthgr(g γthgr(g NOUN ejpam-4877 367 4 ) ) PUNCT ejpam-4877 367 5 ≤ ≤ NOUN ejpam-4877 367 6 2γhgr(g 2γhgr(g NUM ejpam-4877 367 7 ) ) PUNCT ejpam-4877 367 8 . . PUNCT ejpam-4877 368 1 j.a j.a PROPN ejpam-4877 368 2 . . PROPN ejpam-4877 368 3 hassan hassan PROPN ejpam-4877 368 4 , , PUNCT ejpam-4877 368 5 s. s. PROPN ejpam-4877 368 6 canoy canoy PROPN ejpam-4877 368 7 / / SYM ejpam-4877 368 8 eur eur PROPN ejpam-4877 368 9 . . PUNCT ejpam-4877 369 1 j. j. PROPN ejpam-4877 369 2 pure pure PROPN ejpam-4877 369 3 appl appl PROPN ejpam-4877 369 4 . . PROPN ejpam-4877 369 5 math math PROPN ejpam-4877 369 6 , , PUNCT ejpam-4877 369 7 16 16 NUM ejpam-4877 369 8 ( ( PUNCT ejpam-4877 369 9 4 4 NUM ejpam-4877 369 10 ) ) PUNCT ejpam-4877 369 11 ( ( PUNCT ejpam-4877 369 12 2023 2023 NUM ejpam-4877 369 13 ) ) PUNCT ejpam-4877 369 14 , , PUNCT ejpam-4877 369 15 2597 2597 NUM ejpam-4877 369 16 - - SYM ejpam-4877 369 17 2612 2612 NUM ejpam-4877 369 18 2605 2605 NUM ejpam-4877 369 19 remark remark NOUN ejpam-4877 369 20 1 1 NUM ejpam-4877 369 21 . . PUNCT ejpam-4877 370 1 the the DET ejpam-4877 370 2 bound bind VERB ejpam-4877 370 3 given give VERB ejpam-4877 370 4 in in ADP ejpam-4877 370 5 theorem theorem ADJ ejpam-4877 370 6 5 5 NUM ejpam-4877 370 7 is be AUX ejpam-4877 370 8 tight tight ADJ ejpam-4877 370 9 . . PUNCT ejpam-4877 371 1 to to PART ejpam-4877 371 2 see see VERB ejpam-4877 371 3 this this PRON ejpam-4877 371 4 , , PUNCT ejpam-4877 371 5 consider consider VERB ejpam-4877 371 6 c4 c4 NOUN ejpam-4877 371 7 in in ADP ejpam-4877 371 8 fig fig NOUN ejpam-4877 371 9 . . PUNCT ejpam-4877 372 1 1 1 X ejpam-4877 372 2 . . X ejpam-4877 372 3 let let VERB ejpam-4877 372 4 s s PRON ejpam-4877 372 5 = = PUNCT ejpam-4877 372 6 ( ( PUNCT ejpam-4877 372 7 u1 u1 PROPN ejpam-4877 372 8 , , PUNCT ejpam-4877 372 9 u2 u2 NOUN ejpam-4877 372 10 , , PUNCT ejpam-4877 372 11 u3 u3 NOUN ejpam-4877 372 12 , , PUNCT ejpam-4877 372 13 u4 u4 PROPN ejpam-4877 372 14 ) ) PUNCT ejpam-4877 372 15 . . PUNCT ejpam-4877 373 1 then then ADV ejpam-4877 373 2 s s VERB ejpam-4877 373 3 is be AUX ejpam-4877 373 4 a a DET ejpam-4877 373 5 γthgr γthgr ADJ ejpam-4877 373 6 - - PUNCT ejpam-4877 373 7 sequence sequence NOUN ejpam-4877 373 8 of of ADP ejpam-4877 373 9 c4 c4 NOUN ejpam-4877 373 10 . . PUNCT ejpam-4877 374 1 thus thus ADV ejpam-4877 374 2 , , PUNCT ejpam-4877 374 3 γthgr(c4 γthgr(c4 NOUN ejpam-4877 374 4 ) ) PUNCT ejpam-4877 375 1 = = PUNCT ejpam-4877 376 1 4 4 X ejpam-4877 376 2 . . PUNCT ejpam-4877 376 3 next next ADV ejpam-4877 376 4 , , PUNCT ejpam-4877 376 5 let let VERB ejpam-4877 376 6 s∗ s∗ PROPN ejpam-4877 376 7 = = SYM ejpam-4877 376 8 ( ( PUNCT ejpam-4877 376 9 u1 u1 PROPN ejpam-4877 376 10 , , PUNCT ejpam-4877 376 11 u2 u2 PROPN ejpam-4877 376 12 ) ) PUNCT ejpam-4877 376 13 . . PUNCT ejpam-4877 377 1 then then ADV ejpam-4877 377 2 s∗ s∗ PROPN ejpam-4877 377 3 is be AUX ejpam-4877 377 4 a a DET ejpam-4877 377 5 γhgr γhgr ADJ ejpam-4877 377 6 - - PUNCT ejpam-4877 377 7 sequence sequence NOUN ejpam-4877 377 8 of of ADP ejpam-4877 377 9 c4 c4 NOUN ejpam-4877 377 10 . . PUNCT ejpam-4877 378 1 hence hence ADV ejpam-4877 378 2 , , PUNCT ejpam-4877 378 3 γhgr(c4 γhgr(c4 NOUN ejpam-4877 378 4 ) ) PUNCT ejpam-4877 378 5 = = SYM ejpam-4877 379 1 2 2 X ejpam-4877 379 2 . . PUNCT ejpam-4877 379 3 consequently consequently ADV ejpam-4877 379 4 , , PUNCT ejpam-4877 379 5 γthgr(c4 γthgr(c4 NOUN ejpam-4877 379 6 ) ) PUNCT ejpam-4877 379 7 = = SYM ejpam-4877 379 8 4 4 NUM ejpam-4877 379 9 = = SYM ejpam-4877 379 10 2γhgr(c4 2γhgr(c4 NOUN ejpam-4877 379 11 ) ) PUNCT ejpam-4877 379 12 . . PUNCT ejpam-4877 380 1 c4 c4 NOUN ejpam-4877 380 2 : : PUNCT ejpam-4877 380 3 u2 u2 PROPN ejpam-4877 380 4 u3 u3 PROPN ejpam-4877 380 5 u4 u4 PROPN ejpam-4877 380 6 u1 u1 PROPN ejpam-4877 380 7 figure figure NOUN ejpam-4877 380 8 1 1 NUM ejpam-4877 380 9 : : PUNCT ejpam-4877 380 10 a a DET ejpam-4877 380 11 graph graph NOUN ejpam-4877 380 12 c4 c4 NOUN ejpam-4877 380 13 with with ADP ejpam-4877 380 14 γth γth NOUN ejpam-4877 380 15 gr(c4 gr(c4 NOUN ejpam-4877 380 16 ) ) PUNCT ejpam-4877 380 17 = = SYM ejpam-4877 380 18 4 4 NUM ejpam-4877 380 19 = = SYM ejpam-4877 380 20 2γh 2γh ADJ ejpam-4877 380 21 gr(c4 gr(c4 NOUN ejpam-4877 380 22 ) ) PUNCT ejpam-4877 380 23 remark remark NOUN ejpam-4877 380 24 2 2 NUM ejpam-4877 380 25 . . PUNCT ejpam-4877 381 1 let let VERB ejpam-4877 381 2 g g PRON ejpam-4877 381 3 be be AUX ejpam-4877 381 4 a a DET ejpam-4877 381 5 graph graph NOUN ejpam-4877 381 6 of of ADP ejpam-4877 381 7 order order NOUN ejpam-4877 381 8 n n PRON ejpam-4877 381 9 with with ADP ejpam-4877 381 10 γ(c γ(c PROPN ejpam-4877 381 11 ) ) PUNCT ejpam-4877 381 12 ̸= ̸= PROPN ejpam-4877 381 13 1 1 NUM ejpam-4877 381 14 for for ADP ejpam-4877 381 15 each each DET ejpam-4877 381 16 component component NOUN ejpam-4877 381 17 c c PROPN ejpam-4877 381 18 of of ADP ejpam-4877 381 19 g. g. PROPN ejpam-4877 381 20 then then ADV ejpam-4877 381 21 γthgr(g γthgr(g PROPN ejpam-4877 381 22 ) ) PUNCT ejpam-4877 381 23 ≥ ≥ X ejpam-4877 381 24 γhgr(g γhgr(g NOUN ejpam-4877 381 25 ) ) PUNCT ejpam-4877 381 26 does do AUX ejpam-4877 381 27 not not PART ejpam-4877 381 28 hold hold VERB ejpam-4877 381 29 in in ADP ejpam-4877 381 30 general general ADJ ejpam-4877 381 31 . . PUNCT ejpam-4877 382 1 to to PART ejpam-4877 382 2 see see VERB ejpam-4877 382 3 this this PRON ejpam-4877 382 4 , , PUNCT ejpam-4877 382 5 consider consider VERB ejpam-4877 382 6 k5 k5 PROPN ejpam-4877 382 7 ◦ ◦ PROPN ejpam-4877 382 8 k2 k2 PROPN ejpam-4877 382 9 in in ADP ejpam-4877 382 10 fig fig NOUN ejpam-4877 382 11 . . PUNCT ejpam-4877 383 1 2 2 X ejpam-4877 383 2 . . X ejpam-4877 383 3 let let VERB ejpam-4877 383 4 s s PRON ejpam-4877 383 5 = = PUNCT ejpam-4877 383 6 ( ( PUNCT ejpam-4877 383 7 v1 v1 PROPN ejpam-4877 383 8 , , PUNCT ejpam-4877 383 9 v2 v2 NOUN ejpam-4877 383 10 , , PUNCT ejpam-4877 383 11 . . PUNCT ejpam-4877 383 12 . . PUNCT ejpam-4877 384 1 . . PUNCT ejpam-4877 385 1 , , PUNCT ejpam-4877 385 2 v10 v10 PROPN ejpam-4877 385 3 ) ) PUNCT ejpam-4877 385 4 . . PUNCT ejpam-4877 386 1 then then ADV ejpam-4877 386 2 s s VERB ejpam-4877 386 3 is be AUX ejpam-4877 386 4 a a DET ejpam-4877 386 5 γhgrsequence γhgrsequence NOUN ejpam-4877 386 6 of of ADP ejpam-4877 386 7 k5 k5 PROPN ejpam-4877 386 8 ◦ ◦ NOUN ejpam-4877 386 9 k2 k2 PROPN ejpam-4877 386 10 , , PUNCT ejpam-4877 386 11 that that ADV ejpam-4877 386 12 is is ADV ejpam-4877 386 13 , , PUNCT ejpam-4877 386 14 γ γ PROPN ejpam-4877 386 15 h h PROPN ejpam-4877 386 16 gr(k5 gr(k5 PROPN ejpam-4877 386 17 ◦ ◦ NOUN ejpam-4877 386 18 k2 k2 NOUN ejpam-4877 386 19 ) ) PUNCT ejpam-4877 386 20 = = SYM ejpam-4877 386 21 10 10 NUM ejpam-4877 386 22 . . PUNCT ejpam-4877 387 1 next next ADJ ejpam-4877 387 2 , , PUNCT ejpam-4877 387 3 let let VERB ejpam-4877 387 4 s′ s′ ADJ ejpam-4877 387 5 = = SYM ejpam-4877 387 6 ( ( PUNCT ejpam-4877 387 7 v1 v1 PROPN ejpam-4877 387 8 , , PUNCT ejpam-4877 387 9 v12 v12 VERB ejpam-4877 387 10 , , PUNCT ejpam-4877 387 11 v3 v3 PROPN ejpam-4877 387 12 , , PUNCT ejpam-4877 387 13 v13 v13 PROPN ejpam-4877 387 14 ) ) PUNCT ejpam-4877 387 15 . . PUNCT ejpam-4877 388 1 then then ADV ejpam-4877 388 2 s′ s′ PROPN ejpam-4877 388 3 is be AUX ejpam-4877 388 4 a a DET ejpam-4877 388 5 γthgr γthgr ADJ ejpam-4877 388 6 - - PUNCT ejpam-4877 388 7 sequence sequence NOUN ejpam-4877 388 8 of of ADP ejpam-4877 388 9 k5 k5 PROPN ejpam-4877 388 10 ◦ ◦ NOUN ejpam-4877 388 11 k2 k2 PROPN ejpam-4877 388 12 . . PUNCT ejpam-4877 389 1 thus thus ADV ejpam-4877 389 2 , , PUNCT ejpam-4877 389 3 γ γ X ejpam-4877 389 4 th th X ejpam-4877 389 5 gr(k5 gr(k5 NOUN ejpam-4877 389 6 ◦ ◦ NOUN ejpam-4877 389 7 k2 k2 NOUN ejpam-4877 389 8 ) ) PUNCT ejpam-4877 389 9 = = SYM ejpam-4877 390 1 4 4 X ejpam-4877 390 2 . . X ejpam-4877 390 3 k5 k5 PROPN ejpam-4877 390 4 ◦ ◦ PROPN ejpam-4877 390 5 k2 k2 PROPN ejpam-4877 390 6 : : PUNCT ejpam-4877 390 7 v2 v2 PROPN ejpam-4877 390 8 v3 v3 PROPN ejpam-4877 390 9 v1 v1 PROPN ejpam-4877 390 10 v4 v4 PROPN ejpam-4877 390 11 v5 v5 PROPN ejpam-4877 390 12 v7 v7 VERB ejpam-4877 390 13 v8 v8 PROPN ejpam-4877 390 14 v6 v6 NOUN ejpam-4877 390 15 v10 v10 PROPN ejpam-4877 390 16 v9 v9 PROPN ejpam-4877 390 17 v11 v11 NOUN ejpam-4877 390 18 v12 v12 VERB ejpam-4877 390 19 v13 v13 NOUN ejpam-4877 390 20 v14v15 v14v15 NOUN ejpam-4877 390 21 figure figure NOUN ejpam-4877 390 22 2 2 NUM ejpam-4877 390 23 : : PUNCT ejpam-4877 390 24 a a DET ejpam-4877 390 25 graph graph NOUN ejpam-4877 390 26 k5 k5 PROPN ejpam-4877 390 27 ◦ ◦ NOUN ejpam-4877 390 28 k2 k2 NOUN ejpam-4877 390 29 with with ADP ejpam-4877 390 30 γth γth PROPN ejpam-4877 390 31 gr(k5 gr(k5 PROPN ejpam-4877 390 32 ◦ ◦ NOUN ejpam-4877 390 33 k2 k2 NOUN ejpam-4877 390 34 ) ) PUNCT ejpam-4877 390 35 = = SYM ejpam-4877 390 36 4 4 NUM ejpam-4877 390 37 < < SYM ejpam-4877 390 38 10 10 NUM ejpam-4877 390 39 = = SYM ejpam-4877 390 40 γh γh PROPN ejpam-4877 390 41 gr(k5 gr(k5 PROPN ejpam-4877 390 42 ◦ ◦ NOUN ejpam-4877 390 43 k2 k2 NOUN ejpam-4877 390 44 ) ) PUNCT ejpam-4877 390 45 lemma lemma PROPN ejpam-4877 390 46 1 1 NUM ejpam-4877 390 47 . . PUNCT ejpam-4877 391 1 let let VERB ejpam-4877 391 2 g g PRON ejpam-4877 391 3 be be AUX ejpam-4877 391 4 a a DET ejpam-4877 391 5 non non ADJ ejpam-4877 391 6 - - ADJ ejpam-4877 391 7 trivial trivial ADJ ejpam-4877 391 8 connected connected ADJ ejpam-4877 391 9 graph graph NOUN ejpam-4877 391 10 and and CCONJ ejpam-4877 391 11 let let VERB ejpam-4877 391 12 g1 g1 PROPN ejpam-4877 391 13 and and CCONJ ejpam-4877 391 14 g2 g2 PROPN ejpam-4877 391 15 be be VERB ejpam-4877 391 16 two two NUM ejpam-4877 391 17 copies copy NOUN ejpam-4877 391 18 of of ADP ejpam-4877 391 19 g g NOUN ejpam-4877 391 20 in in ADP ejpam-4877 391 21 the the DET ejpam-4877 391 22 graph graph NOUN ejpam-4877 391 23 s(g s(g PROPN ejpam-4877 391 24 ) ) PUNCT ejpam-4877 391 25 . . PUNCT ejpam-4877 392 1 if if SCONJ ejpam-4877 392 2 v v NUM ejpam-4877 392 3 ∈ ∈ PROPN ejpam-4877 392 4 v v NOUN ejpam-4877 392 5 ( ( PUNCT ejpam-4877 392 6 g1 g1 PROPN ejpam-4877 392 7 ) ) PUNCT ejpam-4877 392 8 and and CCONJ ejpam-4877 392 9 v′ v′ PROPN ejpam-4877 392 10 ∈ ∈ PROPN ejpam-4877 392 11 v v NOUN ejpam-4877 392 12 ( ( PUNCT ejpam-4877 392 13 g2 g2 PROPN ejpam-4877 392 14 ) ) PUNCT ejpam-4877 392 15 is be AUX ejpam-4877 392 16 the the DET ejpam-4877 392 17 corresponding corresponding ADJ ejpam-4877 392 18 vertex vertex NOUN ejpam-4877 392 19 of of ADP ejpam-4877 392 20 v v NOUN ejpam-4877 392 21 , , PUNCT ejpam-4877 392 22 then then ADV ejpam-4877 392 23 j.a j.a PROPN ejpam-4877 392 24 . . PROPN ejpam-4877 392 25 hassan hassan PROPN ejpam-4877 392 26 , , PUNCT ejpam-4877 392 27 s. s. PROPN ejpam-4877 392 28 canoy canoy PROPN ejpam-4877 392 29 / / SYM ejpam-4877 392 30 eur eur PROPN ejpam-4877 392 31 . . PUNCT ejpam-4877 393 1 j. j. PROPN ejpam-4877 393 2 pure pure PROPN ejpam-4877 393 3 appl appl PROPN ejpam-4877 393 4 . . PROPN ejpam-4877 393 5 math math PROPN ejpam-4877 393 6 , , PUNCT ejpam-4877 393 7 16 16 NUM ejpam-4877 393 8 ( ( PUNCT ejpam-4877 393 9 4 4 NUM ejpam-4877 393 10 ) ) PUNCT ejpam-4877 393 11 ( ( PUNCT ejpam-4877 393 12 2023 2023 NUM ejpam-4877 393 13 ) ) PUNCT ejpam-4877 393 14 , , PUNCT ejpam-4877 393 15 2597 2597 NUM ejpam-4877 393 16 - - SYM ejpam-4877 393 17 2612 2612 NUM ejpam-4877 393 18 2606 2606 NUM ejpam-4877 393 19 ( ( PUNCT ejpam-4877 393 20 i i NOUN ejpam-4877 393 21 ) ) PUNCT ejpam-4877 393 22 n2 n2 NOUN ejpam-4877 393 23 s(g)(v s(g)(v PUNCT ejpam-4877 393 24 ) ) PUNCT ejpam-4877 393 25 = = SYM ejpam-4877 393 26 n2 n2 PROPN ejpam-4877 393 27 g1 g1 PROPN ejpam-4877 393 28 ( ( PUNCT ejpam-4877 393 29 v v NOUN ejpam-4877 393 30 ) ) PUNCT ejpam-4877 394 1 ∪n2 ∪n2 PROPN ejpam-4877 394 2 g2 g2 PROPN ejpam-4877 395 1 [ [ X ejpam-4877 395 2 v′ v′ X ejpam-4877 395 3 ] ] PUNCT ejpam-4877 395 4 and and CCONJ ejpam-4877 395 5 ( ( PUNCT ejpam-4877 395 6 ii ii NOUN ejpam-4877 395 7 ) ) PUNCT ejpam-4877 395 8 n2 n2 NOUN ejpam-4877 395 9 s(g)(v s(g)(v PUNCT ejpam-4877 395 10 ′ ′ NUM ejpam-4877 395 11 ) ) PUNCT ejpam-4877 395 12 = = SYM ejpam-4877 395 13 n2 n2 ADJ ejpam-4877 395 14 g1 g1 PROPN ejpam-4877 395 15 [ [ X ejpam-4877 395 16 v v X ejpam-4877 395 17 ] ] X ejpam-4877 395 18 ∪n2 ∪n2 PROPN ejpam-4877 395 19 g2 g2 PROPN ejpam-4877 395 20 ( ( PUNCT ejpam-4877 395 21 v′ v′ PROPN ejpam-4877 395 22 ) ) PUNCT ejpam-4877 395 23 . . PUNCT ejpam-4877 396 1 proof proof NOUN ejpam-4877 396 2 . . PUNCT ejpam-4877 397 1 ( ( PUNCT ejpam-4877 397 2 i i NOUN ejpam-4877 397 3 ) ) PUNCT ejpam-4877 397 4 let let VERB ejpam-4877 397 5 a a DET ejpam-4877 397 6 ∈ ∈ PROPN ejpam-4877 397 7 n2 n2 NOUN ejpam-4877 397 8 s(g)(v s(g)(v NOUN ejpam-4877 397 9 ) ) PUNCT ejpam-4877 397 10 . . PUNCT ejpam-4877 398 1 then then ADV ejpam-4877 398 2 ds(g)(a ds(g)(a PROPN ejpam-4877 398 3 , , PUNCT ejpam-4877 398 4 v v NOUN ejpam-4877 398 5 ) ) PUNCT ejpam-4877 398 6 = = SYM ejpam-4877 399 1 2 2 X ejpam-4877 399 2 . . X ejpam-4877 399 3 if if SCONJ ejpam-4877 399 4 a a DET ejpam-4877 399 5 ∈ ∈ PROPN ejpam-4877 399 6 v v NOUN ejpam-4877 399 7 ( ( PUNCT ejpam-4877 399 8 g1 g1 PROPN ejpam-4877 399 9 ) ) PUNCT ejpam-4877 399 10 , , PUNCT ejpam-4877 399 11 then then ADV ejpam-4877 399 12 a a DET ejpam-4877 399 13 ∈ ∈ PROPN ejpam-4877 399 14 n2 n2 NOUN ejpam-4877 399 15 g1 g1 PROPN ejpam-4877 399 16 ( ( PUNCT ejpam-4877 399 17 v v NOUN ejpam-4877 399 18 ) ) PUNCT ejpam-4877 399 19 . . PUNCT ejpam-4877 400 1 suppose suppose VERB ejpam-4877 400 2 a a DET ejpam-4877 400 3 ∈ ∈ PROPN ejpam-4877 400 4 v v NOUN ejpam-4877 400 5 ( ( PUNCT ejpam-4877 400 6 g2 g2 PROPN ejpam-4877 400 7 ) ) PUNCT ejpam-4877 400 8 . . PUNCT ejpam-4877 401 1 by by ADP ejpam-4877 401 2 assumption assumption NOUN ejpam-4877 401 3 , , PUNCT ejpam-4877 401 4 it it PRON ejpam-4877 401 5 follows follow VERB ejpam-4877 401 6 that that SCONJ ejpam-4877 401 7 dg2(a dg2(a PROPN ejpam-4877 401 8 , , PUNCT ejpam-4877 401 9 v v NOUN ejpam-4877 401 10 ′ ′ NOUN ejpam-4877 401 11 ) ) PUNCT ejpam-4877 401 12 = = SYM ejpam-4877 402 1 2 2 X ejpam-4877 402 2 . . PUNCT ejpam-4877 403 1 thus thus ADV ejpam-4877 403 2 , , PUNCT ejpam-4877 403 3 a a DET ejpam-4877 403 4 ∈ ∈ PROPN ejpam-4877 403 5 n2 n2 NOUN ejpam-4877 403 6 g2 g2 PROPN ejpam-4877 403 7 ( ( PUNCT ejpam-4877 403 8 v′ v′ PROPN ejpam-4877 403 9 ) ) PUNCT ejpam-4877 403 10 . . PUNCT ejpam-4877 404 1 hence hence ADV ejpam-4877 404 2 , , PUNCT ejpam-4877 404 3 n2 n2 PROPN ejpam-4877 404 4 s(g)(v s(g)(v NUM ejpam-4877 404 5 ) ) PUNCT ejpam-4877 404 6 ⊆ ⊆ NUM ejpam-4877 404 7 n2 n2 ADJ ejpam-4877 404 8 g1 g1 PROPN ejpam-4877 404 9 ( ( PUNCT ejpam-4877 404 10 v)∪n2 v)∪n2 PROPN ejpam-4877 404 11 g2 g2 PROPN ejpam-4877 405 1 [ [ X ejpam-4877 405 2 v′ v′ X ejpam-4877 405 3 ] ] PUNCT ejpam-4877 405 4 . . PUNCT ejpam-4877 406 1 clearly clearly ADV ejpam-4877 406 2 , , PUNCT ejpam-4877 406 3 n2 n2 ADJ ejpam-4877 406 4 g1 g1 PROPN ejpam-4877 406 5 ( ( PUNCT ejpam-4877 406 6 v)∪n2 v)∪n2 PROPN ejpam-4877 406 7 g2 g2 PROPN ejpam-4877 407 1 [ [ X ejpam-4877 407 2 v′ v′ X ejpam-4877 407 3 ] ] X ejpam-4877 407 4 ⊆ ⊆ NUM ejpam-4877 407 5 n2 n2 NOUN ejpam-4877 407 6 s(g)(v s(g)(v NUM ejpam-4877 407 7 ) ) PUNCT ejpam-4877 407 8 . . PUNCT ejpam-4877 408 1 consequently consequently ADV ejpam-4877 408 2 , , PUNCT ejpam-4877 408 3 n2 n2 ADJ ejpam-4877 408 4 s(g)(v s(g)(v NUM ejpam-4877 408 5 ) ) PUNCT ejpam-4877 408 6 = = SYM ejpam-4877 408 7 n2 n2 PROPN ejpam-4877 408 8 g1 g1 PROPN ejpam-4877 408 9 ( ( PUNCT ejpam-4877 408 10 v v NOUN ejpam-4877 408 11 ) ) PUNCT ejpam-4877 408 12 ∪n2 ∪n2 PROPN ejpam-4877 408 13 g2 g2 PROPN ejpam-4877 409 1 [ [ X ejpam-4877 409 2 v′ v′ X ejpam-4877 409 3 ] ] PUNCT ejpam-4877 409 4 . . PUNCT ejpam-4877 410 1 ( ( PUNCT ejpam-4877 410 2 ii ii NOUN ejpam-4877 410 3 ) ) PUNCT ejpam-4877 410 4 can can AUX ejpam-4877 410 5 be be AUX ejpam-4877 410 6 proved prove VERB ejpam-4877 410 7 similarly similarly ADV ejpam-4877 410 8 . . PUNCT ejpam-4877 411 1 theorem theorem ADJ ejpam-4877 411 2 6 6 NUM ejpam-4877 411 3 . . PUNCT ejpam-4877 412 1 let let VERB ejpam-4877 412 2 g g PRON ejpam-4877 412 3 be be AUX ejpam-4877 412 4 a a DET ejpam-4877 412 5 graph graph NOUN ejpam-4877 412 6 of of ADP ejpam-4877 412 7 order order NOUN ejpam-4877 412 8 n n PRON ejpam-4877 412 9 with with ADP ejpam-4877 412 10 γ(c γ(c PROPN ejpam-4877 412 11 ) ) PUNCT ejpam-4877 412 12 ̸= ̸= PROPN ejpam-4877 412 13 1 1 NUM ejpam-4877 412 14 for for ADP ejpam-4877 412 15 each each DET ejpam-4877 412 16 component component NOUN ejpam-4877 412 17 c c PROPN ejpam-4877 412 18 of of ADP ejpam-4877 412 19 g. g. PROPN ejpam-4877 412 20 if if SCONJ ejpam-4877 412 21 s s X ejpam-4877 412 22 is be AUX ejpam-4877 412 23 a a DET ejpam-4877 412 24 grundy grundy PROPN ejpam-4877 412 25 total total NOUN ejpam-4877 412 26 hop hop NOUN ejpam-4877 412 27 dominating dominating NOUN ejpam-4877 412 28 sequence sequence NOUN ejpam-4877 412 29 of of ADP ejpam-4877 412 30 g1 g1 NOUN ejpam-4877 412 31 or or CCONJ ejpam-4877 412 32 g2 g2 PROPN ejpam-4877 412 33 , , PUNCT ejpam-4877 412 34 then then ADV ejpam-4877 412 35 s s VERB ejpam-4877 412 36 is be AUX ejpam-4877 412 37 a a DET ejpam-4877 412 38 grundy grundy PROPN ejpam-4877 412 39 total total NOUN ejpam-4877 412 40 hop hop NOUN ejpam-4877 412 41 dominating dominating NOUN ejpam-4877 412 42 sequence sequence NOUN ejpam-4877 412 43 of of ADP ejpam-4877 412 44 s(g s(g PROPN ejpam-4877 412 45 ) ) PUNCT ejpam-4877 412 46 . . PUNCT ejpam-4877 413 1 moreover moreover ADV ejpam-4877 413 2 , , PUNCT ejpam-4877 413 3 γthgr(g γthgr(g NOUN ejpam-4877 413 4 ) ) PUNCT ejpam-4877 413 5 ≤ ≤ NOUN ejpam-4877 413 6 γthgr(s(g γthgr(s(g PUNCT ejpam-4877 413 7 ) ) PUNCT ejpam-4877 413 8 ) ) PUNCT ejpam-4877 413 9 . . PUNCT ejpam-4877 414 1 proof proof NOUN ejpam-4877 414 2 . . PUNCT ejpam-4877 415 1 let let VERB ejpam-4877 415 2 g1 g1 PROPN ejpam-4877 415 3 and and CCONJ ejpam-4877 415 4 g2 g2 PROPN ejpam-4877 415 5 be be VERB ejpam-4877 415 6 two two NUM ejpam-4877 415 7 copies copy NOUN ejpam-4877 415 8 of of ADP ejpam-4877 415 9 g. g. PROPN ejpam-4877 415 10 let let VERB ejpam-4877 415 11 s s AUX ejpam-4877 415 12 = = PUNCT ejpam-4877 415 13 ( ( PUNCT ejpam-4877 415 14 v1 v1 PROPN ejpam-4877 415 15 , , PUNCT ejpam-4877 415 16 v2 v2 NOUN ejpam-4877 415 17 , , PUNCT ejpam-4877 415 18 . . PUNCT ejpam-4877 415 19 . . PUNCT ejpam-4877 416 1 . . PUNCT ejpam-4877 417 1 , , PUNCT ejpam-4877 417 2 vk vk AUX ejpam-4877 417 3 ) ) PUNCT ejpam-4877 417 4 be be VERB ejpam-4877 417 5 a a DET ejpam-4877 417 6 grundy grundy PROPN ejpam-4877 417 7 total total NOUN ejpam-4877 417 8 hop hop NOUN ejpam-4877 417 9 dominating dominating NOUN ejpam-4877 417 10 sequence sequence NOUN ejpam-4877 417 11 in in ADP ejpam-4877 417 12 g1 g1 PROPN ejpam-4877 417 13 and and CCONJ ejpam-4877 417 14 let let VERB ejpam-4877 417 15 v′ v′ NOUN ejpam-4877 417 16 ∈ ∈ PROPN ejpam-4877 417 17 v v NOUN ejpam-4877 417 18 ( ( PUNCT ejpam-4877 417 19 g2 g2 PROPN ejpam-4877 417 20 ) ) PUNCT ejpam-4877 417 21 . . PUNCT ejpam-4877 418 1 then then ADV ejpam-4877 418 2 n2 n2 PROPN ejpam-4877 418 3 g1 g1 PROPN ejpam-4877 418 4 ( ( PUNCT ejpam-4877 418 5 vi vi NOUN ejpam-4877 418 6 ) ) PUNCT ejpam-4877 418 7 \ \ NOUN ejpam-4877 418 8 i−1⋃ i−1⋃ PROPN ejpam-4877 418 9 j=1 j=1 PROPN ejpam-4877 418 10 n2 n2 PROPN ejpam-4877 418 11 g1 g1 PROPN ejpam-4877 418 12 ( ( PUNCT ejpam-4877 418 13 vj vj ADJ ejpam-4877 418 14 ) ) PUNCT ejpam-4877 418 15 ̸= ̸= PROPN ejpam-4877 418 16 ∅ ∅ NOUN ejpam-4877 418 17 for for ADP ejpam-4877 418 18 each each DET ejpam-4877 418 19 i i PRON ejpam-4877 418 20 ∈ ∈ PROPN ejpam-4877 418 21 { { PUNCT ejpam-4877 418 22 2 2 NUM ejpam-4877 418 23 , , PUNCT ejpam-4877 418 24 3 3 NUM ejpam-4877 418 25 , , PUNCT ejpam-4877 418 26 . . PUNCT ejpam-4877 418 27 . . PUNCT ejpam-4877 418 28 . . PUNCT ejpam-4877 418 29 , , PUNCT ejpam-4877 418 30 k k X ejpam-4877 418 31 } } PUNCT ejpam-4877 418 32 . . PUNCT ejpam-4877 419 1 thus thus ADV ejpam-4877 419 2 , , PUNCT ejpam-4877 419 3 by by ADP ejpam-4877 419 4 lemma lemma PROPN ejpam-4877 419 5 1 1 NUM ejpam-4877 419 6 n2 n2 NOUN ejpam-4877 419 7 s(g)(vi s(g)(vi NUM ejpam-4877 419 8 ) ) PUNCT ejpam-4877 419 9 \ \ PROPN ejpam-4877 419 10 i−1⋃ i−1⋃ PROPN ejpam-4877 419 11 j=1 j=1 PROPN ejpam-4877 419 12 n2 n2 PROPN ejpam-4877 419 13 s(g)(vj s(g)(vj PROPN ejpam-4877 419 14 ) ) PUNCT ejpam-4877 419 15 = = PUNCT ejpam-4877 420 1 [ [ PUNCT ejpam-4877 420 2 n2 n2 ADJ ejpam-4877 420 3 g1 g1 PROPN ejpam-4877 420 4 ( ( PUNCT ejpam-4877 420 5 vi vi NOUN ejpam-4877 420 6 ) ) PUNCT ejpam-4877 420 7 ∪n2 ∪n2 NOUN ejpam-4877 420 8 g2 g2 PROPN ejpam-4877 421 1 [ [ X ejpam-4877 421 2 v′i v′i X ejpam-4877 421 3 ] ] X ejpam-4877 421 4 ] ] PUNCT ejpam-4877 421 5 \ \ PROPN ejpam-4877 421 6 i−1⋃ i−1⋃ PROPN ejpam-4877 421 7 j=1 j=1 PROPN ejpam-4877 421 8 [ [ PUNCT ejpam-4877 421 9 n2 n2 ADJ ejpam-4877 421 10 g1 g1 PROPN ejpam-4877 421 11 ( ( PUNCT ejpam-4877 421 12 vj vj INTJ ejpam-4877 421 13 ) ) PUNCT ejpam-4877 421 14 ∪n2 ∪n2 PROPN ejpam-4877 421 15 g2 g2 PROPN ejpam-4877 422 1 [ [ X ejpam-4877 422 2 v′j v′j X ejpam-4877 422 3 ] ] X ejpam-4877 422 4 ] ] PUNCT ejpam-4877 423 1 = = SYM ejpam-4877 423 2 (n2 (n2 PROPN ejpam-4877 423 3 g1 g1 PROPN ejpam-4877 423 4 ( ( PUNCT ejpam-4877 423 5 vi vi NOUN ejpam-4877 423 6 ) ) PUNCT ejpam-4877 423 7 ∪n2 ∪n2 NOUN ejpam-4877 423 8 g2 g2 PROPN ejpam-4877 424 1 [ [ X ejpam-4877 424 2 v′i v′i X ejpam-4877 424 3 ] ] PUNCT ejpam-4877 424 4 ) ) PUNCT ejpam-4877 424 5 \ \ PROPN ejpam-4877 424 6 i−1⋃ i−1⋃ PROPN ejpam-4877 424 7 j=1 j=1 PROPN ejpam-4877 424 8 n2 n2 PROPN ejpam-4877 424 9 g1 g1 PROPN ejpam-4877 424 10 ( ( PUNCT ejpam-4877 424 11 vj vj INTJ ejpam-4877 424 12 ) ) PUNCT ejpam-4877 424 13   NOUN ejpam-4877 424 14 ∪ ∪ VERB ejpam-4877 424 15 (n2 (n2 PROPN ejpam-4877 424 16 g1 g1 NOUN ejpam-4877 424 17 ( ( PUNCT ejpam-4877 424 18 vi vi NOUN ejpam-4877 424 19 ) ) PUNCT ejpam-4877 424 20 ∪n2 ∪n2 NOUN ejpam-4877 424 21 g2 g2 PROPN ejpam-4877 425 1 [ [ X ejpam-4877 425 2 v′i v′i X ejpam-4877 425 3 ] ] PUNCT ejpam-4877 425 4 ) ) PUNCT ejpam-4877 425 5 \ \ PROPN ejpam-4877 425 6 i−1⋃ i−1⋃ PROPN ejpam-4877 425 7 j=1 j=1 PROPN ejpam-4877 425 8 n2 n2 PROPN ejpam-4877 425 9 g2 g2 PROPN ejpam-4877 426 1 [ [ X ejpam-4877 426 2 v′j v′j X ejpam-4877 426 3 ] ] PUNCT ejpam-4877 426 4   NOUN ejpam-4877 426 5 ̸= ̸= NOUN ejpam-4877 426 6 ∅ ∅ NOUN ejpam-4877 426 7 for for ADP ejpam-4877 426 8 each each DET ejpam-4877 426 9 i i PRON ejpam-4877 426 10 ∈ ∈ PROPN ejpam-4877 426 11 { { PUNCT ejpam-4877 426 12 2 2 NUM ejpam-4877 426 13 , , PUNCT ejpam-4877 426 14 . . PUNCT ejpam-4877 426 15 . . PUNCT ejpam-4877 426 16 . . PUNCT ejpam-4877 427 1 , , PUNCT ejpam-4877 427 2 k k X ejpam-4877 427 3 } } PUNCT ejpam-4877 427 4 . . PUNCT ejpam-4877 428 1 since since SCONJ ejpam-4877 428 2 ŝ ŝ NUM ejpam-4877 428 3 is be AUX ejpam-4877 428 4 a a DET ejpam-4877 428 5 total total ADJ ejpam-4877 428 6 hop hop NOUN ejpam-4877 428 7 dominating dominating NOUN ejpam-4877 428 8 set set NOUN ejpam-4877 428 9 of of ADP ejpam-4877 428 10 g1 g1 NOUN ejpam-4877 428 11 , , PUNCT ejpam-4877 428 12 there there PRON ejpam-4877 428 13 exists exist VERB ejpam-4877 428 14 w w PROPN ejpam-4877 428 15 ∈ ∈ PROPN ejpam-4877 428 16 ŝ ŝ NOUN ejpam-4877 428 17 ∩n2 ∩n2 PROPN ejpam-4877 428 18 g1 g1 PROPN ejpam-4877 428 19 ( ( PUNCT ejpam-4877 428 20 v v NOUN ejpam-4877 428 21 ) ) PUNCT ejpam-4877 428 22 . . PUNCT ejpam-4877 429 1 by by ADP ejpam-4877 429 2 lemma lemma PROPN ejpam-4877 429 3 1 1 NUM ejpam-4877 429 4 , , PUNCT ejpam-4877 429 5 w w PROPN ejpam-4877 429 6 ∈ ∈ PROPN ejpam-4877 429 7 ŝ ŝ NOUN ejpam-4877 429 8 ∩ ∩ PROPN ejpam-4877 429 9 n2 n2 NOUN ejpam-4877 429 10 s(g)(v s(g)(v PUNCT ejpam-4877 429 11 ′ ′ NUM ejpam-4877 429 12 ) ) PUNCT ejpam-4877 429 13 , , PUNCT ejpam-4877 429 14 i.e. i.e. X ejpam-4877 429 15 , , PUNCT ejpam-4877 429 16 w w PROPN ejpam-4877 429 17 ∈ ∈ PROPN ejpam-4877 429 18 ŝ ŝ NOUN ejpam-4877 429 19 and and CCONJ ejpam-4877 429 20 ds(g)(v ds(g)(v NOUN ejpam-4877 429 21 ′ ′ NOUN ejpam-4877 429 22 , , PUNCT ejpam-4877 429 23 w w NOUN ejpam-4877 429 24 ) ) PUNCT ejpam-4877 429 25 = = SYM ejpam-4877 429 26 2 2 X ejpam-4877 429 27 . . PUNCT ejpam-4877 429 28 consequently consequently ADV ejpam-4877 429 29 , , PUNCT ejpam-4877 429 30 ŝ ŝ X ejpam-4877 429 31 is be AUX ejpam-4877 429 32 a a DET ejpam-4877 429 33 grundy grundy PROPN ejpam-4877 429 34 total total NOUN ejpam-4877 429 35 hop hop NOUN ejpam-4877 429 36 dominating dominating NOUN ejpam-4877 429 37 sequence sequence NOUN ejpam-4877 429 38 of of ADP ejpam-4877 429 39 s(g s(g PROPN ejpam-4877 429 40 ) ) PUNCT ejpam-4877 429 41 . . PUNCT ejpam-4877 430 1 remark remark PROPN ejpam-4877 430 2 3 3 NUM ejpam-4877 430 3 . . PUNCT ejpam-4877 431 1 the the DET ejpam-4877 431 2 bound bind VERB ejpam-4877 431 3 given give VERB ejpam-4877 431 4 in in ADP ejpam-4877 431 5 theorem theorem NOUN ejpam-4877 431 6 6 6 NUM ejpam-4877 431 7 is be AUX ejpam-4877 431 8 tight tight ADJ ejpam-4877 431 9 . . PUNCT ejpam-4877 432 1 moreover moreover ADV ejpam-4877 432 2 , , PUNCT ejpam-4877 432 3 strict strict ADJ ejpam-4877 432 4 inequality inequality NOUN ejpam-4877 432 5 can can AUX ejpam-4877 432 6 also also ADV ejpam-4877 432 7 be be AUX ejpam-4877 432 8 attained attain VERB ejpam-4877 432 9 . . PUNCT ejpam-4877 433 1 for for ADP ejpam-4877 433 2 equality equality NOUN ejpam-4877 433 3 , , PUNCT ejpam-4877 433 4 consider consider VERB ejpam-4877 433 5 c4 c4 NOUN ejpam-4877 433 6 . . PUNCT ejpam-4877 434 1 then then ADV ejpam-4877 434 2 γthgr(c4 γthgr(c4 VERB ejpam-4877 434 3 ) ) PUNCT ejpam-4877 435 1 = = PUNCT ejpam-4877 436 1 4 4 X ejpam-4877 436 2 . . PUNCT ejpam-4877 436 3 now now ADV ejpam-4877 436 4 , , PUNCT ejpam-4877 436 5 consider consider VERB ejpam-4877 436 6 the the DET ejpam-4877 436 7 shadow shadow NOUN ejpam-4877 436 8 graph graph NOUN ejpam-4877 436 9 of of ADP ejpam-4877 436 10 c4 c4 NOUN ejpam-4877 436 11 given give VERB ejpam-4877 436 12 in in ADP ejpam-4877 436 13 fig fig NOUN ejpam-4877 436 14 . . PUNCT ejpam-4877 437 1 3 3 X ejpam-4877 437 2 . . X ejpam-4877 437 3 let let VERB ejpam-4877 437 4 s s VERB ejpam-4877 437 5 = = PUNCT ejpam-4877 437 6 ( ( PUNCT ejpam-4877 437 7 a a PRON ejpam-4877 437 8 , , PUNCT ejpam-4877 437 9 a′ a′ PROPN ejpam-4877 437 10 , , PUNCT ejpam-4877 437 11 b b NOUN ejpam-4877 437 12 , , PUNCT ejpam-4877 437 13 b′ b′ NUM ejpam-4877 437 14 ) ) PUNCT ejpam-4877 437 15 . . PUNCT ejpam-4877 438 1 observe observe VERB ejpam-4877 438 2 that that SCONJ ejpam-4877 438 3 s s VERB ejpam-4877 438 4 is be AUX ejpam-4877 438 5 a a DET ejpam-4877 438 6 grundy grundy PROPN ejpam-4877 438 7 total total NOUN ejpam-4877 438 8 hop hop NOUN ejpam-4877 438 9 dominating dominating NOUN ejpam-4877 438 10 sequence sequence NOUN ejpam-4877 438 11 of of ADP ejpam-4877 438 12 s(c4 s(c4 NOUN ejpam-4877 438 13 ) ) PUNCT ejpam-4877 438 14 . . PUNCT ejpam-4877 439 1 moreover moreover ADV ejpam-4877 439 2 , , PUNCT ejpam-4877 439 3 it it PRON ejpam-4877 439 4 can can AUX ejpam-4877 439 5 be be AUX ejpam-4877 439 6 verified verify VERB ejpam-4877 439 7 that that SCONJ ejpam-4877 439 8 γthgr(s(c4 γthgr(s(c4 NOUN ejpam-4877 439 9 ) ) PUNCT ejpam-4877 439 10 ) ) PUNCT ejpam-4877 440 1 = = PUNCT ejpam-4877 440 2 4 4 X ejpam-4877 440 3 . . PUNCT ejpam-4877 440 4 consequently consequently ADV ejpam-4877 440 5 , , PUNCT ejpam-4877 440 6 γthgr(s(c4 γthgr(s(c4 NOUN ejpam-4877 440 7 ) ) PUNCT ejpam-4877 440 8 ) ) PUNCT ejpam-4877 440 9 = = SYM ejpam-4877 440 10 4 4 NUM ejpam-4877 440 11 = = SYM ejpam-4877 440 12 γthgr(s(c4 γthgr(s(c4 NOUN ejpam-4877 440 13 ) ) PUNCT ejpam-4877 440 14 ) ) PUNCT ejpam-4877 440 15 . . PUNCT ejpam-4877 441 1 j.a j.a PROPN ejpam-4877 441 2 . . PROPN ejpam-4877 441 3 hassan hassan PROPN ejpam-4877 441 4 , , PUNCT ejpam-4877 441 5 s. s. PROPN ejpam-4877 441 6 canoy canoy PROPN ejpam-4877 441 7 / / SYM ejpam-4877 441 8 eur eur PROPN ejpam-4877 441 9 . . PUNCT ejpam-4877 442 1 j. j. PROPN ejpam-4877 442 2 pure pure PROPN ejpam-4877 442 3 appl appl PROPN ejpam-4877 442 4 . . PROPN ejpam-4877 442 5 math math PROPN ejpam-4877 442 6 , , PUNCT ejpam-4877 442 7 16 16 NUM ejpam-4877 442 8 ( ( PUNCT ejpam-4877 442 9 4 4 NUM ejpam-4877 442 10 ) ) PUNCT ejpam-4877 442 11 ( ( PUNCT ejpam-4877 442 12 2023 2023 NUM ejpam-4877 442 13 ) ) PUNCT ejpam-4877 442 14 , , PUNCT ejpam-4877 442 15 2597 2597 NUM ejpam-4877 442 16 - - SYM ejpam-4877 442 17 2612 2612 NUM ejpam-4877 442 18 2607 2607 NUM ejpam-4877 443 1 a a PRON ejpam-4877 443 2 b b NOUN ejpam-4877 443 3 c c NOUN ejpam-4877 443 4 d d X ejpam-4877 443 5 a′ a′ PROPN ejpam-4877 443 6 b′ b′ NUM ejpam-4877 443 7 c′ c′ NOUN ejpam-4877 443 8 d′ d′ NUM ejpam-4877 443 9 s(c4 s(c4 NOUN ejpam-4877 443 10 ) ) PUNCT ejpam-4877 443 11 : : PUNCT ejpam-4877 443 12 figure figure VERB ejpam-4877 443 13 3 3 NUM ejpam-4877 443 14 : : PUNCT ejpam-4877 443 15 a a DET ejpam-4877 443 16 graph graph NOUN ejpam-4877 443 17 c4 c4 NOUN ejpam-4877 443 18 with with ADP ejpam-4877 443 19 γth γth NOUN ejpam-4877 443 20 gr(c4 gr(c4 NOUN ejpam-4877 443 21 ) ) PUNCT ejpam-4877 443 22 = = PUNCT ejpam-4877 444 1 γth γth PROPN ejpam-4877 444 2 gr(s(c4 gr(s(c4 NOUN ejpam-4877 444 3 ) ) PUNCT ejpam-4877 444 4 ) ) PUNCT ejpam-4877 445 1 for for ADP ejpam-4877 445 2 strict strict ADJ ejpam-4877 445 3 inequality inequality NOUN ejpam-4877 445 4 , , PUNCT ejpam-4877 445 5 consider consider VERB ejpam-4877 445 6 p5 p5 ADJ ejpam-4877 445 7 . . PUNCT ejpam-4877 446 1 then then ADV ejpam-4877 446 2 γthgr(p5 γthgr(p5 NOUN ejpam-4877 446 3 ) ) PUNCT ejpam-4877 446 4 = = SYM ejpam-4877 447 1 4 4 X ejpam-4877 447 2 . . PUNCT ejpam-4877 447 3 now now ADV ejpam-4877 447 4 , , PUNCT ejpam-4877 447 5 consider consider VERB ejpam-4877 447 6 the the DET ejpam-4877 447 7 shadow shadow NOUN ejpam-4877 447 8 graph graph NOUN ejpam-4877 447 9 of of ADP ejpam-4877 447 10 p5 p5 NOUN ejpam-4877 447 11 given give VERB ejpam-4877 447 12 in in ADP ejpam-4877 447 13 fig fig NOUN ejpam-4877 447 14 . . PUNCT ejpam-4877 448 1 4 4 X ejpam-4877 448 2 . . X ejpam-4877 448 3 let let VERB ejpam-4877 448 4 s s VERB ejpam-4877 448 5 = = PUNCT ejpam-4877 448 6 ( ( PUNCT ejpam-4877 448 7 a a PRON ejpam-4877 448 8 , , PUNCT ejpam-4877 448 9 a′ a′ PROPN ejpam-4877 448 10 , , PUNCT ejpam-4877 448 11 e e NOUN ejpam-4877 448 12 , , PUNCT ejpam-4877 448 13 e′ e′ PROPN ejpam-4877 448 14 , , PUNCT ejpam-4877 448 15 d d NOUN ejpam-4877 448 16 , , PUNCT ejpam-4877 448 17 d′ d′ NUM ejpam-4877 448 18 ) ) PUNCT ejpam-4877 448 19 . . PUNCT ejpam-4877 449 1 observe observe VERB ejpam-4877 449 2 that that SCONJ ejpam-4877 449 3 s s VERB ejpam-4877 449 4 is be AUX ejpam-4877 449 5 a a DET ejpam-4877 449 6 grundy grundy PROPN ejpam-4877 449 7 total total NOUN ejpam-4877 449 8 hop hop NOUN ejpam-4877 449 9 dominating dominating NOUN ejpam-4877 449 10 sequence sequence NOUN ejpam-4877 449 11 of of ADP ejpam-4877 449 12 s(p5 s(p5 NOUN ejpam-4877 449 13 ) ) PUNCT ejpam-4877 449 14 . . PUNCT ejpam-4877 450 1 moreover moreover ADV ejpam-4877 450 2 , , PUNCT ejpam-4877 450 3 it it PRON ejpam-4877 450 4 can can AUX ejpam-4877 450 5 be be AUX ejpam-4877 450 6 verified verify VERB ejpam-4877 450 7 that that SCONJ ejpam-4877 450 8 γthgr(s(p5 γthgr(s(p5 ADJ ejpam-4877 450 9 ) ) PUNCT ejpam-4877 450 10 ) ) PUNCT ejpam-4877 451 1 = = PUNCT ejpam-4877 451 2 6 6 NUM ejpam-4877 451 3 . . PUNCT ejpam-4877 452 1 hence hence ADV ejpam-4877 452 2 , , PUNCT ejpam-4877 452 3 γthgr(s(p5 γthgr(s(p5 ADJ ejpam-4877 452 4 ) ) PUNCT ejpam-4877 452 5 ) ) PUNCT ejpam-4877 453 1 = = PUNCT ejpam-4877 453 2 4 4 NUM ejpam-4877 453 3 < < SYM ejpam-4877 453 4 6 6 NUM ejpam-4877 453 5 = = SYM ejpam-4877 453 6 γthgr(s(p5 γthgr(s(p5 ADJ ejpam-4877 453 7 ) ) PUNCT ejpam-4877 453 8 ) ) PUNCT ejpam-4877 453 9 . . PUNCT ejpam-4877 454 1 a a DET ejpam-4877 454 2 b b X ejpam-4877 454 3 s(p5 s(p5 NOUN ejpam-4877 454 4 ) ) PUNCT ejpam-4877 454 5 : : PUNCT ejpam-4877 455 1 a′ a′ PROPN ejpam-4877 455 2 b′ b′ NUM ejpam-4877 455 3 c c NOUN ejpam-4877 455 4 c′ c′ NOUN ejpam-4877 456 1 d d X ejpam-4877 456 2 d′ d′ X ejpam-4877 456 3 e e X ejpam-4877 456 4 e′ e′ NOUN ejpam-4877 456 5 figure figure NOUN ejpam-4877 456 6 4 4 NUM ejpam-4877 456 7 : : PUNCT ejpam-4877 456 8 a a DET ejpam-4877 456 9 graph graph NOUN ejpam-4877 456 10 g g NOUN ejpam-4877 456 11 with with ADP ejpam-4877 456 12 γth γth NOUN ejpam-4877 456 13 gr(g gr(g PUNCT ejpam-4877 456 14 ) ) PUNCT ejpam-4877 456 15 < < X ejpam-4877 456 16 γth γth X ejpam-4877 456 17 gr(s(g gr(s(g NUM ejpam-4877 456 18 ) ) PUNCT ejpam-4877 456 19 ) ) PUNCT ejpam-4877 457 1 lemma lemma PROPN ejpam-4877 457 2 2 2 X ejpam-4877 457 3 . . PUNCT ejpam-4877 458 1 let let VERB ejpam-4877 458 2 g g PRON ejpam-4877 458 3 be be AUX ejpam-4877 458 4 a a DET ejpam-4877 458 5 graph graph NOUN ejpam-4877 458 6 of of ADP ejpam-4877 458 7 order order NOUN ejpam-4877 458 8 n n NOUN ejpam-4877 458 9 with with ADP ejpam-4877 458 10 no no DET ejpam-4877 458 11 isolated isolated ADJ ejpam-4877 458 12 vertices vertex NOUN ejpam-4877 458 13 . . PUNCT ejpam-4877 459 1 if if SCONJ ejpam-4877 459 2 |ng(v)| |ng(v)| PROPN ejpam-4877 459 3 ≥ ≥ NOUN ejpam-4877 459 4 l l NOUN ejpam-4877 459 5 for for ADP ejpam-4877 459 6 every every PRON ejpam-4877 459 7 v v NUM ejpam-4877 459 8 ∈ ∈ NOUN ejpam-4877 459 9 v v NOUN ejpam-4877 459 10 ( ( PUNCT ejpam-4877 459 11 g g NOUN ejpam-4877 459 12 ) ) PUNCT ejpam-4877 459 13 , , PUNCT ejpam-4877 459 14 then then ADV ejpam-4877 459 15 γtgr(g γtgr(g NOUN ejpam-4877 459 16 ) ) PUNCT ejpam-4877 459 17 ≤ ≤ NUM ejpam-4877 459 18 n− n− NOUN ejpam-4877 459 19 ( ( PUNCT ejpam-4877 459 20 l l NOUN ejpam-4877 459 21 − − NOUN ejpam-4877 459 22 1 1 NUM ejpam-4877 459 23 ) ) PUNCT ejpam-4877 459 24 . . PUNCT ejpam-4877 460 1 proof proof NOUN ejpam-4877 460 2 . . PUNCT ejpam-4877 461 1 suppose suppose VERB ejpam-4877 461 2 γtgr(g γtgr(g NOUN ejpam-4877 461 3 ) ) PUNCT ejpam-4877 461 4 = = SYM ejpam-4877 462 1 k k NOUN ejpam-4877 462 2 , , PUNCT ejpam-4877 462 3 say say VERB ejpam-4877 462 4 s s X ejpam-4877 462 5 = = PUNCT ejpam-4877 462 6 ( ( PUNCT ejpam-4877 462 7 w1 w1 NOUN ejpam-4877 462 8 , , PUNCT ejpam-4877 462 9 w2 w2 NOUN ejpam-4877 462 10 , , PUNCT ejpam-4877 462 11 · · PUNCT ejpam-4877 462 12 · · PUNCT ejpam-4877 462 13 · · PUNCT ejpam-4877 462 14 , , PUNCT ejpam-4877 462 15 wk wk X ejpam-4877 462 16 ) ) PUNCT ejpam-4877 462 17 is be AUX ejpam-4877 462 18 a a DET ejpam-4877 462 19 grundy grundy PROPN ejpam-4877 462 20 total total ADJ ejpam-4877 462 21 dominating dominating NOUN ejpam-4877 462 22 sequence sequence NOUN ejpam-4877 462 23 of of ADP ejpam-4877 462 24 g. g. PROPN ejpam-4877 462 25 assume assume VERB ejpam-4877 462 26 w1 w1 PROPN ejpam-4877 462 27 = = SYM ejpam-4877 462 28 vi vi PROPN ejpam-4877 462 29 for for ADP ejpam-4877 462 30 some some DET ejpam-4877 462 31 i i PRON ejpam-4877 462 32 ∈ ∈ PROPN ejpam-4877 462 33 { { PUNCT ejpam-4877 462 34 1 1 NUM ejpam-4877 462 35 , , PUNCT ejpam-4877 462 36 . . PUNCT ejpam-4877 462 37 . . PUNCT ejpam-4877 462 38 . . PUNCT ejpam-4877 462 39 , , PUNCT ejpam-4877 462 40 n n CCONJ ejpam-4877 462 41 } } PUNCT ejpam-4877 462 42 . . PUNCT ejpam-4877 463 1 then then ADV ejpam-4877 463 2 |ng(w1)| |ng(w1)| PROPN ejpam-4877 463 3 = = PUNCT ejpam-4877 463 4 |ng(vi)| |ng(vi)| X ejpam-4877 463 5 ≥ ≥ NOUN ejpam-4877 463 6 l l NOUN ejpam-4877 463 7 for for ADP ejpam-4877 463 8 some some DET ejpam-4877 463 9 i i PRON ejpam-4877 463 10 ∈ ∈ PROPN ejpam-4877 463 11 { { PUNCT ejpam-4877 463 12 1 1 NUM ejpam-4877 463 13 , , PUNCT ejpam-4877 463 14 . . PUNCT ejpam-4877 463 15 . . PUNCT ejpam-4877 464 1 . . PUNCT ejpam-4877 464 2 , , PUNCT ejpam-4877 465 1 n n CCONJ ejpam-4877 465 2 } } PUNCT ejpam-4877 465 3 . . PUNCT ejpam-4877 466 1 it it PRON ejpam-4877 466 2 follows follow VERB ejpam-4877 466 3 that that SCONJ ejpam-4877 466 4 there there PRON ejpam-4877 466 5 are be VERB ejpam-4877 466 6 at at ADP ejpam-4877 466 7 most most ADJ ejpam-4877 466 8 n n ADP ejpam-4877 466 9 − − PROPN ejpam-4877 466 10 l l NOUN ejpam-4877 466 11 remaining remain VERB ejpam-4877 466 12 vertices vertex NOUN ejpam-4877 466 13 of of ADP ejpam-4877 466 14 g g NOUN ejpam-4877 466 15 that that PRON ejpam-4877 466 16 could could AUX ejpam-4877 466 17 be be AUX ejpam-4877 466 18 footprinted footprinte VERB ejpam-4877 466 19 by by ADP ejpam-4877 466 20 the the DET ejpam-4877 466 21 next next ADJ ejpam-4877 466 22 terms term NOUN ejpam-4877 466 23 of of ADP ejpam-4877 466 24 s. s. PROPN ejpam-4877 466 25 therefore therefore ADV ejpam-4877 466 26 , , PUNCT ejpam-4877 466 27 γtgr(g γtgr(g NOUN ejpam-4877 466 28 ) ) PUNCT ejpam-4877 466 29 = = SYM ejpam-4877 467 1 k k X ejpam-4877 467 2 ≤ ≤ NUM ejpam-4877 467 3 n− n− NOUN ejpam-4877 467 4 l l PROPN ejpam-4877 467 5 + + CCONJ ejpam-4877 467 6 |{vi}| |{vi}| PROPN ejpam-4877 467 7 = = SYM ejpam-4877 467 8 n− n− NOUN ejpam-4877 467 9 l l NOUN ejpam-4877 468 1 + + NOUN ejpam-4877 468 2 1 1 X ejpam-4877 468 3 = = SYM ejpam-4877 468 4 n− n− PROPN ejpam-4877 468 5 ( ( PUNCT ejpam-4877 468 6 l l NOUN ejpam-4877 468 7 − − NOUN ejpam-4877 468 8 1 1 NUM ejpam-4877 468 9 ) ) PUNCT ejpam-4877 468 10 . . PUNCT ejpam-4877 469 1 proposition proposition NOUN ejpam-4877 469 2 4 4 NUM ejpam-4877 469 3 . . PUNCT ejpam-4877 470 1 let let VERB ejpam-4877 470 2 n n PRON ejpam-4877 470 3 ≥ ≥ X ejpam-4877 470 4 4 4 NUM ejpam-4877 470 5 be be AUX ejpam-4877 470 6 any any DET ejpam-4877 470 7 positive positive ADJ ejpam-4877 470 8 integer integer NOUN ejpam-4877 470 9 . . PUNCT ejpam-4877 471 1 then then ADV ejpam-4877 471 2 γtgr(pn γtgr(pn NOUN ejpam-4877 471 3 ) ) PUNCT ejpam-4877 471 4 = = SYM ejpam-4877 471 5 4 4 NUM ejpam-4877 471 6 = = SYM ejpam-4877 471 7 γtgr(cn γtgr(cn PROPN ejpam-4877 471 8 ) ) PUNCT ejpam-4877 471 9 . . PUNCT ejpam-4877 472 1 j.a j.a PROPN ejpam-4877 472 2 . . PROPN ejpam-4877 472 3 hassan hassan PROPN ejpam-4877 472 4 , , PUNCT ejpam-4877 472 5 s. s. PROPN ejpam-4877 472 6 canoy canoy PROPN ejpam-4877 472 7 / / SYM ejpam-4877 472 8 eur eur PROPN ejpam-4877 472 9 . . PUNCT ejpam-4877 473 1 j. j. PROPN ejpam-4877 473 2 pure pure PROPN ejpam-4877 473 3 appl appl PROPN ejpam-4877 473 4 . . PROPN ejpam-4877 473 5 math math PROPN ejpam-4877 473 6 , , PUNCT ejpam-4877 473 7 16 16 NUM ejpam-4877 473 8 ( ( PUNCT ejpam-4877 473 9 4 4 NUM ejpam-4877 473 10 ) ) PUNCT ejpam-4877 473 11 ( ( PUNCT ejpam-4877 473 12 2023 2023 NUM ejpam-4877 473 13 ) ) PUNCT ejpam-4877 473 14 , , PUNCT ejpam-4877 473 15 2597 2597 NUM ejpam-4877 473 16 - - SYM ejpam-4877 473 17 2612 2612 NUM ejpam-4877 473 18 2608 2608 NUM ejpam-4877 473 19 proof proof NOUN ejpam-4877 473 20 . . PUNCT ejpam-4877 474 1 clearly clearly ADV ejpam-4877 474 2 , , PUNCT ejpam-4877 474 3 γtgr(p γtgr(p NOUN ejpam-4877 474 4 4 4 NUM ejpam-4877 474 5 ) ) PUNCT ejpam-4877 474 6 = = SYM ejpam-4877 474 7 4 4 X ejpam-4877 474 8 . . NOUN ejpam-4877 474 9 for for ADP ejpam-4877 474 10 n n X ejpam-4877 474 11 ≥ ≥ NUM ejpam-4877 474 12 5 5 NUM ejpam-4877 474 13 , , PUNCT ejpam-4877 474 14 let let VERB ejpam-4877 474 15 pn pn VERB ejpam-4877 474 16 = = PUNCT ejpam-4877 475 1 [ [ X ejpam-4877 475 2 u1 u1 NOUN ejpam-4877 475 3 , , PUNCT ejpam-4877 475 4 u2 u2 NOUN ejpam-4877 475 5 , , PUNCT ejpam-4877 475 6 · · PUNCT ejpam-4877 475 7 · · PUNCT ejpam-4877 475 8 · · PUNCT ejpam-4877 475 9 , , PUNCT ejpam-4877 475 10 un un PROPN ejpam-4877 475 11 ] ] X ejpam-4877 475 12 and and CCONJ ejpam-4877 475 13 s s NOUN ejpam-4877 475 14 = = SYM ejpam-4877 475 15 ( ( PUNCT ejpam-4877 475 16 u3 u3 PROPN ejpam-4877 475 17 , , PUNCT ejpam-4877 475 18 u2 u2 NOUN ejpam-4877 475 19 , , PUNCT ejpam-4877 475 20 u1 u1 NOUN ejpam-4877 475 21 , , PUNCT ejpam-4877 475 22 un un PROPN ejpam-4877 475 23 ) ) PUNCT ejpam-4877 475 24 . . PUNCT ejpam-4877 476 1 notice notice VERB ejpam-4877 476 2 that that SCONJ ejpam-4877 476 3 ŝ ŝ VERB ejpam-4877 476 4 is be AUX ejpam-4877 476 5 a a DET ejpam-4877 476 6 total total ADJ ejpam-4877 476 7 dominating dominating NOUN ejpam-4877 476 8 set set NOUN ejpam-4877 476 9 of of ADP ejpam-4877 476 10 pn pn PROPN ejpam-4877 476 11 . . PUNCT ejpam-4877 477 1 now now ADV ejpam-4877 477 2 , , PUNCT ejpam-4877 477 3 observe observe VERB ejpam-4877 477 4 that that SCONJ ejpam-4877 477 5 v4 v4 PROPN ejpam-4877 477 6 ∈ ∈ PROPN ejpam-4877 477 7 npn npn X ejpam-4877 477 8 ( ( PUNCT ejpam-4877 477 9 v2 v2 PROPN ejpam-4877 477 10 ) ) PUNCT ejpam-4877 477 11 \npn \npn PROPN ejpam-4877 477 12 ( ( PUNCT ejpam-4877 477 13 v3 v3 PROPN ejpam-4877 477 14 ) ) PUNCT ejpam-4877 477 15 , , PUNCT ejpam-4877 477 16 v3 v3 PROPN ejpam-4877 477 17 ∈ ∈ PROPN ejpam-4877 477 18 npn npn X ejpam-4877 477 19 ( ( PUNCT ejpam-4877 477 20 v1 v1 NOUN ejpam-4877 477 21 ) ) PUNCT ejpam-4877 477 22 \ \ PUNCT ejpam-4877 477 23 ( ( PUNCT ejpam-4877 477 24 npn npn X ejpam-4877 477 25 ( ( PUNCT ejpam-4877 477 26 v2)∪npn v2)∪npn PROPN ejpam-4877 477 27 ( ( PUNCT ejpam-4877 477 28 v3 v3 PROPN ejpam-4877 477 29 ) ) PUNCT ejpam-4877 477 30 ) ) PUNCT ejpam-4877 477 31 , , PUNCT ejpam-4877 477 32 and and CCONJ ejpam-4877 477 33 v2 v2 PROPN ejpam-4877 477 34 ∈ ∈ PROPN ejpam-4877 477 35 npn npn X ejpam-4877 477 36 ( ( PUNCT ejpam-4877 477 37 vn vn NOUN ejpam-4877 477 38 ) ) PUNCT ejpam-4877 477 39 \ \ PUNCT ejpam-4877 478 1 ( ( PUNCT ejpam-4877 478 2 npn npn X ejpam-4877 478 3 ( ( PUNCT ejpam-4877 478 4 v1)∪npn v1)∪npn PROPN ejpam-4877 478 5 ( ( PUNCT ejpam-4877 478 6 v2)∪npn v2)∪npn PROPN ejpam-4877 478 7 ( ( PUNCT ejpam-4877 478 8 v3 v3 PROPN ejpam-4877 478 9 ) ) PUNCT ejpam-4877 478 10 ) ) PUNCT ejpam-4877 478 11 . . PUNCT ejpam-4877 479 1 it it PRON ejpam-4877 479 2 follow follow VERB ejpam-4877 479 3 that that PRON ejpam-4877 479 4 s s VERB ejpam-4877 479 5 is be AUX ejpam-4877 479 6 a a DET ejpam-4877 479 7 grundy grundy PROPN ejpam-4877 479 8 total total ADJ ejpam-4877 479 9 dominating dominating NOUN ejpam-4877 479 10 sequence sequence NOUN ejpam-4877 479 11 of of ADP ejpam-4877 479 12 pn pn PROPN ejpam-4877 479 13 . . PROPN ejpam-4877 480 1 hence hence ADV ejpam-4877 480 2 , , PUNCT ejpam-4877 480 3 γtgr(pn γtgr(pn NOUN ejpam-4877 480 4 ) ) PUNCT ejpam-4877 480 5 ≥ ≥ NOUN ejpam-4877 480 6 4 4 NUM ejpam-4877 480 7 . . PUNCT ejpam-4877 481 1 on on ADP ejpam-4877 481 2 the the DET ejpam-4877 481 3 other other ADJ ejpam-4877 481 4 hand hand NOUN ejpam-4877 481 5 , , PUNCT ejpam-4877 481 6 suppose suppose VERB ejpam-4877 481 7 γtgr(pn γtgr(pn NOUN ejpam-4877 481 8 ) ) PUNCT ejpam-4877 482 1 = = SYM ejpam-4877 482 2 k k NOUN ejpam-4877 482 3 , , PUNCT ejpam-4877 482 4 say say VERB ejpam-4877 482 5 s s X ejpam-4877 482 6 = = PUNCT ejpam-4877 482 7 ( ( PUNCT ejpam-4877 482 8 w1 w1 NOUN ejpam-4877 482 9 , , PUNCT ejpam-4877 482 10 w2 w2 NOUN ejpam-4877 482 11 , , PUNCT ejpam-4877 482 12 · · PUNCT ejpam-4877 482 13 · · PUNCT ejpam-4877 482 14 · · PUNCT ejpam-4877 482 15 , , PUNCT ejpam-4877 482 16 wk wk X ejpam-4877 482 17 ) ) PUNCT ejpam-4877 482 18 is be AUX ejpam-4877 482 19 a a DET ejpam-4877 482 20 grundy grundy PROPN ejpam-4877 482 21 total total ADJ ejpam-4877 482 22 dominating dominating NOUN ejpam-4877 482 23 sequence sequence NOUN ejpam-4877 482 24 of of ADP ejpam-4877 482 25 pn pn PROPN ejpam-4877 482 26 . . PROPN ejpam-4877 483 1 since since SCONJ ejpam-4877 483 2 |npn |npn NOUN ejpam-4877 483 3 ( ( PUNCT ejpam-4877 483 4 vj)| vj)| NOUN ejpam-4877 483 5 ≥ ≥ X ejpam-4877 483 6 n n CCONJ ejpam-4877 483 7 − − NOUN ejpam-4877 483 8 3 3 NUM ejpam-4877 483 9 for for ADP ejpam-4877 483 10 every every DET ejpam-4877 483 11 j j PROPN ejpam-4877 483 12 ∈ ∈ PROPN ejpam-4877 483 13 { { PUNCT ejpam-4877 483 14 1 1 NUM ejpam-4877 483 15 , , PUNCT ejpam-4877 483 16 . . PUNCT ejpam-4877 483 17 . . PUNCT ejpam-4877 483 18 . . PUNCT ejpam-4877 484 1 , , PUNCT ejpam-4877 484 2 n n CCONJ ejpam-4877 484 3 } } PUNCT ejpam-4877 484 4 , , PUNCT ejpam-4877 484 5 it it PRON ejpam-4877 484 6 follows follow VERB ejpam-4877 484 7 that that DET ejpam-4877 484 8 γtgr(pn γtgr(pn NOUN ejpam-4877 484 9 ) ) PUNCT ejpam-4877 484 10 = = SYM ejpam-4877 484 11 k k PROPN ejpam-4877 484 12 ≤ ≤ ADJ ejpam-4877 484 13 n− n− NOUN ejpam-4877 484 14 ( ( PUNCT ejpam-4877 484 15 n− n− NOUN ejpam-4877 484 16 3− 3− NUM ejpam-4877 484 17 1 1 NUM ejpam-4877 484 18 ) ) PUNCT ejpam-4877 484 19 = = SYM ejpam-4877 484 20 4 4 NUM ejpam-4877 484 21 by by ADP ejpam-4877 484 22 lemma lemma PROPN ejpam-4877 484 23 2 2 NUM ejpam-4877 484 24 . . PUNCT ejpam-4877 484 25 therefore therefore ADV ejpam-4877 484 26 , , PUNCT ejpam-4877 484 27 γtgr(pn γtgr(pn NOUN ejpam-4877 484 28 ) ) PUNCT ejpam-4877 484 29 = = SYM ejpam-4877 485 1 4 4 X ejpam-4877 485 2 . . PUNCT ejpam-4877 486 1 next next ADV ejpam-4877 486 2 , , PUNCT ejpam-4877 486 3 let let VERB ejpam-4877 486 4 cn cn PROPN ejpam-4877 486 5 = = PUNCT ejpam-4877 487 1 [ [ X ejpam-4877 487 2 v1 v1 NOUN ejpam-4877 487 3 , , PUNCT ejpam-4877 487 4 v2 v2 PROPN ejpam-4877 487 5 , , PUNCT ejpam-4877 487 6 · · PUNCT ejpam-4877 487 7 · · PUNCT ejpam-4877 487 8 · · PUNCT ejpam-4877 487 9 , , PUNCT ejpam-4877 487 10 vn vn X ejpam-4877 487 11 , , PUNCT ejpam-4877 487 12 v1 v1 PROPN ejpam-4877 487 13 ] ] PUNCT ejpam-4877 487 14 and and CCONJ ejpam-4877 487 15 s s NOUN ejpam-4877 487 16 = = PUNCT ejpam-4877 487 17 ( ( PUNCT ejpam-4877 487 18 v1 v1 PROPN ejpam-4877 487 19 , , PUNCT ejpam-4877 487 20 v2 v2 PROPN ejpam-4877 487 21 , , PUNCT ejpam-4877 487 22 v3 v3 PROPN ejpam-4877 487 23 , , PUNCT ejpam-4877 487 24 v4 v4 PROPN ejpam-4877 487 25 ) ) PUNCT ejpam-4877 487 26 . . PUNCT ejpam-4877 488 1 clearly clearly ADV ejpam-4877 488 2 , , PUNCT ejpam-4877 488 3 s s VERB ejpam-4877 488 4 is be AUX ejpam-4877 488 5 a a DET ejpam-4877 488 6 total total ADJ ejpam-4877 488 7 dominating dominating NOUN ejpam-4877 488 8 set set NOUN ejpam-4877 488 9 of of ADP ejpam-4877 488 10 cn cn PROPN ejpam-4877 488 11 . . PROPN ejpam-4877 488 12 observe observe VERB ejpam-4877 488 13 that that SCONJ ejpam-4877 488 14 vn vn PROPN ejpam-4877 488 15 ∈ ∈ PROPN ejpam-4877 488 16 ncn ncn PROPN ejpam-4877 488 17 ( ( PUNCT ejpam-4877 488 18 v2 v2 PROPN ejpam-4877 488 19 ) ) PUNCT ejpam-4877 488 20 \ncn \ncn NOUN ejpam-4877 488 21 ( ( PUNCT ejpam-4877 488 22 v1 v1 NOUN ejpam-4877 488 23 ) ) PUNCT ejpam-4877 488 24 , , PUNCT ejpam-4877 488 25 v1 v1 PROPN ejpam-4877 488 26 ∈ ∈ PROPN ejpam-4877 488 27 ncn ncn NOUN ejpam-4877 488 28 ( ( PUNCT ejpam-4877 488 29 v3 v3 PROPN ejpam-4877 488 30 ) ) PUNCT ejpam-4877 488 31 \ \ PUNCT ejpam-4877 489 1 ( ( PUNCT ejpam-4877 489 2 ncn ncn NOUN ejpam-4877 489 3 ( ( PUNCT ejpam-4877 489 4 v2)∪ncn v2)∪ncn PROPN ejpam-4877 489 5 ( ( PUNCT ejpam-4877 489 6 v1 v1 NOUN ejpam-4877 489 7 ) ) PUNCT ejpam-4877 489 8 ) ) PUNCT ejpam-4877 489 9 , , PUNCT ejpam-4877 489 10 and and CCONJ ejpam-4877 489 11 v2 v2 PROPN ejpam-4877 489 12 ∈ ∈ PROPN ejpam-4877 489 13 ncn ncn NOUN ejpam-4877 489 14 ( ( PUNCT ejpam-4877 489 15 v4 v4 PROPN ejpam-4877 489 16 ) ) PUNCT ejpam-4877 489 17 \ \ PUNCT ejpam-4877 489 18 ( ( PUNCT ejpam-4877 489 19 ncn ncn NOUN ejpam-4877 489 20 ( ( PUNCT ejpam-4877 489 21 v3 v3 PROPN ejpam-4877 489 22 ) ) PUNCT ejpam-4877 489 23 ∪ ∪ PROPN ejpam-4877 489 24 ncn ncn NOUN ejpam-4877 489 25 ( ( PUNCT ejpam-4877 489 26 v2 v2 PROPN ejpam-4877 489 27 ) ) PUNCT ejpam-4877 489 28 ∪ ∪ NOUN ejpam-4877 489 29 ncn ncn NOUN ejpam-4877 489 30 ( ( PUNCT ejpam-4877 489 31 v1 v1 NOUN ejpam-4877 489 32 ) ) PUNCT ejpam-4877 489 33 ) ) PUNCT ejpam-4877 489 34 . . PUNCT ejpam-4877 490 1 thus thus ADV ejpam-4877 490 2 , , PUNCT ejpam-4877 490 3 s s VERB ejpam-4877 490 4 is be AUX ejpam-4877 490 5 a a DET ejpam-4877 490 6 grundy grundy PROPN ejpam-4877 490 7 total total ADJ ejpam-4877 490 8 dominating dominating NOUN ejpam-4877 490 9 sequence sequence NOUN ejpam-4877 490 10 of of ADP ejpam-4877 490 11 cn cn PROPN ejpam-4877 490 12 and and CCONJ ejpam-4877 490 13 γtgr(cn γtgr(cn PROPN ejpam-4877 490 14 ) ) PUNCT ejpam-4877 490 15 ≥ ≥ NOUN ejpam-4877 490 16 4 4 NUM ejpam-4877 490 17 . . PUNCT ejpam-4877 491 1 on on ADP ejpam-4877 491 2 the the DET ejpam-4877 491 3 other other ADJ ejpam-4877 491 4 hand hand NOUN ejpam-4877 491 5 , , PUNCT ejpam-4877 491 6 suppose suppose VERB ejpam-4877 491 7 γtgr(cn γtgr(cn NOUN ejpam-4877 491 8 ) ) PUNCT ejpam-4877 491 9 = = SYM ejpam-4877 492 1 k k NOUN ejpam-4877 492 2 , , PUNCT ejpam-4877 492 3 say say VERB ejpam-4877 492 4 s s X ejpam-4877 492 5 = = PUNCT ejpam-4877 492 6 ( ( PUNCT ejpam-4877 492 7 u1 u1 PROPN ejpam-4877 492 8 , , PUNCT ejpam-4877 492 9 u2 u2 PROPN ejpam-4877 492 10 , , PUNCT ejpam-4877 492 11 · · PUNCT ejpam-4877 492 12 · · PUNCT ejpam-4877 492 13 · · PUNCT ejpam-4877 492 14 , , PUNCT ejpam-4877 492 15 uk uk PROPN ejpam-4877 492 16 ) ) PUNCT ejpam-4877 492 17 is be AUX ejpam-4877 492 18 a a DET ejpam-4877 492 19 grundy grundy PROPN ejpam-4877 492 20 total total ADJ ejpam-4877 492 21 dominating dominating NOUN ejpam-4877 492 22 sequence sequence NOUN ejpam-4877 492 23 of of ADP ejpam-4877 492 24 cn cn PROPN ejpam-4877 492 25 . . PUNCT ejpam-4877 493 1 we we PRON ejpam-4877 493 2 may may AUX ejpam-4877 493 3 assume assume VERB ejpam-4877 493 4 that that SCONJ ejpam-4877 493 5 u1 u1 NOUN ejpam-4877 493 6 = = SYM ejpam-4877 493 7 v1 v1 PROPN ejpam-4877 493 8 . . PUNCT ejpam-4877 494 1 then then ADV ejpam-4877 494 2 |ncn |ncn PROPN ejpam-4877 494 3 ( ( PUNCT ejpam-4877 494 4 u1)| u1)| PROPN ejpam-4877 494 5 = = SYM ejpam-4877 494 6 |ncn |ncn NOUN ejpam-4877 494 7 ( ( PUNCT ejpam-4877 494 8 v1)| v1)| NOUN ejpam-4877 494 9 = = SYM ejpam-4877 494 10 n− n− NOUN ejpam-4877 494 11 3 3 NUM ejpam-4877 494 12 . . PUNCT ejpam-4877 495 1 thus thus ADV ejpam-4877 495 2 , , PUNCT ejpam-4877 495 3 γtgr(cn γtgr(cn NOUN ejpam-4877 495 4 ) ) PUNCT ejpam-4877 495 5 = = PUNCT ejpam-4877 496 1 k k PROPN ejpam-4877 496 2 ≤ ≤ ADJ ejpam-4877 496 3 n− n− NOUN ejpam-4877 496 4 ( ( PUNCT ejpam-4877 496 5 n− n− NOUN ejpam-4877 496 6 3− 3− NUM ejpam-4877 496 7 1 1 NUM ejpam-4877 496 8 ) ) PUNCT ejpam-4877 496 9 = = SYM ejpam-4877 496 10 4 4 NUM ejpam-4877 496 11 by by ADP ejpam-4877 496 12 lemma lemma PROPN ejpam-4877 496 13 2 2 NUM ejpam-4877 496 14 . . PUNCT ejpam-4877 496 15 therefore therefore ADV ejpam-4877 496 16 , , PUNCT ejpam-4877 496 17 γtgr(cn γtgr(cn PROPN ejpam-4877 496 18 ) ) PUNCT ejpam-4877 496 19 = = PUNCT ejpam-4877 496 20 4 4 X ejpam-4877 496 21 . . X ejpam-4877 497 1 throughout throughout ADP ejpam-4877 497 2 , , PUNCT ejpam-4877 497 3 [ [ X ejpam-4877 497 4 n n X ejpam-4877 497 5 ] ] X ejpam-4877 497 6 = = PUNCT ejpam-4877 497 7 { { PUNCT ejpam-4877 497 8 1 1 NUM ejpam-4877 497 9 , , PUNCT ejpam-4877 497 10 2 2 NUM ejpam-4877 497 11 , , PUNCT ejpam-4877 497 12 . . PUNCT ejpam-4877 497 13 . . PUNCT ejpam-4877 498 1 . . PUNCT ejpam-4877 499 1 , , PUNCT ejpam-4877 499 2 n n CCONJ ejpam-4877 499 3 } } PUNCT ejpam-4877 499 4 for for SCONJ ejpam-4877 499 5 each each DET ejpam-4877 499 6 positive positive ADJ ejpam-4877 499 7 integer integer NOUN ejpam-4877 499 8 n. n. PROPN ejpam-4877 499 9 lemma lemma PROPN ejpam-4877 499 10 3 3 X ejpam-4877 499 11 . . PUNCT ejpam-4877 499 12 let let VERB ejpam-4877 499 13 g g PRON ejpam-4877 499 14 be be AUX ejpam-4877 499 15 a a DET ejpam-4877 499 16 graph graph NOUN ejpam-4877 499 17 such such ADJ ejpam-4877 499 18 that that PRON ejpam-4877 499 19 γ(g γ(g PROPN ejpam-4877 499 20 ) ) PUNCT ejpam-4877 499 21 ̸= ̸= PROPN ejpam-4877 499 22 1 1 NUM ejpam-4877 499 23 . . PUNCT ejpam-4877 500 1 a a DET ejpam-4877 500 2 sequence sequence NOUN ejpam-4877 500 3 s s PART ejpam-4877 500 4 is be AUX ejpam-4877 500 5 a a DET ejpam-4877 500 6 co co ADJ ejpam-4877 500 7 - - ADJ ejpam-4877 500 8 legal legal ADJ ejpam-4877 500 9 open open ADJ ejpam-4877 500 10 neighborhood neighborhood NOUN ejpam-4877 500 11 sequence sequence NOUN ejpam-4877 500 12 in in ADP ejpam-4877 500 13 g g PROPN ejpam-4877 500 14 if if SCONJ ejpam-4877 501 1 and and CCONJ ejpam-4877 501 2 only only ADV ejpam-4877 501 3 if if SCONJ ejpam-4877 501 4 s s NOUN ejpam-4877 501 5 is be AUX ejpam-4877 501 6 a a DET ejpam-4877 501 7 legal legal ADJ ejpam-4877 501 8 open open ADJ ejpam-4877 501 9 neighborhood neighborhood NOUN ejpam-4877 501 10 sequence sequence NOUN ejpam-4877 501 11 in in ADP ejpam-4877 501 12 g. g. PROPN ejpam-4877 501 13 moreover moreover ADV ejpam-4877 501 14 , , PUNCT ejpam-4877 501 15 s s VERB ejpam-4877 501 16 is be AUX ejpam-4877 501 17 a a DET ejpam-4877 501 18 co co ADJ ejpam-4877 501 19 - - ADJ ejpam-4877 501 20 grundy grundy ADJ ejpam-4877 501 21 total total ADJ ejpam-4877 501 22 dominating dominating NOUN ejpam-4877 501 23 sequence sequence NOUN ejpam-4877 501 24 in in ADP ejpam-4877 501 25 g g PROPN ejpam-4877 501 26 if if SCONJ ejpam-4877 502 1 and and CCONJ ejpam-4877 502 2 only only ADV ejpam-4877 502 3 if if SCONJ ejpam-4877 502 4 it it PRON ejpam-4877 502 5 is be AUX ejpam-4877 502 6 a a DET ejpam-4877 502 7 grundy grundy PROPN ejpam-4877 502 8 total total ADJ ejpam-4877 502 9 dominating dominating NOUN ejpam-4877 502 10 sequence sequence NOUN ejpam-4877 502 11 in in ADP ejpam-4877 502 12 g. g. PROPN ejpam-4877 502 13 in in ADP ejpam-4877 502 14 particular particular ADJ ejpam-4877 502 15 , , PUNCT ejpam-4877 502 16 γtcogr(g γtcogr(g NOUN ejpam-4877 502 17 ) ) PUNCT ejpam-4877 502 18 = = SYM ejpam-4877 502 19 γtgr(g γtgr(g PROPN ejpam-4877 502 20 ) ) PUNCT ejpam-4877 502 21 . . PUNCT ejpam-4877 503 1 proof proof NOUN ejpam-4877 503 2 . . PUNCT ejpam-4877 504 1 let let VERB ejpam-4877 504 2 s s PRON ejpam-4877 504 3 = = PUNCT ejpam-4877 504 4 ( ( PUNCT ejpam-4877 504 5 u1 u1 PROPN ejpam-4877 504 6 , , PUNCT ejpam-4877 504 7 · · PUNCT ejpam-4877 504 8 · · PUNCT ejpam-4877 504 9 · · PUNCT ejpam-4877 504 10 , , PUNCT ejpam-4877 504 11 ut ut AUX ejpam-4877 504 12 ) ) PUNCT ejpam-4877 504 13 be be AUX ejpam-4877 504 14 a a DET ejpam-4877 504 15 sequence sequence NOUN ejpam-4877 504 16 in in ADP ejpam-4877 504 17 g. g. PROPN ejpam-4877 504 18 notice notice VERB ejpam-4877 504 19 that that SCONJ ejpam-4877 504 20 v v X ejpam-4877 504 21 ( ( PUNCT ejpam-4877 504 22 g g NOUN ejpam-4877 504 23 ) ) PUNCT ejpam-4877 504 24 \ng[ui \ng[ui X ejpam-4877 504 25 ] ] PUNCT ejpam-4877 505 1 = = SYM ejpam-4877 505 2 ng(ui ng(ui NOUN ejpam-4877 505 3 ) ) PUNCT ejpam-4877 505 4 for for ADP ejpam-4877 505 5 each each DET ejpam-4877 505 6 i i PRON ejpam-4877 505 7 ∈ ∈ PROPN ejpam-4877 506 1 [ [ X ejpam-4877 506 2 t t X ejpam-4877 506 3 ] ] PUNCT ejpam-4877 506 4 . . PUNCT ejpam-4877 507 1 so so ADV ejpam-4877 507 2 , , PUNCT ejpam-4877 507 3 [ [ X ejpam-4877 507 4 v v X ejpam-4877 507 5 ( ( PUNCT ejpam-4877 507 6 g g NOUN ejpam-4877 507 7 ) ) PUNCT ejpam-4877 507 8 \ng[ui \ng[ui PROPN ejpam-4877 507 9 ] ] X ejpam-4877 507 10 ] ] PUNCT ejpam-4877 507 11 \ \ X ejpam-4877 507 12 ∪i−1 ∪i−1 PROPN ejpam-4877 507 13 j=1[v j=1[v X ejpam-4877 507 14 ( ( PUNCT ejpam-4877 507 15 g g NOUN ejpam-4877 507 16 ) ) PUNCT ejpam-4877 507 17 \ng[uj \ng[uj NOUN ejpam-4877 507 18 ] ] PUNCT ejpam-4877 507 19 ] ] X ejpam-4877 507 20 = = SYM ejpam-4877 507 21 ng(ui ng(ui PROPN ejpam-4877 507 22 ) ) PUNCT ejpam-4877 507 23 \ \ PROPN ejpam-4877 507 24 ∪ ∪ ADP ejpam-4877 507 25 i−1 i−1 PROPN ejpam-4877 507 26 j=1ng(uj j=1ng(uj PROPN ejpam-4877 507 27 ) ) PUNCT ejpam-4877 507 28 . . PUNCT ejpam-4877 508 1 hence hence ADV ejpam-4877 508 2 , , PUNCT ejpam-4877 508 3 s s VERB ejpam-4877 508 4 is be AUX ejpam-4877 508 5 a a DET ejpam-4877 508 6 co co ADJ ejpam-4877 508 7 - - ADJ ejpam-4877 508 8 legal legal ADJ ejpam-4877 508 9 open open ADJ ejpam-4877 508 10 neighborhood neighborhood NOUN ejpam-4877 508 11 sequence sequence NOUN ejpam-4877 508 12 in in ADP ejpam-4877 508 13 g g PROPN ejpam-4877 508 14 if if SCONJ ejpam-4877 509 1 and and CCONJ ejpam-4877 509 2 only only ADV ejpam-4877 509 3 if if SCONJ ejpam-4877 509 4 it it PRON ejpam-4877 509 5 is be AUX ejpam-4877 509 6 a a DET ejpam-4877 509 7 legal legal ADJ ejpam-4877 509 8 open open ADJ ejpam-4877 509 9 neighborhood neighborhood NOUN ejpam-4877 509 10 sequence sequence NOUN ejpam-4877 509 11 in in ADP ejpam-4877 509 12 g. g. PROPN ejpam-4877 509 13 clearly clearly ADV ejpam-4877 509 14 , , PUNCT ejpam-4877 509 15 a a DET ejpam-4877 509 16 co co ADJ ejpam-4877 509 17 - - ADJ ejpam-4877 509 18 legal legal ADJ ejpam-4877 509 19 open open ADJ ejpam-4877 509 20 neighborhood neighborhood NOUN ejpam-4877 509 21 sequence sequence NOUN ejpam-4877 509 22 in in ADP ejpam-4877 509 23 g g PROPN ejpam-4877 509 24 is be AUX ejpam-4877 509 25 a a DET ejpam-4877 509 26 cogrundy cogrundy NOUN ejpam-4877 509 27 total total ADJ ejpam-4877 509 28 dominating dominating NOUN ejpam-4877 509 29 sequence sequence NOUN ejpam-4877 509 30 if if SCONJ ejpam-4877 509 31 and and CCONJ ejpam-4877 509 32 only only ADV ejpam-4877 509 33 if if SCONJ ejpam-4877 509 34 it it PRON ejpam-4877 509 35 is be AUX ejpam-4877 509 36 a a DET ejpam-4877 509 37 grundy grundy PROPN ejpam-4877 509 38 total total ADJ ejpam-4877 509 39 dominating dominating NOUN ejpam-4877 509 40 sequence sequence NOUN ejpam-4877 509 41 in in ADP ejpam-4877 509 42 g. g. PROPN ejpam-4877 509 43 consequently consequently ADV ejpam-4877 509 44 , , PUNCT ejpam-4877 509 45 γtcogr(g γtcogr(g X ejpam-4877 509 46 ) ) PUNCT ejpam-4877 509 47 = = SYM ejpam-4877 509 48 γtgr(g γtgr(g PROPN ejpam-4877 509 49 ) ) PUNCT ejpam-4877 509 50 . . PUNCT ejpam-4877 510 1 theorem theorem ADJ ejpam-4877 510 2 7 7 NUM ejpam-4877 510 3 . . PUNCT ejpam-4877 511 1 let let VERB ejpam-4877 511 2 g g NOUN ejpam-4877 512 1 and and CCONJ ejpam-4877 512 2 h h NOUN ejpam-4877 512 3 be be VERB ejpam-4877 512 4 any any DET ejpam-4877 512 5 two two NUM ejpam-4877 512 6 graphs graph NOUN ejpam-4877 512 7 such such ADJ ejpam-4877 512 8 that that PRON ejpam-4877 512 9 γ(g γ(g PROPN ejpam-4877 512 10 ) ) PUNCT ejpam-4877 512 11 ̸= ̸= PROPN ejpam-4877 512 12 1 1 NUM ejpam-4877 512 13 and and CCONJ ejpam-4877 512 14 γ(h γ(h NOUN ejpam-4877 512 15 ) ) PUNCT ejpam-4877 512 16 ̸= ̸= PROPN ejpam-4877 512 17 1 1 NUM ejpam-4877 512 18 . . PUNCT ejpam-4877 513 1 a a DET ejpam-4877 513 2 sequence sequence NOUN ejpam-4877 513 3 s s VERB ejpam-4877 513 4 of of ADP ejpam-4877 513 5 distinct distinct ADJ ejpam-4877 513 6 vertices vertex NOUN ejpam-4877 513 7 of of ADP ejpam-4877 513 8 g+h g+h PROPN ejpam-4877 513 9 is be AUX ejpam-4877 513 10 a a DET ejpam-4877 513 11 legal legal ADJ ejpam-4877 513 12 open open ADJ ejpam-4877 513 13 hop hop NOUN ejpam-4877 513 14 neighborhood neighborhood NOUN ejpam-4877 513 15 sequence sequence NOUN ejpam-4877 513 16 if if SCONJ ejpam-4877 513 17 and and CCONJ ejpam-4877 513 18 only only ADV ejpam-4877 513 19 if if SCONJ ejpam-4877 513 20 one one NUM ejpam-4877 513 21 of of ADP ejpam-4877 513 22 the the DET ejpam-4877 513 23 following follow VERB ejpam-4877 513 24 holds hold VERB ejpam-4877 513 25 : : PUNCT ejpam-4877 513 26 ( ( PUNCT ejpam-4877 513 27 i i NOUN ejpam-4877 513 28 ) ) PUNCT ejpam-4877 513 29 s s VERB ejpam-4877 513 30 is be AUX ejpam-4877 513 31 a a DET ejpam-4877 513 32 co co ADJ ejpam-4877 513 33 - - ADJ ejpam-4877 513 34 legal legal ADJ ejpam-4877 513 35 open open ADJ ejpam-4877 513 36 neighborhood neighborhood NOUN ejpam-4877 513 37 sequence sequence NOUN ejpam-4877 513 38 in in ADP ejpam-4877 513 39 g g PROPN ejpam-4877 513 40 ( ( PUNCT ejpam-4877 513 41 legal legal ADJ ejpam-4877 513 42 open open ADJ ejpam-4877 513 43 neighborhood neighborhood NOUN ejpam-4877 513 44 sequence sequence NOUN ejpam-4877 513 45 in in ADP ejpam-4877 513 46 g g NOUN ejpam-4877 513 47 ) ) PUNCT ejpam-4877 513 48 . . PUNCT ejpam-4877 514 1 ( ( PUNCT ejpam-4877 514 2 ii ii X ejpam-4877 514 3 ) ) PUNCT ejpam-4877 514 4 s s VERB ejpam-4877 514 5 is be AUX ejpam-4877 514 6 a a DET ejpam-4877 514 7 co co ADJ ejpam-4877 514 8 - - ADJ ejpam-4877 514 9 legal legal ADJ ejpam-4877 514 10 open open ADJ ejpam-4877 514 11 neighborhood neighborhood NOUN ejpam-4877 514 12 sequence sequence NOUN ejpam-4877 514 13 in in ADP ejpam-4877 514 14 h h PROPN ejpam-4877 514 15 ( ( PUNCT ejpam-4877 514 16 legal legal ADJ ejpam-4877 514 17 open open ADJ ejpam-4877 514 18 neighborhood neighborhood NOUN ejpam-4877 514 19 sequence sequence NOUN ejpam-4877 514 20 in in ADP ejpam-4877 514 21 h h NOUN ejpam-4877 514 22 ) ) PUNCT ejpam-4877 514 23 . . PUNCT ejpam-4877 515 1 ( ( PUNCT ejpam-4877 515 2 iii iii X ejpam-4877 515 3 ) ) PUNCT ejpam-4877 515 4 s s VERB ejpam-4877 515 5 is be AUX ejpam-4877 515 6 loh loh NOUN ejpam-4877 515 7 - - PUNCT ejpam-4877 515 8 identical identical ADJ ejpam-4877 515 9 to to ADP ejpam-4877 515 10 s′ s′ ADJ ejpam-4877 515 11 = = NOUN ejpam-4877 515 12 sg⊕sh sg⊕sh NOUN ejpam-4877 515 13 , , PUNCT ejpam-4877 515 14 where where SCONJ ejpam-4877 515 15 sg sg PROPN ejpam-4877 515 16 and and CCONJ ejpam-4877 515 17 sh sh PROPN ejpam-4877 515 18 are be AUX ejpam-4877 515 19 co co ADJ ejpam-4877 515 20 - - ADJ ejpam-4877 515 21 legal legal ADJ ejpam-4877 515 22 open open ADJ ejpam-4877 515 23 neighborhood neighborhood NOUN ejpam-4877 515 24 sequences sequence NOUN ejpam-4877 515 25 in in ADP ejpam-4877 515 26 g g PROPN ejpam-4877 515 27 and and CCONJ ejpam-4877 515 28 h h NOUN ejpam-4877 515 29 , , PUNCT ejpam-4877 515 30 respectively respectively ADV ejpam-4877 515 31 . . PUNCT ejpam-4877 516 1 j.a j.a PROPN ejpam-4877 516 2 . . PROPN ejpam-4877 516 3 hassan hassan PROPN ejpam-4877 516 4 , , PUNCT ejpam-4877 516 5 s. s. PROPN ejpam-4877 516 6 canoy canoy PROPN ejpam-4877 516 7 / / SYM ejpam-4877 516 8 eur eur PROPN ejpam-4877 516 9 . . PUNCT ejpam-4877 517 1 j. j. PROPN ejpam-4877 517 2 pure pure PROPN ejpam-4877 517 3 appl appl PROPN ejpam-4877 517 4 . . PROPN ejpam-4877 517 5 math math PROPN ejpam-4877 517 6 , , PUNCT ejpam-4877 517 7 16 16 NUM ejpam-4877 517 8 ( ( PUNCT ejpam-4877 517 9 4 4 NUM ejpam-4877 517 10 ) ) PUNCT ejpam-4877 517 11 ( ( PUNCT ejpam-4877 517 12 2023 2023 NUM ejpam-4877 517 13 ) ) PUNCT ejpam-4877 517 14 , , PUNCT ejpam-4877 517 15 2597 2597 NUM ejpam-4877 517 16 - - SYM ejpam-4877 517 17 2612 2612 NUM ejpam-4877 517 18 2609 2609 NUM ejpam-4877 517 19 proof proof NOUN ejpam-4877 517 20 . . PUNCT ejpam-4877 517 21 suppose suppose VERB ejpam-4877 517 22 that that SCONJ ejpam-4877 517 23 s s VERB ejpam-4877 517 24 = = PUNCT ejpam-4877 517 25 ( ( PUNCT ejpam-4877 517 26 u1 u1 PROPN ejpam-4877 517 27 , , PUNCT ejpam-4877 517 28 · · PUNCT ejpam-4877 517 29 · · PUNCT ejpam-4877 517 30 · · PUNCT ejpam-4877 517 31 , , PUNCT ejpam-4877 517 32 ut ut PROPN ejpam-4877 517 33 ) ) PUNCT ejpam-4877 517 34 is be AUX ejpam-4877 517 35 a a DET ejpam-4877 517 36 legal legal ADJ ejpam-4877 517 37 open open ADJ ejpam-4877 517 38 hop hop NOUN ejpam-4877 517 39 neighborhood neighborhood NOUN ejpam-4877 517 40 sequence sequence NOUN ejpam-4877 517 41 in in ADP ejpam-4877 517 42 g g PROPN ejpam-4877 517 43 + + NOUN ejpam-4877 517 44 h h NOUN ejpam-4877 517 45 and and CCONJ ejpam-4877 517 46 let let VERB ejpam-4877 517 47 ŝ ŝ NOUN ejpam-4877 517 48 be be AUX ejpam-4877 517 49 the the DET ejpam-4877 517 50 corresponding corresponding ADJ ejpam-4877 517 51 set set NOUN ejpam-4877 517 52 of of ADP ejpam-4877 517 53 s. s. PROPN ejpam-4877 517 54 suppose suppose VERB ejpam-4877 517 55 further far ADV ejpam-4877 517 56 that that PRON ejpam-4877 517 57 ŝ ŝ VERB ejpam-4877 517 58 ⊆ ⊆ NUM ejpam-4877 517 59 v v NOUN ejpam-4877 517 60 ( ( PUNCT ejpam-4877 517 61 g g NOUN ejpam-4877 517 62 ) ) PUNCT ejpam-4877 517 63 . . PUNCT ejpam-4877 518 1 then then ADV ejpam-4877 518 2 by by ADP ejpam-4877 518 3 the the DET ejpam-4877 518 4 legality legality NOUN ejpam-4877 518 5 condition condition NOUN ejpam-4877 518 6 in in ADP ejpam-4877 518 7 s s PROPN ejpam-4877 518 8 , , PUNCT ejpam-4877 518 9 we we PRON ejpam-4877 518 10 have have VERB ejpam-4877 518 11 n2 n2 PROPN ejpam-4877 518 12 g+h(ui g+h(ui PROPN ejpam-4877 518 13 ) ) PUNCT ejpam-4877 518 14 \ \ NOUN ejpam-4877 519 1 ∪i−1 ∪i−1 AUX ejpam-4877 519 2 j=1n j=1n NOUN ejpam-4877 519 3 2 2 NUM ejpam-4877 519 4 g+h(uj g+h(uj NOUN ejpam-4877 519 5 ) ) PUNCT ejpam-4877 519 6 ̸= ̸= NOUN ejpam-4877 519 7 ∅ ∅ NOUN ejpam-4877 519 8 for for ADP ejpam-4877 519 9 each each DET ejpam-4877 519 10 i i PRON ejpam-4877 519 11 ∈ ∈ PROPN ejpam-4877 519 12 { { PUNCT ejpam-4877 519 13 2 2 NUM ejpam-4877 519 14 , , PUNCT ejpam-4877 519 15 3 3 NUM ejpam-4877 519 16 , , PUNCT ejpam-4877 519 17 . . PUNCT ejpam-4877 519 18 . . PUNCT ejpam-4877 520 1 . . PUNCT ejpam-4877 521 1 , , PUNCT ejpam-4877 521 2 t t PROPN ejpam-4877 521 3 } } PUNCT ejpam-4877 521 4 . . PUNCT ejpam-4877 522 1 since since SCONJ ejpam-4877 522 2 n2 n2 PROPN ejpam-4877 522 3 g+h(ui g+h(ui PROPN ejpam-4877 522 4 ) ) PUNCT ejpam-4877 522 5 = = SYM ejpam-4877 522 6 v v NOUN ejpam-4877 522 7 ( ( PUNCT ejpam-4877 522 8 g g NOUN ejpam-4877 522 9 ) ) PUNCT ejpam-4877 522 10 \ng[ui \ng[ui PROPN ejpam-4877 522 11 ] ] PUNCT ejpam-4877 522 12 for for ADP ejpam-4877 522 13 each each DET ejpam-4877 522 14 i i PRON ejpam-4877 522 15 ∈ ∈ PROPN ejpam-4877 522 16 [ [ X ejpam-4877 522 17 t t X ejpam-4877 522 18 ] ] PUNCT ejpam-4877 522 19 , , PUNCT ejpam-4877 522 20 it it PRON ejpam-4877 522 21 follows follow VERB ejpam-4877 522 22 that that SCONJ ejpam-4877 522 23 [ [ X ejpam-4877 522 24 v v X ejpam-4877 522 25 ( ( PUNCT ejpam-4877 522 26 g g NOUN ejpam-4877 522 27 ) ) PUNCT ejpam-4877 522 28 \ng[ui \ng[ui PROPN ejpam-4877 523 1 ] ] X ejpam-4877 523 2 ] ] PUNCT ejpam-4877 523 3 \ \ X ejpam-4877 523 4 ∪i−1 ∪i−1 PROPN ejpam-4877 523 5 j=1[v j=1[v X ejpam-4877 523 6 ( ( PUNCT ejpam-4877 523 7 g g NOUN ejpam-4877 523 8 ) ) PUNCT ejpam-4877 523 9 \ng[uj \ng[uj NOUN ejpam-4877 523 10 ] ] X ejpam-4877 523 11 ] ] X ejpam-4877 523 12 ̸= ̸= NOUN ejpam-4877 523 13 ∅ ∅ NOUN ejpam-4877 523 14 for for ADP ejpam-4877 523 15 each each DET ejpam-4877 523 16 i i PRON ejpam-4877 523 17 ∈ ∈ PROPN ejpam-4877 523 18 { { PUNCT ejpam-4877 523 19 2 2 NUM ejpam-4877 523 20 , , PUNCT ejpam-4877 523 21 3 3 NUM ejpam-4877 523 22 , , PUNCT ejpam-4877 523 23 . . PUNCT ejpam-4877 523 24 . . PUNCT ejpam-4877 524 1 . . PUNCT ejpam-4877 525 1 , , PUNCT ejpam-4877 525 2 t t PROPN ejpam-4877 525 3 } } PUNCT ejpam-4877 525 4 . . PUNCT ejpam-4877 526 1 hence hence ADV ejpam-4877 526 2 , , PUNCT ejpam-4877 526 3 s s VERB ejpam-4877 526 4 is be AUX ejpam-4877 526 5 a a DET ejpam-4877 526 6 co co ADJ ejpam-4877 526 7 - - ADJ ejpam-4877 526 8 legal legal ADJ ejpam-4877 526 9 open open ADJ ejpam-4877 526 10 neighborhood neighborhood NOUN ejpam-4877 526 11 sequence sequence NOUN ejpam-4877 526 12 in in ADP ejpam-4877 526 13 g g NOUN ejpam-4877 526 14 , , PUNCT ejpam-4877 526 15 and and CCONJ ejpam-4877 526 16 so so ADV ejpam-4877 526 17 ( ( PUNCT ejpam-4877 526 18 i i NOUN ejpam-4877 526 19 ) ) PUNCT ejpam-4877 526 20 holds hold VERB ejpam-4877 526 21 . . PUNCT ejpam-4877 527 1 similarly similarly ADV ejpam-4877 527 2 , , PUNCT ejpam-4877 527 3 ( ( PUNCT ejpam-4877 527 4 ii ii NOUN ejpam-4877 527 5 ) ) PUNCT ejpam-4877 527 6 holds hold VERB ejpam-4877 527 7 if if SCONJ ejpam-4877 527 8 ŝ ŝ ADP ejpam-4877 527 9 ⊆ ⊆ NUM ejpam-4877 527 10 v v NOUN ejpam-4877 527 11 ( ( PUNCT ejpam-4877 527 12 h h NOUN ejpam-4877 527 13 ) ) PUNCT ejpam-4877 527 14 . . PUNCT ejpam-4877 528 1 next next ADV ejpam-4877 528 2 , , PUNCT ejpam-4877 528 3 suppose suppose VERB ejpam-4877 528 4 that that SCONJ ejpam-4877 528 5 ŝ ŝ VERB ejpam-4877 528 6 ∩ ∩ PROPN ejpam-4877 528 7 v v ADP ejpam-4877 528 8 ( ( PUNCT ejpam-4877 528 9 g g NOUN ejpam-4877 528 10 ) ) PUNCT ejpam-4877 528 11 ̸= ̸= PROPN ejpam-4877 528 12 ∅ ∅ NOUN ejpam-4877 528 13 and and CCONJ ejpam-4877 528 14 ŝ ŝ VERB ejpam-4877 528 15 ∩ ∩ ADJ ejpam-4877 528 16 v v X ejpam-4877 528 17 ( ( PUNCT ejpam-4877 528 18 h h NOUN ejpam-4877 528 19 ) ) PUNCT ejpam-4877 528 20 ̸= ̸= PROPN ejpam-4877 528 21 ∅. ∅. ADV ejpam-4877 528 22 since since SCONJ ejpam-4877 528 23 n2 n2 PROPN ejpam-4877 528 24 g+h(uj g+h(uj PROPN ejpam-4877 528 25 ) ) PUNCT ejpam-4877 528 26 ⊆ ⊆ NUM ejpam-4877 528 27 v v NOUN ejpam-4877 528 28 ( ( PUNCT ejpam-4877 528 29 g g NOUN ejpam-4877 528 30 ) ) PUNCT ejpam-4877 528 31 for for ADP ejpam-4877 528 32 all all DET ejpam-4877 528 33 uj uj PROPN ejpam-4877 528 34 ∈ ∈ PROPN ejpam-4877 528 35 ŝ ŝ VERB ejpam-4877 528 36 ∩v ∩v PROPN ejpam-4877 528 37 ( ( PUNCT ejpam-4877 528 38 g g NOUN ejpam-4877 528 39 ) ) PUNCT ejpam-4877 528 40 and and CCONJ ejpam-4877 528 41 n2 n2 PROPN ejpam-4877 528 42 g+h(us g+h(us PROPN ejpam-4877 528 43 ) ) PUNCT ejpam-4877 528 44 ⊆ ⊆ NUM ejpam-4877 528 45 v v NOUN ejpam-4877 528 46 ( ( PUNCT ejpam-4877 528 47 h h NOUN ejpam-4877 528 48 ) ) PUNCT ejpam-4877 528 49 for for ADP ejpam-4877 528 50 all all PRON ejpam-4877 528 51 us us PROPN ejpam-4877 528 52 ∈ ∈ PROPN ejpam-4877 528 53 ŝ ŝ NOUN ejpam-4877 528 54 ∩v ∩v PROPN ejpam-4877 528 55 ( ( PUNCT ejpam-4877 528 56 h h NOUN ejpam-4877 528 57 ) ) PUNCT ejpam-4877 528 58 , , PUNCT ejpam-4877 528 59 s s VERB ejpam-4877 528 60 is be AUX ejpam-4877 528 61 loh loh NOUN ejpam-4877 528 62 - - PUNCT ejpam-4877 528 63 identical identical ADJ ejpam-4877 528 64 to to ADP ejpam-4877 528 65 s′ s′ ADJ ejpam-4877 528 66 = = PUNCT ejpam-4877 528 67 sg sg ADP ejpam-4877 528 68 ⊕ ⊕ PROPN ejpam-4877 528 69 sh sh INTJ ejpam-4877 528 70 , , PUNCT ejpam-4877 528 71 where where SCONJ ejpam-4877 528 72 ŝg ŝg NOUN ejpam-4877 528 73 = = SYM ejpam-4877 528 74 ŝ ŝ X ejpam-4877 528 75 ∩ ∩ PROPN ejpam-4877 528 76 v v X ejpam-4877 528 77 ( ( PUNCT ejpam-4877 528 78 g g NOUN ejpam-4877 528 79 ) ) PUNCT ejpam-4877 528 80 = = SYM ejpam-4877 528 81 { { PUNCT ejpam-4877 528 82 u1 u1 NOUN ejpam-4877 528 83 , , PUNCT ejpam-4877 528 84 u2 u2 NOUN ejpam-4877 528 85 , , PUNCT ejpam-4877 528 86 . . PUNCT ejpam-4877 528 87 . . PUNCT ejpam-4877 528 88 . . PUNCT ejpam-4877 529 1 , , PUNCT ejpam-4877 529 2 um um INTJ ejpam-4877 529 3 } } PUNCT ejpam-4877 529 4 , , PUNCT ejpam-4877 529 5 ŝh ŝh PRON ejpam-4877 529 6 = = SYM ejpam-4877 529 7 ŝ ŝ X ejpam-4877 529 8 ∩ ∩ PROPN ejpam-4877 529 9 v v X ejpam-4877 529 10 ( ( PUNCT ejpam-4877 529 11 h h NOUN ejpam-4877 529 12 ) ) PUNCT ejpam-4877 529 13 = = SYM ejpam-4877 529 14 { { PUNCT ejpam-4877 529 15 w1 w1 NOUN ejpam-4877 529 16 , , PUNCT ejpam-4877 529 17 w2 w2 NOUN ejpam-4877 529 18 , , PUNCT ejpam-4877 529 19 . . PUNCT ejpam-4877 529 20 . . PUNCT ejpam-4877 530 1 . . PUNCT ejpam-4877 531 1 , , PUNCT ejpam-4877 531 2 wt wt PROPN ejpam-4877 531 3 } } PUNCT ejpam-4877 531 4 , , PUNCT ejpam-4877 531 5 |ŝ| |ŝ| PROPN ejpam-4877 531 6 = = SYM ejpam-4877 531 7 m+t m+t NUM ejpam-4877 531 8 , , PUNCT ejpam-4877 531 9 and and CCONJ ejpam-4877 531 10 the the DET ejpam-4877 531 11 orders order NOUN ejpam-4877 531 12 of of ADP ejpam-4877 531 13 appearances appearance NOUN ejpam-4877 531 14 of of ADP ejpam-4877 531 15 the the DET ejpam-4877 531 16 terms term NOUN ejpam-4877 531 17 of of ADP ejpam-4877 531 18 both both DET ejpam-4877 531 19 sequences sequence NOUN ejpam-4877 531 20 in in ADP ejpam-4877 531 21 s s NOUN ejpam-4877 531 22 are be AUX ejpam-4877 531 23 retained retain VERB ejpam-4877 531 24 . . PUNCT ejpam-4877 532 1 since since SCONJ ejpam-4877 532 2 s s PROPN ejpam-4877 532 3 is be AUX ejpam-4877 532 4 a a DET ejpam-4877 532 5 legal legal ADJ ejpam-4877 532 6 open open ADJ ejpam-4877 532 7 hop hop NOUN ejpam-4877 532 8 neighborhood neighborhood NOUN ejpam-4877 532 9 sequence sequence NOUN ejpam-4877 532 10 , , PUNCT ejpam-4877 532 11 it it PRON ejpam-4877 532 12 follows follow VERB ejpam-4877 532 13 that that SCONJ ejpam-4877 532 14 [ [ X ejpam-4877 532 15 v v X ejpam-4877 532 16 ( ( PUNCT ejpam-4877 532 17 g g NOUN ejpam-4877 532 18 ) ) PUNCT ejpam-4877 532 19 \ng[ui \ng[ui PROPN ejpam-4877 533 1 ] ] X ejpam-4877 533 2 ] ] PUNCT ejpam-4877 533 3 \ \ X ejpam-4877 533 4 ∪i−1 ∪i−1 PROPN ejpam-4877 533 5 j=1[v j=1[v X ejpam-4877 533 6 ( ( PUNCT ejpam-4877 533 7 g g NOUN ejpam-4877 533 8 ) ) PUNCT ejpam-4877 533 9 \ng[uj \ng[uj NOUN ejpam-4877 533 10 ] ] PUNCT ejpam-4877 533 11 ] ] PUNCT ejpam-4877 533 12 = = SYM ejpam-4877 533 13 n2 n2 PROPN ejpam-4877 533 14 g+h(ui g+h(ui PROPN ejpam-4877 533 15 ) ) PUNCT ejpam-4877 533 16 \ \ NOUN ejpam-4877 533 17 ∪i−1 ∪i−1 PROPN ejpam-4877 533 18 j=1n j=1n NOUN ejpam-4877 533 19 2 2 NUM ejpam-4877 533 20 g+h(uj g+h(uj NOUN ejpam-4877 533 21 ) ) PUNCT ejpam-4877 533 22 ̸= ̸= NOUN ejpam-4877 533 23 ∅ ∅ NOUN ejpam-4877 533 24 for for ADP ejpam-4877 533 25 each each DET ejpam-4877 533 26 i i PRON ejpam-4877 533 27 ∈ ∈ PROPN ejpam-4877 533 28 { { PUNCT ejpam-4877 533 29 2 2 NUM ejpam-4877 533 30 , , PUNCT ejpam-4877 533 31 3 3 NUM ejpam-4877 533 32 , , PUNCT ejpam-4877 533 33 . . PUNCT ejpam-4877 533 34 . . PUNCT ejpam-4877 533 35 . . PUNCT ejpam-4877 534 1 , , PUNCT ejpam-4877 534 2 m m VERB ejpam-4877 534 3 } } PUNCT ejpam-4877 534 4 . . PUNCT ejpam-4877 535 1 thus thus ADV ejpam-4877 535 2 , , PUNCT ejpam-4877 535 3 sg sg PROPN ejpam-4877 535 4 is be AUX ejpam-4877 535 5 a a DET ejpam-4877 535 6 co co ADJ ejpam-4877 535 7 - - ADJ ejpam-4877 535 8 legal legal ADJ ejpam-4877 535 9 open open ADJ ejpam-4877 535 10 neighborhood neighborhood NOUN ejpam-4877 535 11 sequence sequence NOUN ejpam-4877 535 12 in in ADP ejpam-4877 535 13 g. g. PROPN ejpam-4877 535 14 similarly similarly ADV ejpam-4877 535 15 , , PUNCT ejpam-4877 535 16 sh sh PROPN ejpam-4877 535 17 is be AUX ejpam-4877 535 18 a a DET ejpam-4877 535 19 co co ADJ ejpam-4877 535 20 - - ADJ ejpam-4877 535 21 legal legal ADJ ejpam-4877 535 22 open open ADJ ejpam-4877 535 23 neighborhood neighborhood NOUN ejpam-4877 535 24 sequence sequence NOUN ejpam-4877 535 25 in in ADP ejpam-4877 535 26 h h NOUN ejpam-4877 535 27 , , PUNCT ejpam-4877 535 28 and and CCONJ ejpam-4877 535 29 so so ADV ejpam-4877 535 30 ( ( PUNCT ejpam-4877 535 31 iii iii NOUN ejpam-4877 535 32 ) ) PUNCT ejpam-4877 535 33 holds hold VERB ejpam-4877 535 34 . . PUNCT ejpam-4877 536 1 the the DET ejpam-4877 536 2 converse converse NOUN ejpam-4877 536 3 is be AUX ejpam-4877 536 4 clear clear ADJ ejpam-4877 536 5 . . PUNCT ejpam-4877 537 1 the the DET ejpam-4877 537 2 next next ADJ ejpam-4877 537 3 result result NOUN ejpam-4877 537 4 follows follow VERB ejpam-4877 537 5 from from ADP ejpam-4877 537 6 lemma lemma PROPN ejpam-4877 537 7 3 3 NUM ejpam-4877 537 8 and and CCONJ ejpam-4877 537 9 theorem theorem VERB ejpam-4877 537 10 7 7 NUM ejpam-4877 537 11 . . PUNCT ejpam-4877 537 12 corollary corollary ADJ ejpam-4877 537 13 3 3 X ejpam-4877 537 14 . . PUNCT ejpam-4877 538 1 let let VERB ejpam-4877 538 2 g g NOUN ejpam-4877 539 1 and and CCONJ ejpam-4877 539 2 h h NOUN ejpam-4877 539 3 be be VERB ejpam-4877 539 4 any any DET ejpam-4877 539 5 two two NUM ejpam-4877 539 6 graphs graph NOUN ejpam-4877 539 7 such such ADJ ejpam-4877 539 8 that that PRON ejpam-4877 539 9 γ(g γ(g PROPN ejpam-4877 539 10 ) ) PUNCT ejpam-4877 539 11 ̸= ̸= PROPN ejpam-4877 539 12 1 1 NUM ejpam-4877 539 13 and and CCONJ ejpam-4877 539 14 γ(h γ(h NOUN ejpam-4877 539 15 ) ) PUNCT ejpam-4877 539 16 ̸= ̸= PROPN ejpam-4877 539 17 1 1 NUM ejpam-4877 539 18 . . PUNCT ejpam-4877 540 1 a a DET ejpam-4877 540 2 sequence sequence NOUN ejpam-4877 540 3 s s VERB ejpam-4877 540 4 of of ADP ejpam-4877 540 5 distinct distinct ADJ ejpam-4877 540 6 vertices vertex NOUN ejpam-4877 540 7 of of ADP ejpam-4877 540 8 g g PROPN ejpam-4877 540 9 + + CCONJ ejpam-4877 540 10 h h NOUN ejpam-4877 540 11 is be AUX ejpam-4877 540 12 a a DET ejpam-4877 540 13 grundy grundy PROPN ejpam-4877 540 14 total total NOUN ejpam-4877 540 15 hop hop NOUN ejpam-4877 540 16 dominating dominating NOUN ejpam-4877 540 17 sequence sequence NOUN ejpam-4877 540 18 in in ADP ejpam-4877 540 19 g+h g+h PROPN ejpam-4877 541 1 if if SCONJ ejpam-4877 541 2 and and CCONJ ejpam-4877 541 3 only only ADV ejpam-4877 541 4 if if SCONJ ejpam-4877 541 5 s s NOUN ejpam-4877 541 6 is be AUX ejpam-4877 541 7 loh loh NOUN ejpam-4877 541 8 - - PUNCT ejpam-4877 541 9 identical identical ADJ ejpam-4877 541 10 to to ADP ejpam-4877 541 11 s′ s′ ADJ ejpam-4877 541 12 = = PUNCT ejpam-4877 541 13 sg sg PART ejpam-4877 541 14 ⊕sh ⊕sh NOUN ejpam-4877 541 15 , , PUNCT ejpam-4877 541 16 where where SCONJ ejpam-4877 541 17 sg sg PROPN ejpam-4877 541 18 and and CCONJ ejpam-4877 541 19 sh sh PROPN ejpam-4877 541 20 are be AUX ejpam-4877 541 21 co co ADJ ejpam-4877 541 22 - - ADJ ejpam-4877 541 23 grundy grundy ADJ ejpam-4877 541 24 total total ADJ ejpam-4877 541 25 dominating dominating NOUN ejpam-4877 541 26 sequences sequence NOUN ejpam-4877 541 27 in in ADP ejpam-4877 541 28 g g PROPN ejpam-4877 541 29 and and CCONJ ejpam-4877 541 30 h h NOUN ejpam-4877 541 31 , , PUNCT ejpam-4877 541 32 respectively respectively ADV ejpam-4877 541 33 ( ( PUNCT ejpam-4877 541 34 grundy grundy PROPN ejpam-4877 541 35 total total NOUN ejpam-4877 541 36 dominating dominating NOUN ejpam-4877 541 37 sequences sequence NOUN ejpam-4877 541 38 in in ADP ejpam-4877 541 39 g g PROPN ejpam-4877 541 40 and and CCONJ ejpam-4877 541 41 h h NOUN ejpam-4877 541 42 , , PUNCT ejpam-4877 541 43 respectively respectively ADV ejpam-4877 541 44 ) ) PUNCT ejpam-4877 541 45 . . PUNCT ejpam-4877 542 1 moreover moreover ADV ejpam-4877 542 2 , , PUNCT ejpam-4877 542 3 γthgr(g+h γthgr(g+h PROPN ejpam-4877 542 4 ) ) PUNCT ejpam-4877 543 1 = = PUNCT ejpam-4877 543 2 γtcogr(g γtcogr(g X ejpam-4877 543 3 ) ) PUNCT ejpam-4877 543 4 + + NUM ejpam-4877 543 5 γtcogr(h γtcogr(h NOUN ejpam-4877 543 6 ) ) PUNCT ejpam-4877 543 7 = = SYM ejpam-4877 543 8 γtgr(g γtgr(g NOUN ejpam-4877 543 9 ) ) PUNCT ejpam-4877 543 10 + + X ejpam-4877 543 11 γtgr(h γtgr(h NOUN ejpam-4877 543 12 ) ) PUNCT ejpam-4877 543 13 . . PUNCT ejpam-4877 544 1 in in ADP ejpam-4877 544 2 particular particular ADJ ejpam-4877 544 3 , , PUNCT ejpam-4877 544 4 we we PRON ejpam-4877 544 5 have have VERB ejpam-4877 544 6 ( ( PUNCT ejpam-4877 544 7 i i NOUN ejpam-4877 544 8 ) ) PUNCT ejpam-4877 544 9 γthgr(km γthgr(km PROPN ejpam-4877 544 10 , , PUNCT ejpam-4877 544 11 n n CCONJ ejpam-4877 544 12 ) ) PUNCT ejpam-4877 544 13 = = SYM ejpam-4877 545 1 γthgr(km γthgr(km PROPN ejpam-4877 545 2 + + PROPN ejpam-4877 545 3 kn kn PROPN ejpam-4877 545 4 ) ) PUNCT ejpam-4877 545 5 = = SYM ejpam-4877 545 6 γtgr(km γtgr(km PROPN ejpam-4877 545 7 ) ) PUNCT ejpam-4877 546 1 + + X ejpam-4877 546 2 γtgr(kn γtgr(kn ADJ ejpam-4877 546 3 ) ) PUNCT ejpam-4877 546 4 = = SYM ejpam-4877 546 5 4 4 NUM ejpam-4877 546 6 for for ADP ejpam-4877 546 7 any any DET ejpam-4877 546 8 m m NOUN ejpam-4877 546 9 , , PUNCT ejpam-4877 546 10 n n PRON ejpam-4877 546 11 ≥ ≥ NOUN ejpam-4877 546 12 2 2 NUM ejpam-4877 546 13 , , PUNCT ejpam-4877 546 14 ( ( PUNCT ejpam-4877 546 15 ii ii NOUN ejpam-4877 546 16 ) ) PUNCT ejpam-4877 546 17 γthgr(kn γthgr(kn NOUN ejpam-4877 546 18 + + CCONJ ejpam-4877 546 19 pm pm NOUN ejpam-4877 546 20 ) ) PUNCT ejpam-4877 547 1 = = NOUN ejpam-4877 547 2 γtgr(kn γtgr(kn ADJ ejpam-4877 547 3 ) ) PUNCT ejpam-4877 547 4 + + CCONJ ejpam-4877 547 5 γtgr(pm γtgr(pm ADJ ejpam-4877 547 6 ) ) PUNCT ejpam-4877 547 7 = = SYM ejpam-4877 547 8 6 6 NUM ejpam-4877 547 9 for for ADP ejpam-4877 547 10 any any DET ejpam-4877 547 11 n n PRON ejpam-4877 547 12 ≥ ≥ NOUN ejpam-4877 547 13 2 2 NUM ejpam-4877 547 14 and and CCONJ ejpam-4877 547 15 m m PROPN ejpam-4877 547 16 ≥ ≥ NOUN ejpam-4877 547 17 4 4 NUM ejpam-4877 547 18 , , PUNCT ejpam-4877 547 19 ( ( PUNCT ejpam-4877 547 20 iii iii X ejpam-4877 547 21 ) ) PUNCT ejpam-4877 547 22 γthgr(kn γthgr(kn NOUN ejpam-4877 547 23 + + CCONJ ejpam-4877 547 24 cm cm NOUN ejpam-4877 547 25 ) ) PUNCT ejpam-4877 548 1 = = NOUN ejpam-4877 548 2 γtgr(kn γtgr(kn ADJ ejpam-4877 548 3 ) ) PUNCT ejpam-4877 548 4 + + NUM ejpam-4877 548 5 γtgr(cm γtgr(cm NOUN ejpam-4877 548 6 ) ) PUNCT ejpam-4877 548 7 = = NOUN ejpam-4877 548 8 6 6 NUM ejpam-4877 548 9 for for ADP ejpam-4877 548 10 any any DET ejpam-4877 548 11 n n PRON ejpam-4877 548 12 ≥ ≥ NOUN ejpam-4877 548 13 2 2 NUM ejpam-4877 548 14 and and CCONJ ejpam-4877 548 15 m m PROPN ejpam-4877 548 16 ≥ ≥ NOUN ejpam-4877 548 17 4 4 NUM ejpam-4877 548 18 , , PUNCT ejpam-4877 548 19 ( ( PUNCT ejpam-4877 548 20 iv iv X ejpam-4877 548 21 ) ) PUNCT ejpam-4877 548 22 γthgr(pn γthgr(pn NOUN ejpam-4877 548 23 + + CCONJ ejpam-4877 548 24 pm pm NOUN ejpam-4877 548 25 ) ) PUNCT ejpam-4877 548 26 = = SYM ejpam-4877 548 27 γtgr(pn γtgr(pn NOUN ejpam-4877 548 28 ) ) PUNCT ejpam-4877 548 29 + + CCONJ ejpam-4877 548 30 γtgr(pm γtgr(pm ADJ ejpam-4877 548 31 ) ) PUNCT ejpam-4877 548 32 = = SYM ejpam-4877 548 33 8 8 NUM ejpam-4877 548 34 for for ADP ejpam-4877 548 35 any any DET ejpam-4877 548 36 n n CCONJ ejpam-4877 548 37 , , PUNCT ejpam-4877 548 38 m m VERB ejpam-4877 548 39 ≥ ≥ NOUN ejpam-4877 548 40 4 4 NUM ejpam-4877 548 41 , , PUNCT ejpam-4877 548 42 ( ( PUNCT ejpam-4877 548 43 v v NOUN ejpam-4877 548 44 ) ) PUNCT ejpam-4877 548 45 γthgr(pn γthgr(pn NOUN ejpam-4877 548 46 + + CCONJ ejpam-4877 548 47 cm cm NOUN ejpam-4877 548 48 ) ) PUNCT ejpam-4877 548 49 = = SYM ejpam-4877 548 50 γtgr(pn γtgr(pn NOUN ejpam-4877 548 51 ) ) PUNCT ejpam-4877 548 52 + + NUM ejpam-4877 548 53 γtgr(cm γtgr(cm NOUN ejpam-4877 548 54 ) ) PUNCT ejpam-4877 548 55 = = SYM ejpam-4877 548 56 8 8 NUM ejpam-4877 548 57 for for ADP ejpam-4877 548 58 any any DET ejpam-4877 548 59 n n CCONJ ejpam-4877 548 60 , , PUNCT ejpam-4877 548 61 m m VERB ejpam-4877 548 62 ≥ ≥ NOUN ejpam-4877 548 63 4 4 NUM ejpam-4877 548 64 , , PUNCT ejpam-4877 548 65 and and CCONJ ejpam-4877 548 66 ( ( PUNCT ejpam-4877 548 67 vi vi NOUN ejpam-4877 548 68 ) ) PUNCT ejpam-4877 548 69 γthgr(cn γthgr(cn NOUN ejpam-4877 548 70 + + CCONJ ejpam-4877 548 71 cm cm NOUN ejpam-4877 548 72 ) ) PUNCT ejpam-4877 548 73 = = SYM ejpam-4877 548 74 γtgr(cn γtgr(cn PROPN ejpam-4877 548 75 ) ) PUNCT ejpam-4877 548 76 + + NUM ejpam-4877 548 77 γtgr(cm γtgr(cm NOUN ejpam-4877 548 78 ) ) PUNCT ejpam-4877 548 79 = = SYM ejpam-4877 548 80 8 8 NUM ejpam-4877 548 81 for for ADP ejpam-4877 548 82 any any DET ejpam-4877 548 83 n n CCONJ ejpam-4877 548 84 , , PUNCT ejpam-4877 548 85 m m VERB ejpam-4877 548 86 ≥ ≥ NOUN ejpam-4877 548 87 4 4 NUM ejpam-4877 548 88 . . PUNCT ejpam-4877 549 1 j.a j.a PROPN ejpam-4877 549 2 . . PROPN ejpam-4877 549 3 hassan hassan PROPN ejpam-4877 549 4 , , PUNCT ejpam-4877 549 5 s. s. PROPN ejpam-4877 549 6 canoy canoy PROPN ejpam-4877 549 7 / / SYM ejpam-4877 549 8 eur eur PROPN ejpam-4877 549 9 . . PUNCT ejpam-4877 550 1 j. j. PROPN ejpam-4877 550 2 pure pure PROPN ejpam-4877 550 3 appl appl PROPN ejpam-4877 550 4 . . PROPN ejpam-4877 550 5 math math PROPN ejpam-4877 550 6 , , PUNCT ejpam-4877 550 7 16 16 NUM ejpam-4877 550 8 ( ( PUNCT ejpam-4877 550 9 4 4 NUM ejpam-4877 550 10 ) ) PUNCT ejpam-4877 550 11 ( ( PUNCT ejpam-4877 550 12 2023 2023 NUM ejpam-4877 550 13 ) ) PUNCT ejpam-4877 550 14 , , PUNCT ejpam-4877 550 15 2597 2597 NUM ejpam-4877 550 16 - - SYM ejpam-4877 550 17 2612 2612 NUM ejpam-4877 550 18 2610 2610 NUM ejpam-4877 550 19 theorem theorem NOUN ejpam-4877 550 20 8 8 NUM ejpam-4877 550 21 . . PUNCT ejpam-4877 551 1 let let VERB ejpam-4877 551 2 g g PRON ejpam-4877 551 3 be be AUX ejpam-4877 551 4 a a DET ejpam-4877 551 5 non non ADJ ejpam-4877 551 6 - - ADJ ejpam-4877 551 7 trivial trivial ADJ ejpam-4877 551 8 connected connected ADJ ejpam-4877 551 9 graph graph NOUN ejpam-4877 551 10 on on ADP ejpam-4877 551 11 n n DET ejpam-4877 551 12 vertices vertex NOUN ejpam-4877 551 13 and and CCONJ ejpam-4877 551 14 let let VERB ejpam-4877 551 15 h h NOUN ejpam-4877 551 16 be be AUX ejpam-4877 551 17 any any DET ejpam-4877 551 18 graph graph NOUN ejpam-4877 551 19 such such ADJ ejpam-4877 551 20 that that DET ejpam-4877 551 21 γ(h γ(h NOUN ejpam-4877 551 22 ) ) PUNCT ejpam-4877 551 23 ̸= ̸= PROPN ejpam-4877 551 24 1 1 NUM ejpam-4877 551 25 . . PUNCT ejpam-4877 552 1 then then ADV ejpam-4877 552 2 γthgr(g γthgr(g PROPN ejpam-4877 552 3 ◦ ◦ NOUN ejpam-4877 552 4 h h NOUN ejpam-4877 552 5 ) ) PUNCT ejpam-4877 552 6 ≥ ≥ PROPN ejpam-4877 552 7 n n PROPN ejpam-4877 552 8 · · PUNCT ejpam-4877 552 9 γtcogr(h γtcogr(h NOUN ejpam-4877 552 10 ) ) PUNCT ejpam-4877 552 11 = = PUNCT ejpam-4877 553 1 n n X ejpam-4877 553 2 · · PUNCT ejpam-4877 553 3 γtgr(h γtgr(h NOUN ejpam-4877 553 4 ) ) PUNCT ejpam-4877 553 5 . . PUNCT ejpam-4877 554 1 proof proof NOUN ejpam-4877 554 2 . . PUNCT ejpam-4877 555 1 let let VERB ejpam-4877 555 2 v v X ejpam-4877 555 3 ( ( PUNCT ejpam-4877 555 4 g g NOUN ejpam-4877 555 5 ) ) PUNCT ejpam-4877 555 6 = = SYM ejpam-4877 555 7 { { PUNCT ejpam-4877 555 8 u1 u1 NOUN ejpam-4877 555 9 , , PUNCT ejpam-4877 555 10 u2 u2 NOUN ejpam-4877 555 11 , , PUNCT ejpam-4877 555 12 . . PUNCT ejpam-4877 555 13 . . PUNCT ejpam-4877 556 1 . . PUNCT ejpam-4877 557 1 , , PUNCT ejpam-4877 557 2 un un PROPN ejpam-4877 557 3 } } PUNCT ejpam-4877 557 4 and and CCONJ ejpam-4877 557 5 let let VERB ejpam-4877 557 6 sui sui PROPN ejpam-4877 557 7 = = SYM ejpam-4877 557 8 ( ( PUNCT ejpam-4877 557 9 w1 w1 PROPN ejpam-4877 557 10 ui ui PROPN ejpam-4877 557 11 , , PUNCT ejpam-4877 557 12 w2 w2 NOUN ejpam-4877 557 13 ui ui PROPN ejpam-4877 557 14 , , PUNCT ejpam-4877 557 15 · · PUNCT ejpam-4877 557 16 · · PUNCT ejpam-4877 557 17 · · PUNCT ejpam-4877 557 18 , , PUNCT ejpam-4877 557 19 wk wk X ejpam-4877 557 20 ui ui PROPN ejpam-4877 557 21 ) ) PUNCT ejpam-4877 557 22 be be AUX ejpam-4877 557 23 a a DET ejpam-4877 557 24 co co ADJ ejpam-4877 557 25 - - ADJ ejpam-4877 557 26 grundy grundy ADJ ejpam-4877 557 27 total total ADJ ejpam-4877 557 28 dominating dominating NOUN ejpam-4877 557 29 sequence sequence NOUN ejpam-4877 557 30 in in ADP ejpam-4877 557 31 hui hui PROPN ejpam-4877 557 32 for for ADP ejpam-4877 557 33 each each DET ejpam-4877 557 34 i i PRON ejpam-4877 557 35 ∈ ∈ PROPN ejpam-4877 558 1 [ [ X ejpam-4877 558 2 n n X ejpam-4877 558 3 ] ] X ejpam-4877 558 4 , , PUNCT ejpam-4877 558 5 where where SCONJ ejpam-4877 558 6 k k PROPN ejpam-4877 558 7 = = NOUN ejpam-4877 558 8 γtcogr(h γtcogr(h PROPN ejpam-4877 558 9 ) ) PUNCT ejpam-4877 558 10 . . PUNCT ejpam-4877 559 1 let let VERB ejpam-4877 559 2 s s PRON ejpam-4877 559 3 = = VERB ejpam-4877 559 4 su1 su1 PROPN ejpam-4877 559 5 ⊕ ⊕ PROPN ejpam-4877 559 6 su2 su2 PROPN ejpam-4877 559 7 ⊕ ⊕ PROPN ejpam-4877 559 8 · · PUNCT ejpam-4877 559 9 · · PUNCT ejpam-4877 559 10 · · PUNCT ejpam-4877 559 11 ⊕ ⊕ PROPN ejpam-4877 559 12 sun sun PROPN ejpam-4877 559 13 . . PUNCT ejpam-4877 560 1 let let VERB ejpam-4877 560 2 v v NUM ejpam-4877 560 3 ∈ ∈ PROPN ejpam-4877 560 4 v v NOUN ejpam-4877 560 5 ( ( PUNCT ejpam-4877 560 6 g g PROPN ejpam-4877 560 7 ◦ ◦ NOUN ejpam-4877 560 8 h h NOUN ejpam-4877 560 9 ) ) PUNCT ejpam-4877 560 10 \ \ NOUN ejpam-4877 560 11 ŝ ŝ NOUN ejpam-4877 560 12 and and CCONJ ejpam-4877 560 13 let let VERB ejpam-4877 560 14 ut ut PROPN ejpam-4877 560 15 ∈ ∈ PROPN ejpam-4877 560 16 v v X ejpam-4877 560 17 ( ( PUNCT ejpam-4877 560 18 g g NOUN ejpam-4877 560 19 ) ) PUNCT ejpam-4877 560 20 such such ADJ ejpam-4877 560 21 that that DET ejpam-4877 560 22 v v NUM ejpam-4877 560 23 ∈ ∈ PROPN ejpam-4877 560 24 v v NOUN ejpam-4877 560 25 ( ( PUNCT ejpam-4877 560 26 ut ut PROPN ejpam-4877 560 27 + + NOUN ejpam-4877 560 28 hut hut NOUN ejpam-4877 560 29 ) ) PUNCT ejpam-4877 560 30 for for ADP ejpam-4877 560 31 some some DET ejpam-4877 560 32 t t NOUN ejpam-4877 560 33 ∈ ∈ PROPN ejpam-4877 561 1 [ [ X ejpam-4877 561 2 n n X ejpam-4877 561 3 ] ] PUNCT ejpam-4877 561 4 . . PUNCT ejpam-4877 562 1 suppose suppose VERB ejpam-4877 562 2 first first ADV ejpam-4877 562 3 that that PRON ejpam-4877 562 4 v v NOUN ejpam-4877 562 5 = = SYM ejpam-4877 562 6 ut ut PROPN ejpam-4877 562 7 . . PROPN ejpam-4877 562 8 let let VERB ejpam-4877 562 9 us we PRON ejpam-4877 562 10 ∈ ∈ PROPN ejpam-4877 562 11 ng(ut ng(ut PRON ejpam-4877 562 12 ) ) PUNCT ejpam-4877 562 13 and and CCONJ ejpam-4877 562 14 pick pick VERB ejpam-4877 562 15 any any DET ejpam-4877 562 16 wj wj PROPN ejpam-4877 562 17 us us PROPN ejpam-4877 562 18 ∈ ∈ PROPN ejpam-4877 562 19 ŝus ŝus NOUN ejpam-4877 562 20 for for ADP ejpam-4877 562 21 some some DET ejpam-4877 562 22 s s X ejpam-4877 562 23 ∈ ∈ PROPN ejpam-4877 563 1 [ [ X ejpam-4877 563 2 n n X ejpam-4877 563 3 ] ] PUNCT ejpam-4877 563 4 . . PUNCT ejpam-4877 564 1 then then ADV ejpam-4877 564 2 wj wj VERB ejpam-4877 564 3 us us PROPN ejpam-4877 564 4 ∈ ∈ PROPN ejpam-4877 564 5 ŝ ŝ VERB ejpam-4877 564 6 ∩n2 ∩n2 PROPN ejpam-4877 564 7 g g PROPN ejpam-4877 564 8 ◦ ◦ NOUN ejpam-4877 564 9 h(ut h(ut NOUN ejpam-4877 564 10 ) ) PUNCT ejpam-4877 564 11 . . PUNCT ejpam-4877 565 1 suppose suppose VERB ejpam-4877 565 2 v v ADP ejpam-4877 565 3 ̸= ̸= PROPN ejpam-4877 565 4 ut ut PROPN ejpam-4877 565 5 . . PUNCT ejpam-4877 566 1 then then ADV ejpam-4877 566 2 v v ADP ejpam-4877 566 3 ∈ ∈ PROPN ejpam-4877 566 4 v v NOUN ejpam-4877 566 5 ( ( PUNCT ejpam-4877 566 6 hut hut PROPN ejpam-4877 566 7 ) ) PUNCT ejpam-4877 566 8 \ \ NOUN ejpam-4877 567 1 ŝut ŝut NOUN ejpam-4877 567 2 . . PUNCT ejpam-4877 568 1 since since SCONJ ejpam-4877 568 2 ŝut ŝut ADJ ejpam-4877 568 3 is be AUX ejpam-4877 568 4 a a DET ejpam-4877 568 5 co co ADJ ejpam-4877 568 6 - - ADJ ejpam-4877 568 7 grundy grundy ADJ ejpam-4877 568 8 total total ADJ ejpam-4877 568 9 dominating dominating NOUN ejpam-4877 568 10 sequence sequence NOUN ejpam-4877 568 11 in in ADP ejpam-4877 568 12 hut hut NOUN ejpam-4877 568 13 , , PUNCT ejpam-4877 568 14 it it PRON ejpam-4877 568 15 follows follow VERB ejpam-4877 568 16 that that SCONJ ejpam-4877 568 17 there there PRON ejpam-4877 568 18 exists exist VERB ejpam-4877 568 19 wl wl X ejpam-4877 568 20 ut ut PROPN ejpam-4877 568 21 ∈ ∈ PROPN ejpam-4877 568 22 ŝut ŝut VERB ejpam-4877 568 23 ⊆ ⊆ NUM ejpam-4877 568 24 ŝ ŝ NOUN ejpam-4877 568 25 such such ADJ ejpam-4877 568 26 that that DET ejpam-4877 568 27 dhut dhut NOUN ejpam-4877 568 28 ( ( PUNCT ejpam-4877 568 29 v v NOUN ejpam-4877 568 30 , , PUNCT ejpam-4877 568 31 wl wl PROPN ejpam-4877 568 32 ut ut PROPN ejpam-4877 568 33 ) ) PUNCT ejpam-4877 568 34 ̸= ̸= PROPN ejpam-4877 568 35 1 1 NUM ejpam-4877 568 36 . . PUNCT ejpam-4877 569 1 it it PRON ejpam-4877 569 2 follows follow VERB ejpam-4877 569 3 that that SCONJ ejpam-4877 569 4 dg dg AUX ejpam-4877 569 5 ◦ ◦ NOUN ejpam-4877 569 6 h(v h(v PROPN ejpam-4877 569 7 , , PUNCT ejpam-4877 569 8 wl wl PROPN ejpam-4877 569 9 ut ut PROPN ejpam-4877 569 10 ) ) PUNCT ejpam-4877 570 1 = = PUNCT ejpam-4877 570 2 2 2 X ejpam-4877 570 3 . . PUNCT ejpam-4877 570 4 therefore therefore ADV ejpam-4877 570 5 , , PUNCT ejpam-4877 570 6 ŝ ŝ X ejpam-4877 570 7 is be AUX ejpam-4877 570 8 a a DET ejpam-4877 570 9 total total ADJ ejpam-4877 570 10 hop hop NOUN ejpam-4877 570 11 dominating dominating NOUN ejpam-4877 570 12 set set VERB ejpam-4877 570 13 in in ADP ejpam-4877 570 14 g g PROPN ejpam-4877 570 15 ◦ ◦ NOUN ejpam-4877 570 16 h. h. NOUN ejpam-4877 571 1 now now ADV ejpam-4877 571 2 , , PUNCT ejpam-4877 571 3 we we PRON ejpam-4877 571 4 relabel relabel VERB ejpam-4877 571 5 the the DET ejpam-4877 571 6 terms term NOUN ejpam-4877 571 7 in in ADP ejpam-4877 571 8 s s NOUN ejpam-4877 571 9 , , PUNCT ejpam-4877 571 10 say say VERB ejpam-4877 571 11 s s X ejpam-4877 571 12 = = PUNCT ejpam-4877 571 13 ( ( PUNCT ejpam-4877 571 14 v1 v1 PROPN ejpam-4877 571 15 , , PUNCT ejpam-4877 571 16 v2 v2 PROPN ejpam-4877 571 17 , , PUNCT ejpam-4877 571 18 · · PUNCT ejpam-4877 571 19 · · PUNCT ejpam-4877 571 20 · · PUNCT ejpam-4877 571 21 , , PUNCT ejpam-4877 571 22 vk vk X ejpam-4877 571 23 , , PUNCT ejpam-4877 571 24 · · PUNCT ejpam-4877 571 25 · · PUNCT ejpam-4877 571 26 · · PUNCT ejpam-4877 571 27 , , PUNCT ejpam-4877 571 28 vnk vnk NOUN ejpam-4877 571 29 ) ) PUNCT ejpam-4877 571 30 . . PUNCT ejpam-4877 572 1 let let VERB ejpam-4877 572 2 i i PRON ejpam-4877 572 3 ∈ ∈ PROPN ejpam-4877 573 1 [ [ X ejpam-4877 573 2 nk nk X ejpam-4877 573 3 ] ] X ejpam-4877 573 4 \ \ PUNCT ejpam-4877 573 5 { { PUNCT ejpam-4877 573 6 1 1 NUM ejpam-4877 573 7 } } PUNCT ejpam-4877 573 8 and and CCONJ ejpam-4877 573 9 let let VERB ejpam-4877 573 10 vi vi NOUN ejpam-4877 573 11 = = NOUN ejpam-4877 573 12 wt wt ADP ejpam-4877 573 13 ur ur INTJ ejpam-4877 573 14 for for ADP ejpam-4877 573 15 some some DET ejpam-4877 573 16 r r NOUN ejpam-4877 573 17 ∈ ∈ PROPN ejpam-4877 574 1 [ [ X ejpam-4877 574 2 n n X ejpam-4877 574 3 ] ] PUNCT ejpam-4877 574 4 and and CCONJ ejpam-4877 574 5 t t PROPN ejpam-4877 574 6 ∈ ∈ PROPN ejpam-4877 575 1 [ [ X ejpam-4877 575 2 k k X ejpam-4877 575 3 ] ] X ejpam-4877 575 4 . . PUNCT ejpam-4877 576 1 then then ADV ejpam-4877 576 2 n2 n2 PROPN ejpam-4877 576 3 g g PROPN ejpam-4877 576 4 ◦ ◦ NOUN ejpam-4877 576 5 h(vi h(vi NOUN ejpam-4877 576 6 ) ) PUNCT ejpam-4877 576 7 \ \ NOUN ejpam-4877 577 1 ∪i−1 ∪i−1 X ejpam-4877 577 2 j=1n j=1n NOUN ejpam-4877 577 3 2 2 NUM ejpam-4877 577 4 g g NOUN ejpam-4877 577 5 ◦ ◦ NOUN ejpam-4877 577 6 h(vj h(vj NOUN ejpam-4877 577 7 ) ) PUNCT ejpam-4877 577 8 = = SYM ejpam-4877 577 9 n2 n2 PROPN ejpam-4877 577 10 g g PROPN ejpam-4877 577 11 ◦ ◦ NOUN ejpam-4877 577 12 h(wt h(wt NOUN ejpam-4877 577 13 ur ur NOUN ejpam-4877 577 14 ) ) PUNCT ejpam-4877 577 15 \ \ PUNCT ejpam-4877 578 1 [ [ X ejpam-4877 578 2 ( ( PUNCT ejpam-4877 578 3 ∪t−1 ∪t−1 X ejpam-4877 578 4 s=1n s=1n NOUN ejpam-4877 578 5 2 2 NUM ejpam-4877 578 6 g g NOUN ejpam-4877 578 7 ◦ ◦ NOUN ejpam-4877 578 8 h(ws h(w NOUN ejpam-4877 578 9 ur ur NOUN ejpam-4877 578 10 ) ) PUNCT ejpam-4877 578 11 ) ) PUNCT ejpam-4877 578 12 ∪ ∪ ADV ejpam-4877 578 13 ( ( PUNCT ejpam-4877 578 14 ∪{n2 ∪{n2 NOUN ejpam-4877 578 15 g g ADP ejpam-4877 578 16 ◦ ◦ NOUN ejpam-4877 578 17 h(wp h(wp PROPN ejpam-4877 578 18 uq uq NOUN ejpam-4877 578 19 ) ) PUNCT ejpam-4877 578 20 : : PUNCT ejpam-4877 578 21 p p X ejpam-4877 578 22 ∈ ∈ PROPN ejpam-4877 578 23 [ [ X ejpam-4877 578 24 k k X ejpam-4877 578 25 ] ] X ejpam-4877 578 26 and and CCONJ ejpam-4877 578 27 1 1 NUM ejpam-4877 578 28 ≤ ≤ NUM ejpam-4877 578 29 q q ADJ ejpam-4877 578 30 ≤ ≤ NUM ejpam-4877 578 31 r r NOUN ejpam-4877 578 32 − − NOUN ejpam-4877 578 33 1 1 NUM ejpam-4877 578 34 } } PUNCT ejpam-4877 578 35 ) ) PUNCT ejpam-4877 578 36 ] ] PUNCT ejpam-4877 578 37 . . PUNCT ejpam-4877 579 1 if if SCONJ ejpam-4877 579 2 t t NOUN ejpam-4877 579 3 = = SYM ejpam-4877 579 4 1 1 NUM ejpam-4877 579 5 , , PUNCT ejpam-4877 579 6 then then ADV ejpam-4877 579 7 n2 n2 PROPN ejpam-4877 579 8 g g PROPN ejpam-4877 579 9 ◦ ◦ NOUN ejpam-4877 579 10 h(wt h(wt NOUN ejpam-4877 579 11 ur ur NOUN ejpam-4877 579 12 ) ) PUNCT ejpam-4877 579 13 \ \ PUNCT ejpam-4877 579 14 ( ( PUNCT ejpam-4877 579 15 ∪t−1 ∪t−1 X ejpam-4877 579 16 s=1n s=1n VERB ejpam-4877 579 17 2 2 NUM ejpam-4877 579 18 g g NOUN ejpam-4877 579 19 ◦ ◦ NOUN ejpam-4877 579 20 h(ws h(w NOUN ejpam-4877 579 21 ur ur NOUN ejpam-4877 579 22 ) ) PUNCT ejpam-4877 579 23 ) ) PUNCT ejpam-4877 580 1 = = PUNCT ejpam-4877 580 2 n2 n2 PROPN ejpam-4877 580 3 g g PROPN ejpam-4877 580 4 ◦ ◦ NOUN ejpam-4877 580 5 h(wt h(wt NOUN ejpam-4877 580 6 ur ur INTJ ejpam-4877 580 7 ) ) PUNCT ejpam-4877 580 8 . . PUNCT ejpam-4877 581 1 clearly clearly ADV ejpam-4877 581 2 , , PUNCT ejpam-4877 581 3 wt wt INTJ ejpam-4877 581 4 ur ur NOUN ejpam-4877 581 5 ∈ ∈ PROPN ejpam-4877 581 6 n2 n2 NOUN ejpam-4877 581 7 g g PROPN ejpam-4877 581 8 ◦ ◦ NOUN ejpam-4877 581 9 h(wt h(wt NOUN ejpam-4877 581 10 ur ur NOUN ejpam-4877 581 11 ) ) PUNCT ejpam-4877 581 12 \ \ PUNCT ejpam-4877 582 1 [ [ X ejpam-4877 582 2 ∪{n2 ∪{n2 NOUN ejpam-4877 582 3 g g ADP ejpam-4877 582 4 ◦ ◦ NOUN ejpam-4877 582 5 h(wp h(wp PROPN ejpam-4877 582 6 uq uq NOUN ejpam-4877 582 7 ) ) PUNCT ejpam-4877 582 8 : : PUNCT ejpam-4877 583 1 p p X ejpam-4877 583 2 ∈ ∈ PROPN ejpam-4877 583 3 [ [ X ejpam-4877 583 4 k k X ejpam-4877 583 5 ] ] X ejpam-4877 583 6 and and CCONJ ejpam-4877 583 7 1 1 NUM ejpam-4877 583 8 ≤ ≤ NUM ejpam-4877 583 9 q q ADJ ejpam-4877 583 10 ≤ ≤ NUM ejpam-4877 583 11 r r NOUN ejpam-4877 583 12 − − NOUN ejpam-4877 583 13 1 1 NUM ejpam-4877 583 14 } } PUNCT ejpam-4877 583 15 ] ] PUNCT ejpam-4877 583 16 . . PUNCT ejpam-4877 584 1 suppose suppose VERB ejpam-4877 585 1 t t PROPN ejpam-4877 585 2 ̸= ̸= PROPN ejpam-4877 585 3 1 1 NUM ejpam-4877 585 4 . . PUNCT ejpam-4877 586 1 since since SCONJ ejpam-4877 586 2 sur sur PROPN ejpam-4877 586 3 is be AUX ejpam-4877 586 4 a a DET ejpam-4877 586 5 co co ADJ ejpam-4877 586 6 - - ADJ ejpam-4877 586 7 legal legal ADJ ejpam-4877 586 8 open open ADJ ejpam-4877 586 9 neighborhood neighborhood NOUN ejpam-4877 586 10 sequence sequence NOUN ejpam-4877 586 11 in in ADP ejpam-4877 586 12 hur hur PROPN ejpam-4877 586 13 , , PUNCT ejpam-4877 586 14 n2 n2 PROPN ejpam-4877 586 15 g g PROPN ejpam-4877 586 16 ◦ ◦ NOUN ejpam-4877 586 17 h(wt h(wt NOUN ejpam-4877 586 18 ur ur NOUN ejpam-4877 586 19 ) ) PUNCT ejpam-4877 586 20 \ \ PUNCT ejpam-4877 587 1 ( ( PUNCT ejpam-4877 587 2 ∪t−1 ∪t−1 X ejpam-4877 587 3 s=1n s=1n VERB ejpam-4877 587 4 2 2 NUM ejpam-4877 587 5 g g NOUN ejpam-4877 587 6 ◦ ◦ NOUN ejpam-4877 587 7 h(ws h(w NOUN ejpam-4877 587 8 ur ur NOUN ejpam-4877 587 9 ) ) PUNCT ejpam-4877 587 10 ) ) PUNCT ejpam-4877 588 1 = = PUNCT ejpam-4877 589 1 [ [ X ejpam-4877 589 2 v v X ejpam-4877 589 3 ( ( PUNCT ejpam-4877 589 4 hur hur PROPN ejpam-4877 589 5 ) ) PUNCT ejpam-4877 589 6 \nhur \nhur PROPN ejpam-4877 589 7 ( ( PUNCT ejpam-4877 589 8 wt wt INTJ ejpam-4877 589 9 ur ur INTJ ejpam-4877 589 10 ) ) PUNCT ejpam-4877 589 11 ] ] PUNCT ejpam-4877 589 12 \ \ PUNCT ejpam-4877 590 1 [ [ X ejpam-4877 590 2 ∪t−1 ∪t−1 X ejpam-4877 590 3 s=1(v s=1(v NOUN ejpam-4877 590 4 ( ( PUNCT ejpam-4877 590 5 hur hur PROPN ejpam-4877 590 6 ) ) PUNCT ejpam-4877 590 7 \nhur \nhur PROPN ejpam-4877 590 8 ( ( PUNCT ejpam-4877 590 9 ws ws NOUN ejpam-4877 590 10 ur ur INTJ ejpam-4877 590 11 ) ) PUNCT ejpam-4877 590 12 ) ) PUNCT ejpam-4877 590 13 ] ] PUNCT ejpam-4877 590 14 ̸= ̸= PROPN ejpam-4877 590 15 ∅. ∅. ADV ejpam-4877 590 16 observe observe VERB ejpam-4877 590 17 that that DET ejpam-4877 590 18 n2 n2 NOUN ejpam-4877 590 19 g g PROPN ejpam-4877 590 20 ◦ ◦ NOUN ejpam-4877 590 21 h(wt h(wt NOUN ejpam-4877 590 22 ur ur NOUN ejpam-4877 590 23 ) ) PUNCT ejpam-4877 590 24 \ \ PUNCT ejpam-4877 591 1 ( ( PUNCT ejpam-4877 591 2 ∪t−1 ∪t−1 X ejpam-4877 591 3 s=1n s=1n VERB ejpam-4877 591 4 2 2 NUM ejpam-4877 591 5 g g NOUN ejpam-4877 591 6 ◦ ◦ NOUN ejpam-4877 591 7 h(ws h(w NOUN ejpam-4877 591 8 ur ur NOUN ejpam-4877 591 9 ) ) PUNCT ejpam-4877 591 10 ) ) PUNCT ejpam-4877 592 1 ∩ ∩ NOUN ejpam-4877 593 1 [ [ X ejpam-4877 593 2 ∪{n2 ∪{n2 NOUN ejpam-4877 593 3 g g ADP ejpam-4877 593 4 ◦ ◦ NOUN ejpam-4877 593 5 h(wp h(wp PROPN ejpam-4877 593 6 uq uq NOUN ejpam-4877 593 7 ) ) PUNCT ejpam-4877 593 8 : : PUNCT ejpam-4877 594 1 p p X ejpam-4877 594 2 ∈ ∈ PROPN ejpam-4877 594 3 [ [ X ejpam-4877 594 4 k k X ejpam-4877 594 5 ] ] X ejpam-4877 594 6 and and CCONJ ejpam-4877 594 7 1 1 NUM ejpam-4877 594 8 ≤ ≤ NUM ejpam-4877 594 9 q q ADJ ejpam-4877 594 10 ≤ ≤ NUM ejpam-4877 594 11 r r NOUN ejpam-4877 594 12 − − NOUN ejpam-4877 594 13 1 1 NUM ejpam-4877 594 14 } } PUNCT ejpam-4877 594 15 ] ] PUNCT ejpam-4877 594 16 = = PUNCT ejpam-4877 594 17 ∅. ∅. VERB ejpam-4877 594 18 hence hence ADV ejpam-4877 594 19 , , PUNCT ejpam-4877 594 20 n2 n2 ADJ ejpam-4877 594 21 g g PROPN ejpam-4877 594 22 ◦ ◦ NOUN ejpam-4877 594 23 h(vi)\∪i−1 h(vi)\∪i−1 NOUN ejpam-4877 594 24 j=1n j=1n VERB ejpam-4877 594 25 2 2 NUM ejpam-4877 594 26 g g NOUN ejpam-4877 594 27 ◦ ◦ NOUN ejpam-4877 594 28 h(vj h(vj NOUN ejpam-4877 594 29 ) ) PUNCT ejpam-4877 594 30 ̸= ̸= NOUN ejpam-4877 594 31 ∅ ∅ NOUN ejpam-4877 594 32 for for ADP ejpam-4877 594 33 all all PRON ejpam-4877 594 34 i i PRON ejpam-4877 594 35 ∈ ∈ X ejpam-4877 595 1 [ [ X ejpam-4877 595 2 nk]\{1 nk]\{1 NOUN ejpam-4877 595 3 } } PUNCT ejpam-4877 595 4 and and CCONJ ejpam-4877 595 5 so so ADV ejpam-4877 595 6 s s VERB ejpam-4877 595 7 is be AUX ejpam-4877 595 8 a a DET ejpam-4877 595 9 grundy grundy PROPN ejpam-4877 595 10 total total NOUN ejpam-4877 595 11 hop hop NOUN ejpam-4877 595 12 dominating dominating NOUN ejpam-4877 595 13 sequence sequence NOUN ejpam-4877 595 14 in in ADP ejpam-4877 595 15 g g PROPN ejpam-4877 595 16 ◦ ◦ NOUN ejpam-4877 595 17 h. h. NOUN ejpam-4877 595 18 consequently consequently ADV ejpam-4877 595 19 , , PUNCT ejpam-4877 595 20 γthgr(g γthgr(g PROPN ejpam-4877 595 21 ◦ ◦ NOUN ejpam-4877 595 22 h h NOUN ejpam-4877 595 23 ) ) PUNCT ejpam-4877 595 24 ≥ ≥ NOUN ejpam-4877 595 25 |ŝ| |ŝ| PROPN ejpam-4877 595 26 = = SYM ejpam-4877 595 27 n∑ n∑ NOUN ejpam-4877 595 28 i=1 i=1 PROPN ejpam-4877 595 29 |ŝvi |ŝvi PROPN ejpam-4877 595 30 | | ADV ejpam-4877 595 31 = = SYM ejpam-4877 595 32 n n PROPN ejpam-4877 595 33 · · PUNCT ejpam-4877 595 34 γtcogr(h γtcogr(h NOUN ejpam-4877 595 35 ) ) PUNCT ejpam-4877 595 36 = = PUNCT ejpam-4877 595 37 n n X ejpam-4877 595 38 · · PUNCT ejpam-4877 595 39 γtgr(h γtgr(h NOUN ejpam-4877 595 40 ) ) PUNCT ejpam-4877 595 41 . . PUNCT ejpam-4877 596 1 remark remark PROPN ejpam-4877 596 2 4 4 NUM ejpam-4877 596 3 . . PUNCT ejpam-4877 597 1 the the DET ejpam-4877 597 2 bound bind VERB ejpam-4877 597 3 given give VERB ejpam-4877 597 4 in in ADP ejpam-4877 597 5 theorem theorem ADJ ejpam-4877 597 6 8 8 NUM ejpam-4877 597 7 is be AUX ejpam-4877 597 8 tight tight ADJ ejpam-4877 597 9 . . PUNCT ejpam-4877 598 1 to to PART ejpam-4877 598 2 see see VERB ejpam-4877 598 3 this this PRON ejpam-4877 598 4 , , PUNCT ejpam-4877 598 5 consider consider VERB ejpam-4877 598 6 the the DET ejpam-4877 598 7 graph graph NOUN ejpam-4877 598 8 k5 k5 PROPN ejpam-4877 598 9 ◦ ◦ NOUN ejpam-4877 598 10 p4 p4 ADJ ejpam-4877 598 11 in in ADP ejpam-4877 598 12 fig fig NOUN ejpam-4877 598 13 . . PUNCT ejpam-4877 599 1 5 5 X ejpam-4877 599 2 . . X ejpam-4877 599 3 let let VERB ejpam-4877 599 4 s s PRON ejpam-4877 599 5 = = PUNCT ejpam-4877 599 6 ( ( PUNCT ejpam-4877 599 7 a1 a1 PROPN ejpam-4877 599 8 , , PUNCT ejpam-4877 599 9 a2 a2 PROPN ejpam-4877 599 10 , , PUNCT ejpam-4877 599 11 · · PUNCT ejpam-4877 599 12 · · PUNCT ejpam-4877 599 13 · · PUNCT ejpam-4877 599 14 , , PUNCT ejpam-4877 599 15 a20 a20 PROPN ejpam-4877 599 16 ) ) PUNCT ejpam-4877 599 17 . . PUNCT ejpam-4877 600 1 then then ADV ejpam-4877 600 2 s s VERB ejpam-4877 600 3 is be AUX ejpam-4877 600 4 a a DET ejpam-4877 600 5 grundy grundy PROPN ejpam-4877 600 6 total total NOUN ejpam-4877 600 7 hop hop NOUN ejpam-4877 600 8 dominating dominating NOUN ejpam-4877 600 9 sequence sequence NOUN ejpam-4877 600 10 of of ADP ejpam-4877 600 11 k5 k5 PROPN ejpam-4877 600 12 ◦ ◦ PROPN ejpam-4877 600 13 p4 p4 ADJ ejpam-4877 600 14 . . PUNCT ejpam-4877 601 1 moreover moreover ADV ejpam-4877 601 2 , , PUNCT ejpam-4877 601 3 it it PRON ejpam-4877 601 4 can can AUX ejpam-4877 601 5 be be AUX ejpam-4877 601 6 verified verify VERB ejpam-4877 601 7 that that SCONJ ejpam-4877 601 8 γthgr(k5 γthgr(k5 NOUN ejpam-4877 601 9 ◦ ◦ VERB ejpam-4877 601 10 p4 p4 ADJ ejpam-4877 601 11 ) ) PUNCT ejpam-4877 602 1 = = SYM ejpam-4877 602 2 20 20 NUM ejpam-4877 602 3 . . PUNCT ejpam-4877 603 1 since since SCONJ ejpam-4877 603 2 γtgr(p γtgr(p NOUN ejpam-4877 603 3 4 4 NUM ejpam-4877 603 4 ) ) PUNCT ejpam-4877 603 5 = = SYM ejpam-4877 603 6 4 4 NUM ejpam-4877 603 7 , , PUNCT ejpam-4877 603 8 the the DET ejpam-4877 603 9 assertion assertion NOUN ejpam-4877 603 10 follows follow VERB ejpam-4877 603 11 . . PUNCT ejpam-4877 604 1 references reference NOUN ejpam-4877 604 2 2611 2611 NUM ejpam-4877 604 3 k5 k5 PROPN ejpam-4877 604 4 ◦ ◦ NOUN ejpam-4877 604 5 p4 p4 ADJ ejpam-4877 604 6 : : PUNCT ejpam-4877 604 7 a2a1 a2a1 SYM ejpam-4877 604 8 a3 a3 PROPN ejpam-4877 604 9 a4 a4 PROPN ejpam-4877 604 10 a5 a5 PROPN ejpam-4877 604 11 a6 a6 NOUN ejpam-4877 604 12 a7 a7 PROPN ejpam-4877 604 13 a8 a8 PROPN ejpam-4877 604 14 a9 a9 PROPN ejpam-4877 604 15 a10 a10 PROPN ejpam-4877 604 16 a11 a11 PROPN ejpam-4877 604 17 a12 a12 PROPN ejpam-4877 604 18 a13 a13 PROPN ejpam-4877 604 19 a14 a14 PROPN ejpam-4877 604 20 a15 a15 PROPN ejpam-4877 604 21 a16 a16 PROPN ejpam-4877 604 22 a17 a17 PROPN ejpam-4877 604 23 a18 a18 PROPN ejpam-4877 604 24 a19 a19 PROPN ejpam-4877 604 25 a20 a20 PROPN ejpam-4877 604 26 figure figure NOUN ejpam-4877 604 27 5 5 NUM ejpam-4877 604 28 : : PUNCT ejpam-4877 604 29 a a DET ejpam-4877 604 30 graph graph NOUN ejpam-4877 604 31 k5 k5 PROPN ejpam-4877 604 32 ◦ ◦ NOUN ejpam-4877 604 33 p4 p4 ADJ ejpam-4877 604 34 with with ADP ejpam-4877 604 35 γth γth PROPN ejpam-4877 604 36 gr(k5 gr(k5 NOUN ejpam-4877 604 37 ◦ ◦ NOUN ejpam-4877 604 38 p4 p4 ADJ ejpam-4877 604 39 ) ) PUNCT ejpam-4877 605 1 = = PUNCT ejpam-4877 606 1 |k5|γt |k5|γt PROPN ejpam-4877 606 2 gr(p gr(p NUM ejpam-4877 606 3 4 4 NUM ejpam-4877 606 4 ) ) PUNCT ejpam-4877 606 5 . . PUNCT ejpam-4877 607 1 acknowledgements acknowledgement NOUN ejpam-4877 607 2 the the DET ejpam-4877 607 3 authors author NOUN ejpam-4877 607 4 would would AUX ejpam-4877 607 5 like like VERB ejpam-4877 607 6 to to PART ejpam-4877 607 7 thank thank VERB ejpam-4877 607 8 the the DET ejpam-4877 607 9 referees referee NOUN ejpam-4877 607 10 for for ADP ejpam-4877 607 11 the the DET ejpam-4877 607 12 comments comment NOUN ejpam-4877 607 13 and and CCONJ ejpam-4877 607 14 suggestions suggestion NOUN ejpam-4877 607 15 they they PRON ejpam-4877 607 16 made make VERB ejpam-4877 607 17 in in ADP ejpam-4877 607 18 the the DET ejpam-4877 607 19 initial initial ADJ ejpam-4877 607 20 manuscript manuscript NOUN ejpam-4877 607 21 . . PUNCT ejpam-4877 608 1 also also ADV ejpam-4877 608 2 , , PUNCT ejpam-4877 608 3 special special ADJ ejpam-4877 608 4 thanks thank NOUN ejpam-4877 608 5 must must AUX ejpam-4877 608 6 go go VERB ejpam-4877 608 7 to to ADP ejpam-4877 608 8 the the DET ejpam-4877 608 9 department department NOUN ejpam-4877 608 10 of of ADP ejpam-4877 608 11 science science NOUN ejpam-4877 608 12 and and CCONJ ejpam-4877 608 13 technology technology NOUN ejpam-4877 608 14 accelerated accelerate VERB ejpam-4877 608 15 science science NOUN ejpam-4877 608 16 and and CCONJ ejpam-4877 608 17 technology technology NOUN ejpam-4877 608 18 human human ADJ ejpam-4877 608 19 resource resource NOUN ejpam-4877 608 20 development development NOUN ejpam-4877 608 21 program program NOUN ejpam-4877 608 22 ( ( PUNCT ejpam-4877 608 23 dost dost NOUN ejpam-4877 608 24 - - PUNCT ejpam-4877 608 25 asthrdp)-philippines asthrdp)-philippine NOUN ejpam-4877 608 26 , , PUNCT ejpam-4877 608 27 msu msu PROPN ejpam-4877 608 28 - - PUNCT ejpam-4877 608 29 iligan iligan PROPN ejpam-4877 608 30 institute institute PROPN ejpam-4877 608 31 of of ADP ejpam-4877 608 32 technology technology PROPN ejpam-4877 608 33 ( ( PUNCT ejpam-4877 608 34 philippines philippine NOUN ejpam-4877 608 35 ) ) PUNCT ejpam-4877 608 36 , , PUNCT ejpam-4877 608 37 and and CCONJ ejpam-4877 608 38 msu msu PROPN ejpam-4877 608 39 - - PUNCT ejpam-4877 608 40 tawi tawi NOUN ejpam-4877 608 41 - - PUNCT ejpam-4877 608 42 tawi tawi NOUN ejpam-4877 608 43 college college PROPN ejpam-4877 608 44 of of ADP ejpam-4877 608 45 technology technology NOUN ejpam-4877 608 46 and and CCONJ ejpam-4877 608 47 oceanography oceanography NOUN ejpam-4877 608 48 ( ( PUNCT ejpam-4877 608 49 philippines philippine NOUN ejpam-4877 608 50 ) ) PUNCT ejpam-4877 608 51 for for ADP ejpam-4877 608 52 funding fund VERB ejpam-4877 608 53 this this DET ejpam-4877 608 54 research research NOUN ejpam-4877 608 55 . . PUNCT ejpam-4877 609 1 references reference NOUN ejpam-4877 609 2 [ [ X ejpam-4877 609 3 1 1 NUM ejpam-4877 609 4 ] ] PUNCT ejpam-4877 609 5 s. s. PROPN ejpam-4877 609 6 ayyaswamy ayyaswamy PROPN ejpam-4877 609 7 , , PUNCT ejpam-4877 609 8 b. b. PROPN ejpam-4877 609 9 krishnakumari krishnakumari PROPN ejpam-4877 609 10 , , PUNCT ejpam-4877 609 11 b. b. PROPN ejpam-4877 609 12 natarjan natarjan PROPN ejpam-4877 609 13 , , PUNCT ejpam-4877 609 14 and and CCONJ ejpam-4877 609 15 y. y. PROPN ejpam-4877 609 16 venkatakrishnan venkatakrishnan PROPN ejpam-4877 609 17 . . PUNCT ejpam-4877 610 1 bounds bound NOUN ejpam-4877 610 2 on on ADP ejpam-4877 610 3 the the DET ejpam-4877 610 4 hop hop NOUN ejpam-4877 610 5 domination domination NOUN ejpam-4877 610 6 number number NOUN ejpam-4877 610 7 of of ADP ejpam-4877 610 8 a a DET ejpam-4877 610 9 tree tree NOUN ejpam-4877 610 10 . . PUNCT ejpam-4877 611 1 proceedings proceeding NOUN ejpam-4877 611 2 - - PUNCT ejpam-4877 611 3 mathematical mathematical ADJ ejpam-4877 611 4 sciences science NOUN ejpam-4877 611 5 . . PUNCT ejpam-4877 611 6 , , PUNCT ejpam-4877 611 7 125(4):449–455 125(4):449–455 ADP ejpam-4877 611 8 , , PUNCT ejpam-4877 611 9 2015 2015 NUM ejpam-4877 611 10 . . PUNCT ejpam-4877 612 1 [ [ X ejpam-4877 612 2 2 2 NUM ejpam-4877 612 3 ] ] PUNCT ejpam-4877 612 4 s. s. PROPN ejpam-4877 612 5 ayyaswamy ayyaswamy PROPN ejpam-4877 612 6 , , PUNCT ejpam-4877 612 7 c. c. PROPN ejpam-4877 612 8 natarajan natarajan PROPN ejpam-4877 612 9 , , PUNCT ejpam-4877 612 10 and and CCONJ ejpam-4877 612 11 g. g. PROPN ejpam-4877 612 12 sathiamoorphy sathiamoorphy PROPN ejpam-4877 612 13 . . PUNCT ejpam-4877 613 1 a a DET ejpam-4877 613 2 note note NOUN ejpam-4877 613 3 on on ADP ejpam-4877 613 4 hop hop NOUN ejpam-4877 613 5 domination domination NOUN ejpam-4877 613 6 number number NOUN ejpam-4877 613 7 of of ADP ejpam-4877 613 8 some some DET ejpam-4877 613 9 special special ADJ ejpam-4877 613 10 families family NOUN ejpam-4877 613 11 of of ADP ejpam-4877 613 12 graphs graph NOUN ejpam-4877 613 13 . . PUNCT ejpam-4877 614 1 international international ADJ ejpam-4877 614 2 journal journal NOUN ejpam-4877 614 3 of of ADP ejpam-4877 614 4 pure pure ADJ ejpam-4877 614 5 and and CCONJ ejpam-4877 614 6 applied applied ADJ ejpam-4877 614 7 mathematics mathematic NOUN ejpam-4877 614 8 . . PUNCT ejpam-4877 614 9 , , PUNCT ejpam-4877 614 10 119(12):11465–14171 119(12):11465–14171 NUM ejpam-4877 614 11 , , PUNCT ejpam-4877 614 12 2018 2018 NUM ejpam-4877 614 13 . . PUNCT ejpam-4877 615 1 [ [ X ejpam-4877 615 2 3 3 X ejpam-4877 615 3 ] ] X ejpam-4877 615 4 b. b. PROPN ejpam-4877 615 5 bresar bresar VERB ejpam-4877 615 6 . . PUNCT ejpam-4877 616 1 on on ADP ejpam-4877 616 2 grundy grundy PROPN ejpam-4877 616 3 total total ADJ ejpam-4877 616 4 domination domination NOUN ejpam-4877 616 5 number number NOUN ejpam-4877 616 6 in in ADP ejpam-4877 616 7 product product NOUN ejpam-4877 616 8 graphs graph NOUN ejpam-4877 616 9 . . PUNCT ejpam-4877 617 1 discussiones discussione NOUN ejpam-4877 617 2 math math NOUN ejpam-4877 617 3 . . PUNCT ejpam-4877 617 4 , , PUNCT ejpam-4877 617 5 graph graph NOUN ejpam-4877 617 6 theory theory NOUN ejpam-4877 617 7 , , PUNCT ejpam-4877 617 8 ( ( PUNCT ejpam-4877 617 9 41):225–247 41):225–247 NOUN ejpam-4877 617 10 , , PUNCT ejpam-4877 617 11 2021 2021 NUM ejpam-4877 617 12 . . PUNCT ejpam-4877 618 1 [ [ X ejpam-4877 618 2 4 4 NUM ejpam-4877 618 3 ] ] X ejpam-4877 618 4 b. b. PROPN ejpam-4877 618 5 bresar bresar PROPN ejpam-4877 618 6 , , PUNCT ejpam-4877 618 7 t. t. NOUN ejpam-4877 618 8 gologranc gologranc PROPN ejpam-4877 618 9 , , PUNCT ejpam-4877 618 10 m. m. NOUN ejpam-4877 618 11 milanic milanic PROPN ejpam-4877 618 12 , , PUNCT ejpam-4877 618 13 d. d. PROPN ejpam-4877 618 14 rall rall PROPN ejpam-4877 618 15 , , PUNCT ejpam-4877 618 16 and and CCONJ ejpam-4877 618 17 r. r. PROPN ejpam-4877 618 18 rizzi rizzi PROPN ejpam-4877 618 19 . . PUNCT ejpam-4877 619 1 dominating dominate VERB ejpam-4877 619 2 sequences sequence NOUN ejpam-4877 619 3 in in ADP ejpam-4877 619 4 graphs graph NOUN ejpam-4877 619 5 . . PUNCT ejpam-4877 620 1 discrete discrete ADJ ejpam-4877 620 2 math math NOUN ejpam-4877 620 3 . . PUNCT ejpam-4877 621 1 , , PUNCT ejpam-4877 621 2 ( ( PUNCT ejpam-4877 621 3 336):22–36 336):22–36 NUM ejpam-4877 621 4 , , PUNCT ejpam-4877 621 5 2014 2014 NUM ejpam-4877 621 6 . . PUNCT ejpam-4877 622 1 references reference NOUN ejpam-4877 622 2 2612 2612 NUM ejpam-4877 623 1 [ [ X ejpam-4877 623 2 5 5 NUM ejpam-4877 623 3 ] ] PUNCT ejpam-4877 623 4 b. b. PROPN ejpam-4877 623 5 bresar bresar PROPN ejpam-4877 623 6 , , PUNCT ejpam-4877 623 7 m.a m.a PROPN ejpam-4877 623 8 . . PROPN ejpam-4877 623 9 henning henning PROPN ejpam-4877 623 10 , , PUNCT ejpam-4877 623 11 and and CCONJ ejpam-4877 623 12 d.f d.f PROPN ejpam-4877 623 13 . . PROPN ejpam-4877 623 14 rall rall PROPN ejpam-4877 623 15 . . PUNCT ejpam-4877 624 1 total total ADJ ejpam-4877 624 2 dominating dominating NOUN ejpam-4877 624 3 sequences sequence NOUN ejpam-4877 624 4 in in ADP ejpam-4877 624 5 graphs graph NOUN ejpam-4877 624 6 , , PUNCT ejpam-4877 624 7 . . PUNCT ejpam-4877 625 1 discrete discrete ADJ ejpam-4877 625 2 math math NOUN ejpam-4877 625 3 . . PUNCT ejpam-4877 626 1 , , PUNCT ejpam-4877 626 2 , , PUNCT ejpam-4877 626 3 ( ( PUNCT ejpam-4877 626 4 339):1665–1676 339):1665–1676 NUM ejpam-4877 626 5 , , PUNCT ejpam-4877 626 6 2016 2016 NUM ejpam-4877 626 7 . . PUNCT ejpam-4877 627 1 [ [ X ejpam-4877 627 2 6 6 NUM ejpam-4877 627 3 ] ] PUNCT ejpam-4877 627 4 b. b. PROPN ejpam-4877 627 5 bresar bresar PROPN ejpam-4877 627 6 , , PUNCT ejpam-4877 627 7 t. t. PROPN ejpam-4877 627 8 kos kos PROPN ejpam-4877 627 9 , , PUNCT ejpam-4877 627 10 and and CCONJ ejpam-4877 627 11 p. p. NOUN ejpam-4877 627 12 torres torre NOUN ejpam-4877 627 13 . . PUNCT ejpam-4877 628 1 grundy grundy PROPN ejpam-4877 628 2 domination domination NOUN ejpam-4877 628 3 and and CCONJ ejpam-4877 628 4 zero zero NUM ejpam-4877 628 5 forcing force VERB ejpam-4877 628 6 in in ADP ejpam-4877 628 7 kneser kneser NOUN ejpam-4877 628 8 graphs graph NOUN ejpam-4877 628 9 . . PUNCT ejpam-4877 629 1 ars ar VERB ejpam-4877 629 2 math math PROPN ejpam-4877 629 3 . . PUNCT ejpam-4877 630 1 contemp contemp NOUN ejpam-4877 630 2 . . PUNCT ejpam-4877 631 1 , , PUNCT ejpam-4877 631 2 ( ( PUNCT ejpam-4877 631 3 17):419–430 17):419–430 NUM ejpam-4877 631 4 , , PUNCT ejpam-4877 631 5 2019 2019 NUM ejpam-4877 631 6 . . PUNCT ejpam-4877 632 1 [ [ X ejpam-4877 632 2 7 7 X ejpam-4877 632 3 ] ] X ejpam-4877 632 4 e. e. PROPN ejpam-4877 632 5 cockayne cockayne PROPN ejpam-4877 632 6 , , PUNCT ejpam-4877 632 7 r. r. PROPN ejpam-4877 632 8 dawes dawes PROPN ejpam-4877 632 9 , , PUNCT ejpam-4877 632 10 and and CCONJ ejpam-4877 632 11 s. s. PROPN ejpam-4877 632 12 hedetnieme hedetnieme PROPN ejpam-4877 632 13 . . PUNCT ejpam-4877 633 1 total total ADJ ejpam-4877 633 2 domination domination NOUN ejpam-4877 633 3 in in ADP ejpam-4877 633 4 graphs graph NOUN ejpam-4877 633 5 . . PUNCT ejpam-4877 634 1 networks network NOUN ejpam-4877 634 2 , , PUNCT ejpam-4877 634 3 10:211–219 10:211–219 NUM ejpam-4877 634 4 , , PUNCT ejpam-4877 634 5 1980 1980 NUM ejpam-4877 634 6 . . PUNCT ejpam-4877 635 1 [ [ X ejpam-4877 635 2 8 8 X ejpam-4877 635 3 ] ] PUNCT ejpam-4877 635 4 j. j. PROPN ejpam-4877 635 5 hassan hassan PROPN ejpam-4877 635 6 and and CCONJ ejpam-4877 635 7 s. s. PROPN ejpam-4877 635 8 canoy canoy PROPN ejpam-4877 635 9 jr jr PROPN ejpam-4877 635 10 . . PUNCT ejpam-4877 636 1 grundy grundy PROPN ejpam-4877 636 2 hop hop PROPN ejpam-4877 636 3 domination domination PROPN ejpam-4877 636 4 in in ADP ejpam-4877 636 5 graphs graph NOUN ejpam-4877 636 6 , , PUNCT ejpam-4877 636 7 . . PUNCT ejpam-4877 637 1 eur eur PROPN ejpam-4877 637 2 . . PUNCT ejpam-4877 638 1 j. j. PROPN ejpam-4877 638 2 pure pure PROPN ejpam-4877 638 3 appl appl PROPN ejpam-4877 638 4 . . PUNCT ejpam-4877 638 5 math math PROPN ejpam-4877 638 6 . . PUNCT ejpam-4877 638 7 , , PUNCT ejpam-4877 638 8 15(4):1623–1636 15(4):1623–1636 NUM ejpam-4877 638 9 , , PUNCT ejpam-4877 638 10 2022 2022 NUM ejpam-4877 638 11 . . PUNCT ejpam-4877 639 1 [ [ X ejpam-4877 639 2 9 9 NUM ejpam-4877 639 3 ] ] PUNCT ejpam-4877 639 4 j. j. PROPN ejpam-4877 639 5 hassan hassan PROPN ejpam-4877 639 6 and and CCONJ ejpam-4877 639 7 s. s. PROPN ejpam-4877 639 8 canoy canoy PROPN ejpam-4877 639 9 jr jr PROPN ejpam-4877 639 10 . . PROPN ejpam-4877 639 11 hop hop PROPN ejpam-4877 639 12 independent independent ADJ ejpam-4877 639 13 hop hop NOUN ejpam-4877 639 14 domination domination NOUN ejpam-4877 639 15 in in ADP ejpam-4877 639 16 graphs graph NOUN ejpam-4877 639 17 . . PUNCT ejpam-4877 640 1 , , PUNCT ejpam-4877 640 2 . . PUNCT ejpam-4877 641 1 eur eur PROPN ejpam-4877 641 2 . . PUNCT ejpam-4877 642 1 j. j. PROPN ejpam-4877 642 2 pure pure PROPN ejpam-4877 642 3 appl appl PROPN ejpam-4877 642 4 . . PUNCT ejpam-4877 642 5 math math PROPN ejpam-4877 642 6 . . PUNCT ejpam-4877 642 7 , , PUNCT ejpam-4877 642 8 15(4):1783–1796 15(4):1783–1796 NUM ejpam-4877 642 9 , , PUNCT ejpam-4877 642 10 2022 2022 NUM ejpam-4877 642 11 . . PUNCT ejpam-4877 643 1 [ [ X ejpam-4877 643 2 10 10 NUM ejpam-4877 643 3 ] ] PUNCT ejpam-4877 643 4 j. j. PROPN ejpam-4877 643 5 hassan hassan PROPN ejpam-4877 643 6 and and CCONJ ejpam-4877 643 7 s. s. PROPN ejpam-4877 643 8 canoy canoy PROPN ejpam-4877 643 9 jr jr PROPN ejpam-4877 643 10 . . PROPN ejpam-4877 643 11 connected connect VERB ejpam-4877 643 12 grundy grundy PROPN ejpam-4877 643 13 hop hop NOUN ejpam-4877 643 14 dominating dominate VERB ejpam-4877 643 15 sequences sequence NOUN ejpam-4877 643 16 in in ADP ejpam-4877 643 17 graphs graph NOUN ejpam-4877 643 18 . . PUNCT ejpam-4877 644 1 eur eur PROPN ejpam-4877 644 2 . . PUNCT ejpam-4877 645 1 j. j. PROPN ejpam-4877 645 2 pure pure PROPN ejpam-4877 645 3 appl appl PROPN ejpam-4877 645 4 . . PUNCT ejpam-4877 645 5 math math PROPN ejpam-4877 645 6 . . PUNCT ejpam-4877 645 7 , , PUNCT ejpam-4877 646 1 16(2):1212–1227 16(2):1212–1227 NUM ejpam-4877 646 2 , , PUNCT ejpam-4877 646 3 2023 2023 NUM ejpam-4877 646 4 . . PUNCT ejpam-4877 647 1 [ [ X ejpam-4877 647 2 11 11 NUM ejpam-4877 647 3 ] ] PUNCT ejpam-4877 647 4 j. j. PROPN ejpam-4877 647 5 hassan hassan PROPN ejpam-4877 647 6 and and CCONJ ejpam-4877 647 7 s. s. PROPN ejpam-4877 647 8 canoy canoy PROPN ejpam-4877 647 9 jr jr PROPN ejpam-4877 647 10 . . PROPN ejpam-4877 647 11 convex convex VERB ejpam-4877 647 12 hop hop NOUN ejpam-4877 647 13 domination domination NOUN ejpam-4877 647 14 in in ADP ejpam-4877 647 15 graphs graph NOUN ejpam-4877 647 16 . . PUNCT ejpam-4877 648 1 eur eur PROPN ejpam-4877 648 2 . . PUNCT ejpam-4877 649 1 j. j. PROPN ejpam-4877 649 2 pure pure PROPN ejpam-4877 649 3 appl appl PROPN ejpam-4877 649 4 . . PUNCT ejpam-4877 649 5 math math PROPN ejpam-4877 649 6 . . PUNCT ejpam-4877 649 7 , , PUNCT ejpam-4877 649 8 16(1):319–335 16(1):319–335 NOUN ejpam-4877 649 9 , , PUNCT ejpam-4877 649 10 2023 2023 NUM ejpam-4877 649 11 . . PUNCT ejpam-4877 650 1 [ [ X ejpam-4877 650 2 12 12 NUM ejpam-4877 650 3 ] ] PUNCT ejpam-4877 650 4 m. m. NOUN ejpam-4877 650 5 henning henning PROPN ejpam-4877 650 6 and and CCONJ ejpam-4877 650 7 n. n. PROPN ejpam-4877 650 8 rad rad PROPN ejpam-4877 650 9 . . PROPN ejpam-4877 651 1 on on ADP ejpam-4877 651 2 2 2 NUM ejpam-4877 651 3 - - PUNCT ejpam-4877 651 4 step step NOUN ejpam-4877 651 5 and and CCONJ ejpam-4877 651 6 hop hop NOUN ejpam-4877 651 7 dominating dominating NOUN ejpam-4877 651 8 sets set NOUN ejpam-4877 651 9 in in ADP ejpam-4877 651 10 graphs graph NOUN ejpam-4877 651 11 . . PUNCT ejpam-4877 652 1 graphs graph NOUN ejpam-4877 652 2 and and CCONJ ejpam-4877 652 3 combinatorics combinatoric NOUN ejpam-4877 652 4 . . PUNCT ejpam-4877 652 5 , , PUNCT ejpam-4877 652 6 33(4):913–927 33(4):913–927 PROPN ejpam-4877 652 7 , , PUNCT ejpam-4877 652 8 2017 2017 NUM ejpam-4877 652 9 . . PUNCT ejpam-4877 653 1 [ [ X ejpam-4877 653 2 13 13 NUM ejpam-4877 653 3 ] ] PUNCT ejpam-4877 653 4 s. s. PROPN ejpam-4877 653 5 canoy canoy PROPN ejpam-4877 653 6 jr jr PROPN ejpam-4877 653 7 . . PROPN ejpam-4877 653 8 , , PUNCT ejpam-4877 653 9 r. r. PROPN ejpam-4877 653 10 mollejon mollejon NOUN ejpam-4877 653 11 , , PUNCT ejpam-4877 653 12 and and CCONJ ejpam-4877 653 13 j. j. PROPN ejpam-4877 653 14 g. g. PROPN ejpam-4877 653 15 canoy canoy PROPN ejpam-4877 653 16 . . PUNCT ejpam-4877 654 1 hop hop PROPN ejpam-4877 654 2 dominating dominating NOUN ejpam-4877 654 3 sets set NOUN ejpam-4877 654 4 in in ADP ejpam-4877 654 5 graphs graph NOUN ejpam-4877 654 6 under under ADP ejpam-4877 654 7 binary binary ADJ ejpam-4877 654 8 operations operation NOUN ejpam-4877 654 9 . . PUNCT ejpam-4877 655 1 eur eur PROPN ejpam-4877 655 2 . . PUNCT ejpam-4877 656 1 j. j. PROPN ejpam-4877 656 2 pure pure PROPN ejpam-4877 656 3 appl appl PROPN ejpam-4877 656 4 . . PUNCT ejpam-4877 656 5 math math PROPN ejpam-4877 656 6 . . PUNCT ejpam-4877 656 7 , , PUNCT ejpam-4877 657 1 12(4):1455–1463 12(4):1455–1463 NUM ejpam-4877 657 2 , , PUNCT ejpam-4877 657 3 2019 2019 NUM ejpam-4877 657 4 . . PUNCT ejpam-4877 658 1 [ [ X ejpam-4877 658 2 14 14 NUM ejpam-4877 658 3 ] ] X ejpam-4877 658 4 s. s. PROPN ejpam-4877 658 5 canoy canoy PROPN ejpam-4877 658 6 jr jr PROPN ejpam-4877 658 7 . . PROPN ejpam-4877 658 8 and and CCONJ ejpam-4877 658 9 g. g. PROPN ejpam-4877 658 10 salasalan salasalan NOUN ejpam-4877 658 11 . . PUNCT ejpam-4877 659 1 revisiting revisit VERB ejpam-4877 659 2 domination domination NOUN ejpam-4877 659 3 , , PUNCT ejpam-4877 659 4 hop hop NOUN ejpam-4877 659 5 domination domination NOUN ejpam-4877 659 6 , , PUNCT ejpam-4877 659 7 and and CCONJ ejpam-4877 659 8 global global ADJ ejpam-4877 659 9 hop hop NOUN ejpam-4877 659 10 domination domination NOUN ejpam-4877 659 11 in in ADP ejpam-4877 659 12 graphs graph NOUN ejpam-4877 659 13 , , PUNCT ejpam-4877 659 14 . . PUNCT ejpam-4877 660 1 eur eur PROPN ejpam-4877 660 2 . . PUNCT ejpam-4877 661 1 j. j. PROPN ejpam-4877 661 2 pure pure PROPN ejpam-4877 661 3 appl appl PROPN ejpam-4877 661 4 . . PUNCT ejpam-4877 661 5 math math PROPN ejpam-4877 661 6 . . PUNCT ejpam-4877 661 7 , , PUNCT ejpam-4877 661 8 14:1415–1428 14:1415–1428 NUM ejpam-4877 661 9 , , PUNCT ejpam-4877 661 10 2021 2021 NUM ejpam-4877 661 11 . . PUNCT ejpam-4877 662 1 [ [ X ejpam-4877 662 2 15 15 NUM ejpam-4877 662 3 ] ] X ejpam-4877 662 4 s. s. PROPN ejpam-4877 662 5 pal pal PROPN ejpam-4877 662 6 m. m. PROPN ejpam-4877 662 7 henning henning PROPN ejpam-4877 662 8 and and CCONJ ejpam-4877 662 9 d. d. PROPN ejpam-4877 662 10 pradhan pradhan PROPN ejpam-4877 662 11 . . PUNCT ejpam-4877 663 1 algorithm algorithm PROPN ejpam-4877 663 2 and and CCONJ ejpam-4877 663 3 hardness hardness NOUN ejpam-4877 663 4 results result NOUN ejpam-4877 663 5 on on ADP ejpam-4877 663 6 hop hop NOUN ejpam-4877 663 7 domination domination NOUN ejpam-4877 663 8 in in ADP ejpam-4877 663 9 graphs graph NOUN ejpam-4877 663 10 . . PUNCT ejpam-4877 664 1 inform inform NOUN ejpam-4877 664 2 . . PUNCT ejpam-4877 665 1 process process NOUN ejpam-4877 665 2 . . PUNCT ejpam-4877 666 1 lett lett PROPN ejpam-4877 666 2 . . PROPN ejpam-4877 666 3 , , PUNCT ejpam-4877 666 4 153 153 NUM ejpam-4877 666 5 : : PUNCT ejpam-4877 666 6 doi:10.1016 doi:10.1016 PROPN ejpam-4877 666 7 / / SYM ejpam-4877 666 8 j.ipl.2019.105872 j.ipl.2019.105872 PROPN ejpam-4877 666 9 . . PROPN ejpam-4877 666 10 , , PUNCT ejpam-4877 666 11 2020 2020 NUM ejpam-4877 666 12 . . PUNCT ejpam-4877 667 1 [ [ X ejpam-4877 667 2 16 16 NUM ejpam-4877 667 3 ] ] X ejpam-4877 667 4 g. g. PROPN ejpam-4877 667 5 nasini nasini PROPN ejpam-4877 667 6 and and CCONJ ejpam-4877 667 7 p. p. NOUN ejpam-4877 667 8 torres torre NOUN ejpam-4877 667 9 . . PUNCT ejpam-4877 668 1 grundy grundy PROPN ejpam-4877 668 2 dominating dominate VERB ejpam-4877 668 3 sequences sequence NOUN ejpam-4877 668 4 on on ADP ejpam-4877 668 5 x x ADJ ejpam-4877 668 6 - - ADJ ejpam-4877 668 7 join join ADJ ejpam-4877 668 8 product product NOUN ejpam-4877 668 9 . . PUNCT ejpam-4877 669 1 discrete discrete ADJ ejpam-4877 669 2 applied applied ADJ ejpam-4877 669 3 mathematics mathematic NOUN ejpam-4877 669 4 . . PUNCT ejpam-4877 669 5 , , PUNCT ejpam-4877 669 6 ( ( PUNCT ejpam-4877 669 7 284):138–149 284):138–149 NOUN ejpam-4877 669 8 , , PUNCT ejpam-4877 669 9 2020 2020 NUM ejpam-4877 669 10 . . PUNCT ejpam-4877 670 1 [ [ X ejpam-4877 670 2 17 17 NUM ejpam-4877 670 3 ] ] X ejpam-4877 670 4 c. c. PROPN ejpam-4877 670 5 natarajan natarajan PROPN ejpam-4877 670 6 and and CCONJ ejpam-4877 670 7 s. s. PROPN ejpam-4877 670 8 ayyaswamy ayyaswamy PROPN ejpam-4877 670 9 . . PUNCT ejpam-4877 671 1 hop hop PROPN ejpam-4877 671 2 domination domination NOUN ejpam-4877 671 3 in in ADP ejpam-4877 671 4 graphs graphs PROPN ejpam-4877 671 5 ii ii PROPN ejpam-4877 671 6 . . PUNCT ejpam-4877 671 7 versita versita PROPN ejpam-4877 671 8 , , PUNCT ejpam-4877 671 9 23(2):187 23(2):187 NUM ejpam-4877 671 10 – – PUNCT ejpam-4877 671 11 199 199 NUM ejpam-4877 671 12 , , PUNCT ejpam-4877 671 13 2015 2015 NUM ejpam-4877 671 14 . . PUNCT ejpam-4877 672 1 [ [ X ejpam-4877 672 2 18 18 NUM ejpam-4877 672 3 ] ] PUNCT ejpam-4877 672 4 b. b. NOUN ejpam-4877 672 5 omamalin omamalin PROPN ejpam-4877 672 6 , , PUNCT ejpam-4877 672 7 s. s. PROPN ejpam-4877 672 8 canoy canoy PROPN ejpam-4877 672 9 jr jr PROPN ejpam-4877 672 10 . . PROPN ejpam-4877 672 11 , , PUNCT ejpam-4877 672 12 and and CCONJ ejpam-4877 672 13 h. h. PROPN ejpam-4877 672 14 rara rara PROPN ejpam-4877 672 15 . . PUNCT ejpam-4877 673 1 locating locate VERB ejpam-4877 673 2 total total ADJ ejpam-4877 673 3 dominating dominating NOUN ejpam-4877 673 4 sets set NOUN ejpam-4877 673 5 in in ADP ejpam-4877 673 6 the the DET ejpam-4877 673 7 join join NOUN ejpam-4877 673 8 , , PUNCT ejpam-4877 673 9 corona corona PROPN ejpam-4877 673 10 , , PUNCT ejpam-4877 673 11 and and CCONJ ejpam-4877 673 12 composition composition NOUN ejpam-4877 673 13 of of ADP ejpam-4877 673 14 graphs graph NOUN ejpam-4877 673 15 . . PUNCT ejpam-4877 674 1 applied apply VERB ejpam-4877 674 2 mathematical mathematical ADJ ejpam-4877 674 3 sciences science NOUN ejpam-4877 674 4 , , PUNCT ejpam-4877 674 5 8(48):2363–2374 8(48):2363–2374 NUM ejpam-4877 674 6 , , PUNCT ejpam-4877 674 7 2014 2014 NUM ejpam-4877 674 8 . . PUNCT ejpam-4877 675 1 [ [ X ejpam-4877 675 2 19 19 NUM ejpam-4877 675 3 ] ] X ejpam-4877 675 4 s. s. PROPN ejpam-4877 675 5 divya divya PROPN ejpam-4877 675 6 rashmi rashmi PROPN ejpam-4877 675 7 , , PUNCT ejpam-4877 675 8 s. s. PROPN ejpam-4877 675 9 amurugan amurugan PROPN ejpam-4877 675 10 , , PUNCT ejpam-4877 675 11 and and CCONJ ejpam-4877 675 12 i. i. PROPN ejpam-4877 675 13 venkat venkat PROPN ejpam-4877 675 14 . . PUNCT ejpam-4877 676 1 secure secure ADJ ejpam-4877 676 2 domination domination NOUN ejpam-4877 676 3 in in ADP ejpam-4877 676 4 graphs graph NOUN ejpam-4877 676 5 . . PUNCT ejpam-4877 677 1 int int NOUN ejpam-4877 677 2 . . PUNCT ejpam-4877 678 1 j j PROPN ejpam-4877 678 2 advance advance VERB ejpam-4877 678 3 soft soft ADJ ejpam-4877 678 4 compu compu PROPN ejpam-4877 678 5 . . PUNCT ejpam-4877 679 1 appl appl PROPN ejpam-4877 679 2 , , PUNCT ejpam-4877 679 3 8(2):79–83 8(2):79–83 NUM ejpam-4877 679 4 , , PUNCT ejpam-4877 679 5 2016 2016 NUM ejpam-4877 679 6 . . PUNCT ejpam-4877 680 1 [ [ X ejpam-4877 680 2 20 20 NUM ejpam-4877 680 3 ] ] X ejpam-4877 680 4 g. g. NOUN ejpam-4877 680 5 salasalan salasalan NOUN ejpam-4877 680 6 and and CCONJ ejpam-4877 680 7 s. s. PROPN ejpam-4877 680 8 canoy canoy PROPN ejpam-4877 680 9 jr jr PROPN ejpam-4877 680 10 . . PROPN ejpam-4877 680 11 global global PROPN ejpam-4877 680 12 hop hop PROPN ejpam-4877 680 13 domination domination NOUN ejpam-4877 680 14 of of ADP ejpam-4877 680 15 graphs graph NOUN ejpam-4877 680 16 . . PUNCT ejpam-4877 681 1 eur eur PROPN ejpam-4877 681 2 . . PUNCT ejpam-4877 682 1 j. j. PROPN ejpam-4877 682 2 pure pure PROPN ejpam-4877 682 3 appl appl PROPN ejpam-4877 682 4 . . PUNCT ejpam-4877 682 5 math math PROPN ejpam-4877 682 6 . . PUNCT ejpam-4877 682 7 , , PUNCT ejpam-4877 682 8 14(1):112–125 14(1):112–125 NUM ejpam-4877 682 9 , , PUNCT ejpam-4877 682 10 2021 2021 NUM ejpam-4877 682 11 . . PUNCT ejpam-4877 683 1 [ [ X ejpam-4877 683 2 21 21 NUM ejpam-4877 683 3 ] ] X ejpam-4877 683 4 e. e. PROPN ejpam-4877 683 5 sampathkumar sampathkumar PROPN ejpam-4877 683 6 and and CCONJ ejpam-4877 683 7 h. h. PROPN ejpam-4877 683 8 walikar walikar PROPN ejpam-4877 683 9 . . PUNCT ejpam-4877 684 1 the the DET ejpam-4877 684 2 connected connected ADJ ejpam-4877 684 3 domination domination NOUN ejpam-4877 684 4 number number NOUN ejpam-4877 684 5 of of ADP ejpam-4877 684 6 a a DET ejpam-4877 684 7 graph graph NOUN ejpam-4877 684 8 . . PUNCT ejpam-4877 685 1 j. j. PROPN ejpam-4877 685 2 math math PROPN ejpam-4877 685 3 . . PUNCT ejpam-4877 686 1 phys phy NOUN ejpam-4877 686 2 . . PUNCT ejpam-4877 687 1 sci sci PROPN ejpam-4877 687 2 . . PROPN ejpam-4877 687 3 , , PUNCT ejpam-4877 687 4 13(6):607–613 13(6):607–613 PROPN ejpam-4877 687 5 , , PUNCT ejpam-4877 687 6 1979 1979 NUM ejpam-4877 687 7 . . PUNCT