id sid tid token lemma pos cana-5930 1 1 communications communication NOUN cana-5930 1 2 on on ADP cana-5930 1 3 applied apply VERB cana-5930 1 4 nonlinear nonlinear ADJ cana-5930 1 5 analysis analysis NOUN cana-5930 1 6 issn issn NOUN cana-5930 1 7 : : PUNCT cana-5930 1 8 1074 1074 NUM cana-5930 1 9 - - PUNCT cana-5930 1 10 133x 133x NUM cana-5930 1 11 vol vol VERB cana-5930 1 12 32 32 NUM cana-5930 1 13 no no NOUN cana-5930 1 14 . . PUNCT cana-5930 2 1 10s 10 NOUN cana-5930 2 2 ( ( PUNCT cana-5930 2 3 2025 2025 NUM cana-5930 2 4 ) ) PUNCT cana-5930 2 5 3103 3103 NUM cana-5930 3 1 https://internationalpubls.com https://internationalpubls.com X cana-5930 3 2 more more ADJ cana-5930 3 3 results result NOUN cana-5930 3 4 on on ADP cana-5930 3 5 complementary complementary ADJ cana-5930 3 6 tree tree NOUN cana-5930 3 7 domination domination NOUN cana-5930 3 8 number number NOUN cana-5930 3 9 of of ADP cana-5930 3 10 semi semi ADJ cana-5930 3 11 total total ADJ cana-5930 3 12 point point NOUN cana-5930 3 13 graph graph NOUN cana-5930 3 14 s. s. PROPN cana-5930 3 15 jayalakshmi1 jayalakshmi1 PROPN cana-5930 3 16 and and CCONJ cana-5930 3 17 p. p. NOUN cana-5930 3 18 vidhya2 vidhya2 NOUN cana-5930 4 1 1research 1research NUM cana-5930 4 2 scholar scholar NOUN cana-5930 4 3 ( ( PUNCT cana-5930 4 4 part part NOUN cana-5930 4 5 time time NOUN cana-5930 4 6 ) ) PUNCT cana-5930 4 7 , , PUNCT cana-5930 4 8 school school NOUN cana-5930 4 9 of of ADP cana-5930 4 10 mathematics mathematics PROPN cana-5930 4 11 madurai madurai PROPN cana-5930 4 12 kamaraj kamaraj ADJ cana-5930 4 13 university university NOUN cana-5930 4 14 , , PUNCT cana-5930 4 15 madurai madurai NOUN cana-5930 4 16 625021 625021 NUM cana-5930 4 17 , , PUNCT cana-5930 4 18 tamilnadu tamilnadu NOUN cana-5930 4 19 , , PUNCT cana-5930 4 20 india india PROPN cana-5930 4 21 e e PROPN cana-5930 4 22 - - NOUN cana-5930 4 23 mail mail NOUN cana-5930 4 24 : : PUNCT cana-5930 4 25 jayark83@gmail.com jayark83@gmail.com X cana-5930 4 26 2associate 2associate NUM cana-5930 4 27 professor professor NOUN cana-5930 4 28 , , PUNCT cana-5930 4 29 department department NOUN cana-5930 4 30 of of ADP cana-5930 4 31 mathematics mathematics PROPN cana-5930 4 32 , , PUNCT cana-5930 4 33 emg emg PROPN cana-5930 4 34 yadava yadava PROPN cana-5930 4 35 women women PROPN cana-5930 4 36 ’s ’s PART cana-5930 4 37 college college NOUN cana-5930 4 38 madurai madurai PROPN cana-5930 4 39 625014 625014 NUM cana-5930 4 40 , , PUNCT cana-5930 4 41 tamilnadu tamilnadu NOUN cana-5930 4 42 , , PUNCT cana-5930 4 43 india india PROPN cana-5930 4 44 e e PROPN cana-5930 4 45 - - NOUN cana-5930 4 46 mail mail NOUN cana-5930 4 47 : : PUNCT cana-5930 4 48 vidhyaramman@gmail.com vidhyaramman@gmail.com X cana-5930 4 49 , , PUNCT cana-5930 4 50 p.vidhya-mat@emgywomenscollege.ac.in p.vidhya-mat@emgywomenscollege.ac.in AUX cana-5930 4 51 article article NOUN cana-5930 4 52 history history NOUN cana-5930 4 53 : : PUNCT cana-5930 4 54 received receive VERB cana-5930 4 55 : : PUNCT cana-5930 4 56 02 02 NUM cana-5930 4 57 - - PUNCT cana-5930 4 58 01 01 NUM cana-5930 4 59 - - PUNCT cana-5930 4 60 2025 2025 NUM cana-5930 4 61 revised revise VERB cana-5930 4 62 : : PUNCT cana-5930 4 63 25 25 NUM cana-5930 4 64 - - PUNCT cana-5930 4 65 02 02 NUM cana-5930 4 66 - - PUNCT cana-5930 4 67 2025 2025 NUM cana-5930 4 68 accepted accept VERB cana-5930 4 69 : : PUNCT cana-5930 4 70 20 20 NUM cana-5930 4 71 - - SYM cana-5930 4 72 03 03 NUM cana-5930 4 73 - - PUNCT cana-5930 4 74 2025 2025 NUM cana-5930 4 75 abstract abstract NOUN cana-5930 4 76 : : PUNCT cana-5930 4 77 a a DET cana-5930 4 78 set set NOUN cana-5930 4 79 d d NOUN cana-5930 4 80 βŠ† βŠ† NUM cana-5930 4 81 v v NOUN cana-5930 4 82 of of ADP cana-5930 4 83 a a DET cana-5930 4 84 graph graph NOUN cana-5930 4 85 g g NOUN cana-5930 4 86 = = SYM cana-5930 4 87 ( ( PUNCT cana-5930 4 88 v v NOUN cana-5930 4 89 , , PUNCT cana-5930 4 90 e e NOUN cana-5930 4 91 ) ) PUNCT cana-5930 4 92 is be AUX cana-5930 4 93 a a DET cana-5930 4 94 complementary complementary ADJ cana-5930 4 95 tree tree NOUN cana-5930 4 96 dominating dominating NOUN cana-5930 4 97 set set VERB cana-5930 4 98 if if SCONJ cana-5930 4 99 the the DET cana-5930 4 100 induced induced ADJ cana-5930 4 101 subgraph subgraph NOUN cana-5930 4 102 < < X cana-5930 4 103 𝑉(g 𝑉(g PROPN cana-5930 4 104 ) ) PUNCT cana-5930 4 105 βˆ’ βˆ’ PROPN cana-5930 5 1 d d X cana-5930 5 2 > > X cana-5930 5 3 is be AUX cana-5930 5 4 a a DET cana-5930 5 5 tree tree NOUN cana-5930 5 6 . . PUNCT cana-5930 6 1 the the DET cana-5930 6 2 complementary complementary ADJ cana-5930 6 3 tree tree NOUN cana-5930 6 4 domination domination NOUN cana-5930 6 5 number number NOUN cana-5930 6 6 Ξ³ctd(g Ξ³ctd(g PROPN cana-5930 6 7 ) ) PUNCT cana-5930 6 8 is be AUX cana-5930 6 9 the the DET cana-5930 6 10 minimum minimum ADJ cana-5930 6 11 cardinality cardinality NOUN cana-5930 6 12 of of ADP cana-5930 6 13 a a DET cana-5930 6 14 complementary complementary ADJ cana-5930 6 15 tree tree NOUN cana-5930 6 16 dominating dominating NOUN cana-5930 6 17 set set NOUN cana-5930 6 18 ( ( PUNCT cana-5930 6 19 ctd ctd NOUN cana-5930 6 20 - - PUNCT cana-5930 6 21 set set NOUN cana-5930 6 22 ) ) PUNCT cana-5930 6 23 of of ADP cana-5930 6 24 g. g. PROPN cana-5930 6 25 the the DET cana-5930 6 26 semi semi ADJ cana-5930 6 27 total total ADJ cana-5930 6 28 point point NOUN cana-5930 6 29 graph graph NOUN cana-5930 6 30 t2(g t2(g PRON cana-5930 6 31 ) ) PUNCT cana-5930 6 32 is be AUX cana-5930 6 33 the the DET cana-5930 6 34 graph graph NOUN cana-5930 6 35 g g NOUN cana-5930 6 36 whose whose DET cana-5930 6 37 vertex vertex NOUN cana-5930 6 38 set set NOUN cana-5930 6 39 is be AUX cana-5930 6 40 v(g v(g ADJ cana-5930 6 41 ) ) PUNCT cana-5930 6 42 βˆͺ βˆͺ ADP cana-5930 6 43 e(g e(g PROPN cana-5930 6 44 ) ) PUNCT cana-5930 6 45 . . PUNCT cana-5930 7 1 where where SCONJ cana-5930 7 2 two two NUM cana-5930 7 3 vertices vertex NOUN cana-5930 7 4 are be AUX cana-5930 7 5 adjacent adjacent ADJ cana-5930 7 6 if if SCONJ cana-5930 7 7 and and CCONJ cana-5930 7 8 only only ADV cana-5930 7 9 if if SCONJ cana-5930 7 10 ( ( PUNCT cana-5930 7 11 i i NOUN cana-5930 7 12 ) ) PUNCT cana-5930 7 13 they they PRON cana-5930 7 14 are be AUX cana-5930 7 15 adjacent adjacent ADJ cana-5930 7 16 vertices vertex NOUN cana-5930 7 17 of of ADP cana-5930 7 18 g g PROPN cana-5930 7 19 or or CCONJ cana-5930 7 20 ( ( PUNCT cana-5930 7 21 ii ii NOUN cana-5930 7 22 ) ) PUNCT cana-5930 7 23 one one NOUN cana-5930 7 24 is be AUX cana-5930 7 25 a a DET cana-5930 7 26 vertex vertex NOUN cana-5930 7 27 and and CCONJ cana-5930 7 28 the the DET cana-5930 7 29 other other ADJ cana-5930 7 30 is be AUX cana-5930 7 31 an an DET cana-5930 7 32 edge edge NOUN cana-5930 7 33 of of ADP cana-5930 7 34 g g PROPN cana-5930 7 35 incident incident NOUN cana-5930 7 36 with with ADP cana-5930 7 37 it it PRON cana-5930 7 38 . . PUNCT cana-5930 8 1 in in ADP cana-5930 8 2 this this DET cana-5930 8 3 paper paper NOUN cana-5930 8 4 complementary complementary ADJ cana-5930 8 5 tree tree NOUN cana-5930 8 6 domination domination NOUN cana-5930 8 7 number number NOUN cana-5930 8 8 of of ADP cana-5930 8 9 semi semi ADJ cana-5930 8 10 total total ADJ cana-5930 8 11 point point NOUN cana-5930 8 12 graph graph NOUN cana-5930 8 13 of of ADP cana-5930 8 14 graphs graph NOUN cana-5930 8 15 , , PUNCT cana-5930 8 16 its its PRON cana-5930 8 17 bounds bound NOUN cana-5930 8 18 and and CCONJ cana-5930 8 19 relation relation NOUN cana-5930 8 20 between between ADP cana-5930 8 21 Ξ³ctd(g Ξ³ctd(g PROPN cana-5930 8 22 ) ) PUNCT cana-5930 8 23 and and CCONJ cana-5930 8 24 Ξ³ctd(t2(g Ξ³ctd(t2(g NOUN cana-5930 8 25 ) ) PUNCT cana-5930 8 26 ) ) PUNCT cana-5930 8 27 are be AUX cana-5930 8 28 obtained obtain VERB cana-5930 8 29 . . PUNCT cana-5930 9 1 keywords keyword NOUN cana-5930 9 2 : : PUNCT cana-5930 9 3 dominating dominate VERB cana-5930 9 4 set set NOUN cana-5930 9 5 , , PUNCT cana-5930 9 6 complementary complementary ADJ cana-5930 9 7 tree tree NOUN cana-5930 9 8 dominating dominating NOUN cana-5930 9 9 set set NOUN cana-5930 9 10 , , PUNCT cana-5930 9 11 semi semi ADV cana-5930 9 12 total total ADJ cana-5930 9 13 point point NOUN cana-5930 9 14 graph graph NOUN cana-5930 9 15 . . PUNCT cana-5930 10 1 1 1 X cana-5930 10 2 . . X cana-5930 10 3 introduction introduction NOUN cana-5930 10 4 a a DET cana-5930 10 5 graph graph NOUN cana-5930 10 6 𝐺(𝑉 𝐺(𝑉 PRON cana-5930 10 7 , , PUNCT cana-5930 10 8 𝐸 𝐸 PROPN cana-5930 10 9 ) ) PUNCT cana-5930 10 10 discussed discuss VERB cana-5930 10 11 in in ADP cana-5930 10 12 this this DET cana-5930 10 13 paper paper NOUN cana-5930 10 14 be be AUX cana-5930 10 15 a a DET cana-5930 10 16 simple simple ADJ cana-5930 10 17 , , PUNCT cana-5930 10 18 finite finite ADJ cana-5930 10 19 , , PUNCT cana-5930 10 20 undirected undirected ADJ cana-5930 10 21 , , PUNCT cana-5930 10 22 connected connected ADJ cana-5930 10 23 graph graph NOUN cana-5930 10 24 with with ADP cana-5930 10 25 p p NOUN cana-5930 10 26 vertices vertex NOUN cana-5930 10 27 and and CCONJ cana-5930 10 28 q q NOUN cana-5930 10 29 edges edge NOUN cana-5930 10 30 . . PUNCT cana-5930 11 1 a a DET cana-5930 11 2 set set NOUN cana-5930 11 3 of of ADP cana-5930 11 4 vertices vertex NOUN cana-5930 11 5 in in ADP cana-5930 11 6 a a DET cana-5930 11 7 graph graph NOUN cana-5930 11 8 g g NOUN cana-5930 11 9 is be AUX cana-5930 11 10 independent independent ADJ cana-5930 11 11 , , PUNCT cana-5930 11 12 if if SCONJ cana-5930 11 13 no no DET cana-5930 11 14 two two NUM cana-5930 11 15 vertices vertex NOUN cana-5930 11 16 are be AUX cana-5930 11 17 adjacent adjacent ADJ cana-5930 11 18 . . PUNCT cana-5930 12 1 the the DET cana-5930 12 2 largest large ADJ cana-5930 12 3 number number NOUN cana-5930 12 4 of of ADP cana-5930 12 5 vertices vertex NOUN cana-5930 12 6 in in ADP cana-5930 12 7 such such DET cana-5930 12 8 a a DET cana-5930 12 9 set set NOUN cana-5930 12 10 is be AUX cana-5930 12 11 called call VERB cana-5930 12 12 the the DET cana-5930 12 13 independence independence NOUN cana-5930 12 14 number number NOUN cana-5930 12 15 and and CCONJ cana-5930 12 16 is be AUX cana-5930 12 17 denoted denote VERB cana-5930 12 18 by by ADP cana-5930 12 19 Ξ²0(𝐺 Ξ²0(𝐺 PROPN cana-5930 12 20 ) ) PUNCT cana-5930 12 21 . . PUNCT cana-5930 13 1 the the DET cana-5930 13 2 corona corona PROPN cana-5930 13 3 𝐺1 𝐺1 PROPN cana-5930 13 4 ∘ ∘ PROPN cana-5930 13 5 𝐺2 𝐺2 NOUN cana-5930 13 6 of of ADP cana-5930 13 7 two two NUM cana-5930 13 8 graphs graph NOUN cana-5930 13 9 𝐺1 𝐺1 NOUN cana-5930 13 10 and and CCONJ cana-5930 13 11 𝐺2 𝐺2 NOUN cana-5930 13 12 are be AUX cana-5930 13 13 defined define VERB cana-5930 13 14 as as ADP cana-5930 13 15 the the DET cana-5930 13 16 graph graph NOUN cana-5930 13 17 g g PROPN cana-5930 13 18 obtained obtain VERB cana-5930 13 19 by by ADP cana-5930 13 20 taking take VERB cana-5930 13 21 one one NUM cana-5930 13 22 copy copy NOUN cana-5930 13 23 of of ADP cana-5930 13 24 𝐺1 𝐺1 NOUN cana-5930 13 25 of of ADP cana-5930 13 26 order order NOUN cana-5930 13 27 𝑝1 𝑝1 NOUN cana-5930 13 28 and and CCONJ cana-5930 13 29 𝑝1 𝑝1 NOUN cana-5930 13 30 copies copy NOUN cana-5930 13 31 of of ADP cana-5930 13 32 𝐺2 𝐺2 NOUN cana-5930 13 33 and and CCONJ cana-5930 13 34 then then ADV cana-5930 13 35 joining join VERB cana-5930 13 36 the the DET cana-5930 13 37 π‘–π‘‘β„Ž π‘–π‘‘β„Ž NOUN cana-5930 13 38 vertex vertex NOUN cana-5930 13 39 of of ADP cana-5930 13 40 𝐺1 𝐺1 NOUN cana-5930 13 41 to to ADP cana-5930 13 42 every every DET cana-5930 13 43 vertex vertex NOUN cana-5930 13 44 in in ADP cana-5930 13 45 the the DET cana-5930 13 46 π‘–π‘‘β„Ž π‘–π‘‘β„Ž NOUN cana-5930 13 47 copy copy NOUN cana-5930 13 48 of of ADP cana-5930 13 49 𝐺2 𝐺2 NOUN cana-5930 13 50 . . PUNCT cana-5930 14 1 the the DET cana-5930 14 2 corona corona PROPN cana-5930 14 3 𝐺1 𝐺1 PROPN cana-5930 14 4 ∘ ∘ PROPN cana-5930 14 5 𝐺2 𝐺2 PROPN cana-5930 14 6 has have VERB cana-5930 14 7 𝑝1(1 𝑝1(1 ADJ cana-5930 14 8 + + ADJ cana-5930 14 9 𝑝2 𝑝2 NOUN cana-5930 14 10 ) ) PUNCT cana-5930 14 11 vertices vertex NOUN cana-5930 14 12 and and CCONJ cana-5930 14 13 π‘ž1 π‘ž1 NOUN cana-5930 14 14 + + CCONJ cana-5930 15 1 𝑝1π‘ž2 𝑝1π‘ž2 CCONJ cana-5930 15 2 + + CCONJ cana-5930 15 3 𝑝1𝑝2 𝑝1𝑝2 PRON cana-5930 15 4 edges edge VERB cana-5930 15 5 . . PUNCT cana-5930 16 1 the the DET cana-5930 16 2 graph graph NOUN cana-5930 16 3 𝐢𝑛 𝐢𝑛 PROPN cana-5930 16 4 ( ( PUNCT cana-5930 16 5 𝑑 𝑑 NOUN cana-5930 16 6 ) ) PUNCT cana-5930 16 7 is be AUX cana-5930 16 8 the the DET cana-5930 16 9 one one NUM cana-5930 16 10 point point NOUN cana-5930 16 11 union union NOUN cana-5930 16 12 of of ADP cana-5930 16 13 t t PROPN cana-5930 16 14 cycles cycle NOUN cana-5930 16 15 of of ADP cana-5930 16 16 length length NOUN cana-5930 16 17 n. n. PROPN cana-5930 16 18 a a DET cana-5930 16 19 graph graph NOUN cana-5930 16 20 g g NOUN cana-5930 16 21 is be AUX cana-5930 16 22 unicyclic unicyclic ADJ cana-5930 16 23 if if SCONJ cana-5930 16 24 it it PRON cana-5930 16 25 contains contain VERB cana-5930 16 26 exactly exactly ADV cana-5930 16 27 one one NUM cana-5930 16 28 cycle cycle NOUN cana-5930 16 29 . . PUNCT cana-5930 17 1 a a DET cana-5930 17 2 broom broom NOUN cana-5930 17 3 graph graph NOUN cana-5930 17 4 𝐡𝑛,π‘š 𝐡𝑛,π‘š NOUN cana-5930 17 5 is be AUX cana-5930 17 6 a a DET cana-5930 17 7 graph graph NOUN cana-5930 17 8 of of ADP cana-5930 17 9 n n NOUN cana-5930 17 10 vertices vertex NOUN cana-5930 17 11 which which PRON cana-5930 17 12 have have VERB cana-5930 17 13 a a DET cana-5930 17 14 path path NOUN cana-5930 17 15 π‘ƒπ‘š π‘ƒπ‘š PROPN cana-5930 17 16 and and CCONJ cana-5930 17 17 𝑛 𝑛 DET cana-5930 17 18 βˆ’ βˆ’ NOUN cana-5930 17 19 π‘š π‘š ADP cana-5930 17 20 pendant pendant ADJ cana-5930 17 21 vertices vertex NOUN cana-5930 17 22 , , PUNCT cana-5930 17 23 all all PRON cana-5930 17 24 of of ADP cana-5930 17 25 these these DET cana-5930 17 26 vertices vertex NOUN cana-5930 17 27 are be AUX cana-5930 17 28 adjacent adjacent ADJ cana-5930 17 29 to to ADP cana-5930 17 30 either either CCONJ cana-5930 17 31 the the DET cana-5930 17 32 origin origin NOUN cana-5930 17 33 u u NOUN cana-5930 17 34 or or CCONJ cana-5930 17 35 the the DET cana-5930 17 36 terminus terminus NOUN cana-5930 17 37 v v NOUN cana-5930 17 38 of of ADP cana-5930 17 39 the the DET cana-5930 17 40 path path NOUN cana-5930 17 41 . . PUNCT cana-5930 18 1 any any DET cana-5930 18 2 undefined undefined ADJ cana-5930 18 3 term term NOUN cana-5930 18 4 in in ADP cana-5930 18 5 this this DET cana-5930 18 6 paper paper NOUN cana-5930 18 7 may may AUX cana-5930 18 8 be be AUX cana-5930 18 9 found find VERB cana-5930 18 10 in in ADP cana-5930 18 11 harary harary NOUN cana-5930 18 12 [ [ X cana-5930 18 13 2 2 NUM cana-5930 18 14 ] ] PUNCT cana-5930 18 15 . . PUNCT cana-5930 19 1 the the DET cana-5930 19 2 concept concept NOUN cana-5930 19 3 of of ADP cana-5930 19 4 domination domination NOUN cana-5930 19 5 in in ADP cana-5930 19 6 graphs graph NOUN cana-5930 19 7 was be AUX cana-5930 19 8 introduced introduce VERB cana-5930 19 9 by by ADP cana-5930 19 10 ore ore NOUN cana-5930 19 11 [ [ X cana-5930 19 12 5 5 NUM cana-5930 19 13 ] ] PUNCT cana-5930 19 14 . . PUNCT cana-5930 20 1 a a DET cana-5930 20 2 set set VERB cana-5930 20 3 𝐷 𝐷 NOUN cana-5930 20 4 βŠ† βŠ† NUM cana-5930 20 5 𝑉 𝑉 PROPN cana-5930 20 6 is be AUX cana-5930 20 7 said say VERB cana-5930 20 8 to to PART cana-5930 20 9 be be AUX cana-5930 20 10 a a DET cana-5930 20 11 dominating dominating NOUN cana-5930 20 12 set set NOUN cana-5930 20 13 of of ADP cana-5930 20 14 g g NOUN cana-5930 20 15 , , PUNCT cana-5930 20 16 if if SCONJ cana-5930 20 17 every every DET cana-5930 20 18 vertex vertex NOUN cana-5930 20 19 in in ADP cana-5930 20 20 𝑉 𝑉 PROPN cana-5930 20 21 βˆ’ βˆ’ PROPN cana-5930 20 22 𝐷 𝐷 PROPN cana-5930 20 23 is be AUX cana-5930 20 24 adjacent adjacent ADJ cana-5930 20 25 to to ADP cana-5930 20 26 some some DET cana-5930 20 27 vertex vertex NOUN cana-5930 20 28 in in ADP cana-5930 20 29 d. d. PROPN cana-5930 20 30 the the DET cana-5930 20 31 minimum minimum ADJ cana-5930 20 32 cardinality cardinality NOUN cana-5930 20 33 of of ADP cana-5930 20 34 a a DET cana-5930 20 35 dominating dominating NOUN cana-5930 20 36 set set NOUN cana-5930 20 37 is be AUX cana-5930 20 38 called call VERB cana-5930 20 39 the the DET cana-5930 20 40 domination domination NOUN cana-5930 20 41 number number NOUN cana-5930 20 42 of of ADP cana-5930 20 43 g g NOUN cana-5930 20 44 and and CCONJ cana-5930 20 45 is be AUX cana-5930 20 46 denoted denote VERB cana-5930 20 47 by by ADP cana-5930 20 48 Ξ³(𝐺 Ξ³(𝐺 NOUN cana-5930 20 49 ) ) PUNCT cana-5930 20 50 . . PUNCT cana-5930 21 1 the the DET cana-5930 21 2 complementary complementary ADJ cana-5930 21 3 tree tree NOUN cana-5930 21 4 domination domination NOUN cana-5930 21 5 number number NOUN cana-5930 21 6 of of ADP cana-5930 21 7 a a DET cana-5930 21 8 graph graph NOUN cana-5930 21 9 was be AUX cana-5930 21 10 introduced introduce VERB cana-5930 21 11 bys bys PROPN cana-5930 21 12 . . PUNCT cana-5930 22 1 muthammai muthammai PROPN cana-5930 22 2 , , PUNCT cana-5930 22 3 m. m. NOUN cana-5930 22 4 bhanumathi bhanumathi PROPN cana-5930 22 5 and and CCONJ cana-5930 22 6 p. p. NOUN cana-5930 22 7 vidhya vidhya PROPN cana-5930 23 1 [ [ X cana-5930 23 2 4 4 X cana-5930 23 3 ] ] PUNCT cana-5930 23 4 have have AUX cana-5930 23 5 established establish VERB cana-5930 23 6 some some DET cana-5930 23 7 results result NOUN cana-5930 23 8 on on ADP cana-5930 23 9 complementary complementary ADJ cana-5930 23 10 tree tree NOUN cana-5930 23 11 domination domination NOUN cana-5930 23 12 number number NOUN cana-5930 23 13 of of ADP cana-5930 23 14 graphs graph NOUN cana-5930 23 15 . . PUNCT cana-5930 24 1 a a DET cana-5930 24 2 set set VERB cana-5930 24 3 𝐷 𝐷 NOUN cana-5930 24 4 βŠ† βŠ† NUM cana-5930 24 5 𝑉(𝐺 𝑉(𝐺 NOUN cana-5930 24 6 ) ) PUNCT cana-5930 24 7 is be AUX cana-5930 24 8 said say VERB cana-5930 24 9 to to PART cana-5930 24 10 be be AUX cana-5930 24 11 complementary complementary ADJ cana-5930 24 12 tree tree NOUN cana-5930 24 13 dominating dominating NOUN cana-5930 24 14 set set NOUN cana-5930 24 15 ( ( PUNCT cana-5930 24 16 ctd ctd NOUN cana-5930 24 17 - - PUNCT cana-5930 24 18 set set NOUN cana-5930 24 19 ) ) PUNCT cana-5930 24 20 if if SCONJ cana-5930 24 21 the the DET cana-5930 24 22 communications communication NOUN cana-5930 24 23 on on ADP cana-5930 24 24 applied apply VERB cana-5930 24 25 nonlinear nonlinear ADJ cana-5930 24 26 analysis analysis NOUN cana-5930 24 27 issn issn NOUN cana-5930 24 28 : : PUNCT cana-5930 24 29 1074 1074 NUM cana-5930 24 30 - - PUNCT cana-5930 24 31 133x 133x NUM cana-5930 24 32 vol vol VERB cana-5930 24 33 32 32 NUM cana-5930 24 34 no no NOUN cana-5930 24 35 . . PUNCT cana-5930 25 1 10s 10 NOUN cana-5930 25 2 ( ( PUNCT cana-5930 25 3 2025 2025 NUM cana-5930 25 4 ) ) PUNCT cana-5930 25 5 3104 3104 NUM cana-5930 26 1 https://internationalpubls.com https://internationalpubls.com X cana-5930 26 2 induced induce VERB cana-5930 26 3 subgraph subgraph NOUN cana-5930 26 4 βŸ¨π‘‰(𝐺 βŸ¨π‘‰(𝐺 NOUN cana-5930 26 5 ) ) PUNCT cana-5930 26 6 βˆ’ βˆ’ PROPN cana-5930 27 1 𝐷⟩ 𝐷⟩ PROPN cana-5930 27 2 is be AUX cana-5930 27 3 a a DET cana-5930 27 4 tree tree NOUN cana-5930 27 5 . . PUNCT cana-5930 28 1 the the DET cana-5930 28 2 minimum minimum ADJ cana-5930 28 3 cardinality cardinality NOUN cana-5930 28 4 of of ADP cana-5930 28 5 a a DET cana-5930 28 6 ctd ctd NOUN cana-5930 28 7 - - PUNCT cana-5930 28 8 set set NOUN cana-5930 28 9 is be AUX cana-5930 28 10 called call VERB cana-5930 28 11 the the DET cana-5930 28 12 complementary complementary ADJ cana-5930 28 13 tree tree NOUN cana-5930 28 14 domination domination NOUN cana-5930 28 15 number number NOUN cana-5930 28 16 of of ADP cana-5930 28 17 g g NOUN cana-5930 28 18 and and CCONJ cana-5930 28 19 is be AUX cana-5930 28 20 denoted denote VERB cana-5930 28 21 by by ADP cana-5930 28 22 γ𝑐𝑑𝑑(𝐺 γ𝑐𝑑𝑑(𝐺 PROPN cana-5930 28 23 ) ) PUNCT cana-5930 28 24 . . PUNCT cana-5930 29 1 e. e. PROPN cana-5930 29 2 sampath sampath PROPN cana-5930 29 3 kumar kumar PROPN cana-5930 29 4 and and CCONJ cana-5930 29 5 s.b s.b PROPN cana-5930 29 6 . . PUNCT cana-5930 30 1 chikkodimath[3 chikkodimath[3 PROPN cana-5930 30 2 ] ] PUNCT cana-5930 30 3 introduced introduce VERB cana-5930 30 4 the the DET cana-5930 30 5 concept concept NOUN cana-5930 30 6 of of ADP cana-5930 30 7 semi semi ADJ cana-5930 30 8 total total ADJ cana-5930 30 9 point point NOUN cana-5930 30 10 graphs graph NOUN cana-5930 30 11 of of ADP cana-5930 30 12 a a DET cana-5930 30 13 graph graph NOUN cana-5930 30 14 . . PUNCT cana-5930 31 1 also also ADV cana-5930 31 2 b. b. PROPN cana-5930 31 3 basavanagoud basavanagoud PROPN cana-5930 31 4 , , PUNCT cana-5930 31 5 s.m s.m PROPN cana-5930 31 6 . . PROPN cana-5930 31 7 hosamani hosamani PROPN cana-5930 31 8 and and CCONJ cana-5930 31 9 s.h s.h PROPN cana-5930 31 10 . . PROPN cana-5930 31 11 malghan[1 malghan[1 PROPN cana-5930 31 12 ] ] PUNCT cana-5930 31 13 have have AUX cana-5930 31 14 obtained obtain VERB cana-5930 31 15 some some DET cana-5930 31 16 results result NOUN cana-5930 31 17 on on ADP cana-5930 31 18 domination domination NOUN cana-5930 31 19 number number NOUN cana-5930 31 20 of of ADP cana-5930 31 21 semi semi ADJ cana-5930 31 22 total total ADJ cana-5930 31 23 point point NOUN cana-5930 31 24 graph graph NOUN cana-5930 31 25 . . PUNCT cana-5930 32 1 given give VERB cana-5930 32 2 a a DET cana-5930 32 3 graph graph NOUN cana-5930 32 4 g g NOUN cana-5930 32 5 , , PUNCT cana-5930 32 6 the the DET cana-5930 32 7 semi semi ADJ cana-5930 32 8 total total ADJ cana-5930 32 9 point point NOUN cana-5930 32 10 graph graph NOUN cana-5930 32 11 𝑇2(𝐺 𝑇2(𝐺 PROPN cana-5930 32 12 ) ) PUNCT cana-5930 32 13 of of ADP cana-5930 32 14 g g PROPN cana-5930 32 15 is be AUX cana-5930 32 16 the the DET cana-5930 32 17 graph graph NOUN cana-5930 32 18 whose whose DET cana-5930 32 19 point point NOUN cana-5930 32 20 set set NOUN cana-5930 32 21 is be AUX cana-5930 32 22 𝑉(𝐺 𝑉(𝐺 NOUN cana-5930 32 23 ) ) PUNCT cana-5930 32 24 βˆͺ βˆͺ ADP cana-5930 32 25 𝐸(𝐺 𝐸(𝐺 PROPN cana-5930 32 26 ) ) PUNCT cana-5930 32 27 where where SCONJ cana-5930 32 28 two two NUM cana-5930 32 29 points point NOUN cana-5930 32 30 are be AUX cana-5930 32 31 adjacent adjacent ADJ cana-5930 32 32 if if SCONJ cana-5930 32 33 and and CCONJ cana-5930 32 34 only only ADV cana-5930 32 35 if if SCONJ cana-5930 32 36 ( ( PUNCT cana-5930 32 37 i i NOUN cana-5930 32 38 ) ) PUNCT cana-5930 32 39 they they PRON cana-5930 32 40 are be AUX cana-5930 32 41 adjacent adjacent ADJ cana-5930 32 42 points point NOUN cana-5930 32 43 of of ADP cana-5930 32 44 g g PROPN cana-5930 32 45 or or CCONJ cana-5930 32 46 ( ( PUNCT cana-5930 32 47 ii ii NOUN cana-5930 32 48 ) ) PUNCT cana-5930 32 49 one one NOUN cana-5930 32 50 is be AUX cana-5930 32 51 a a DET cana-5930 32 52 point point NOUN cana-5930 32 53 of of ADP cana-5930 32 54 g g PROPN cana-5930 32 55 and and CCONJ cana-5930 32 56 the the DET cana-5930 32 57 other other ADJ cana-5930 32 58 is be AUX cana-5930 32 59 a a DET cana-5930 32 60 line line NOUN cana-5930 32 61 of of ADP cana-5930 32 62 g g NOUN cana-5930 32 63 , , PUNCT cana-5930 32 64 incident incident NOUN cana-5930 32 65 with with ADP cana-5930 32 66 it it PRON cana-5930 32 67 . . PUNCT cana-5930 33 1 for for ADP cana-5930 33 2 notation notation NOUN cana-5930 33 3 convenience convenience NOUN cana-5930 33 4 , , PUNCT cana-5930 33 5 an an DET cana-5930 33 6 edge edge NOUN cana-5930 33 7 ( ( PUNCT cana-5930 33 8 𝑒1 𝑒1 NOUN cana-5930 33 9 , , PUNCT cana-5930 33 10 𝑒2 𝑒2 PROPN cana-5930 33 11 ) ) PUNCT cana-5930 33 12 ∈ ∈ PROPN cana-5930 33 13 𝐸(𝐺 𝐸(𝐺 PROPN cana-5930 33 14 ) ) PUNCT cana-5930 33 15 then then ADV cana-5930 33 16 its its PRON cana-5930 33 17 corresponding corresponding ADJ cana-5930 33 18 edge edge NOUN cana-5930 33 19 vertex vertex NOUN cana-5930 33 20 is be AUX cana-5930 33 21 denoted denote VERB cana-5930 33 22 by by ADP cana-5930 33 23 𝑒12 𝑒12 NUM cana-5930 33 24 β€² β€² NUM cana-5930 33 25 in in ADP cana-5930 33 26 𝑇2(𝐺 𝑇2(𝐺 PROPN cana-5930 33 27 ) ) PUNCT cana-5930 33 28 . . PUNCT cana-5930 34 1 in in ADP cana-5930 34 2 this this DET cana-5930 34 3 paper paper NOUN cana-5930 34 4 complementary complementary ADJ cana-5930 34 5 tree tree NOUN cana-5930 34 6 domination domination NOUN cana-5930 34 7 number number NOUN cana-5930 34 8 of of ADP cana-5930 34 9 semi semi ADJ cana-5930 34 10 total total ADJ cana-5930 34 11 point point NOUN cana-5930 34 12 graph graph NOUN cana-5930 34 13 of of ADP cana-5930 34 14 graphs graph NOUN cana-5930 34 15 , , PUNCT cana-5930 34 16 its its PRON cana-5930 34 17 bounds bound NOUN cana-5930 34 18 and and CCONJ cana-5930 34 19 relation relation NOUN cana-5930 34 20 between between ADP cana-5930 34 21 γ𝑐𝑑𝑑(𝐺 γ𝑐𝑑𝑑(𝐺 PROPN cana-5930 34 22 ) ) PUNCT cana-5930 34 23 and and CCONJ cana-5930 34 24 γ𝑐𝑑𝑑(𝑇2(𝐺 γ𝑐𝑑𝑑(𝑇2(𝐺 NOUN cana-5930 34 25 ) ) PUNCT cana-5930 34 26 ) ) PUNCT cana-5930 34 27 are be AUX cana-5930 34 28 obtained obtain VERB cana-5930 34 29 . . PUNCT cana-5930 35 1 2 2 X cana-5930 35 2 . . NOUN cana-5930 35 3 prior prior ADJ cana-5930 35 4 results result NOUN cana-5930 35 5 observation observation VERB cana-5930 35 6 2.1 2.1 NUM cana-5930 35 7 . . PUNCT cana-5930 36 1 [ [ X cana-5930 36 2 6 6 NUM cana-5930 36 3 ] ] PUNCT cana-5930 36 4 ( ( PUNCT cana-5930 36 5 i i NOUN cana-5930 36 6 ) ) PUNCT cana-5930 36 7 for for ADP cana-5930 36 8 the the DET cana-5930 36 9 path path NOUN cana-5930 37 1 𝑃𝑝 𝑃𝑝 PROPN cana-5930 37 2 , , PUNCT cana-5930 37 3 Ξ³ Ξ³ PROPN cana-5930 37 4 𝑐𝑑𝑑 𝑐𝑑𝑑 NOUN cana-5930 37 5 ( ( PUNCT cana-5930 37 6 𝑇2(𝑃𝑝 𝑇2(𝑃𝑝 NOUN cana-5930 37 7 ) ) PUNCT cana-5930 37 8 ) ) PUNCT cana-5930 37 9 = = PUNCT cana-5930 38 1 𝑝 𝑝 ADJ cana-5930 38 2 βˆ’ βˆ’ NOUN cana-5930 38 3 1 1 NUM cana-5930 38 4 , , PUNCT cana-5930 38 5 where where SCONJ cana-5930 38 6 ( ( PUNCT cana-5930 38 7 𝑝 𝑝 NOUN cana-5930 38 8 β‰₯ β‰₯ NOUN cana-5930 38 9 2 2 NUM cana-5930 38 10 ) ) PUNCT cana-5930 38 11 . . PUNCT cana-5930 39 1 ( ( PUNCT cana-5930 39 2 ii ii NOUN cana-5930 39 3 ) ) PUNCT cana-5930 39 4 for for ADP cana-5930 39 5 the the DET cana-5930 39 6 cycle cycle NOUN cana-5930 39 7 𝐢𝑝 𝐢𝑝 PROPN cana-5930 39 8 , , PUNCT cana-5930 39 9 Ξ³ Ξ³ X cana-5930 39 10 𝑐𝑑𝑑 𝑐𝑑𝑑 NOUN cana-5930 39 11 ( ( PUNCT cana-5930 39 12 𝑇2(𝐢𝑝 𝑇2(𝐢𝑝 PROPN cana-5930 39 13 ) ) PUNCT cana-5930 39 14 ) ) PUNCT cana-5930 40 1 = = PUNCT cana-5930 40 2 𝑝 𝑝 ADJ cana-5930 40 3 βˆ’ βˆ’ NOUN cana-5930 40 4 1 1 NUM cana-5930 40 5 , , PUNCT cana-5930 40 6 where where SCONJ cana-5930 40 7 ( ( PUNCT cana-5930 40 8 𝑝 𝑝 NOUN cana-5930 40 9 β‰₯ β‰₯ NOUN cana-5930 40 10 3 3 NUM cana-5930 40 11 ) ) PUNCT cana-5930 40 12 . . PUNCT cana-5930 41 1 ( ( PUNCT cana-5930 41 2 iii iii X cana-5930 41 3 ) ) PUNCT cana-5930 41 4 for for ADP cana-5930 41 5 the the DET cana-5930 41 6 star star PROPN cana-5930 41 7 graph graph PROPN cana-5930 41 8 𝐾1,π‘βˆ’1 𝐾1,π‘βˆ’1 PROPN cana-5930 41 9 , , PUNCT cana-5930 41 10 Ξ³ Ξ³ PROPN cana-5930 41 11 𝑐𝑑𝑑 𝑐𝑑𝑑 NOUN cana-5930 41 12 ( ( PUNCT cana-5930 41 13 𝑇2(𝐾1,π‘βˆ’1 𝑇2(𝐾1,π‘βˆ’1 PROPN cana-5930 41 14 ) ) PUNCT cana-5930 41 15 ) ) PUNCT cana-5930 42 1 = = PUNCT cana-5930 42 2 𝑝 𝑝 ADP cana-5930 42 3 βˆ’ βˆ’ NOUN cana-5930 42 4 1 1 NUM cana-5930 42 5 or or CCONJ cana-5930 42 6 π‘ž π‘ž PROPN cana-5930 42 7 , , PUNCT cana-5930 42 8 𝑝 𝑝 PRON cana-5930 42 9 β‰₯ β‰₯ NOUN cana-5930 42 10 2 2 NUM cana-5930 42 11 . . PUNCT cana-5930 42 12 ( ( PUNCT cana-5930 42 13 iv iv X cana-5930 42 14 ) ) PUNCT cana-5930 42 15 for for ADP cana-5930 42 16 a a DET cana-5930 42 17 complete complete ADJ cana-5930 42 18 graph graph NOUN cana-5930 42 19 𝐾𝑝 𝐾𝑝 PROPN cana-5930 42 20 , , PUNCT cana-5930 42 21 𝑝 𝑝 NOUN cana-5930 42 22 β‰₯ β‰₯ NOUN cana-5930 42 23 4 4 NUM cana-5930 42 24 then then ADV cana-5930 42 25 Ξ³ Ξ³ PROPN cana-5930 42 26 𝑐𝑑𝑑 𝑐𝑑𝑑 NOUN cana-5930 42 27 ( ( PUNCT cana-5930 42 28 𝑇2(𝐾𝑝 𝑇2(𝐾𝑝 PROPN cana-5930 42 29 ) ) PUNCT cana-5930 42 30 ) ) PUNCT cana-5930 43 1 = = PRON cana-5930 43 2 𝑝2βˆ’5𝑝+12 𝑝2βˆ’5𝑝+12 X cana-5930 43 3 2 2 X cana-5930 43 4 . . PUNCT cana-5930 44 1 ( ( PUNCT cana-5930 44 2 v v NOUN cana-5930 44 3 ) ) PUNCT cana-5930 44 4 for for ADP cana-5930 44 5 a a DET cana-5930 44 6 wheel wheel NOUN cana-5930 44 7 graph graph NOUN cana-5930 44 8 π‘Šπ‘ π‘Šπ‘ PROPN cana-5930 44 9 , , PUNCT cana-5930 44 10 𝑝 𝑝 PROPN cana-5930 44 11 β‰₯ β‰₯ NOUN cana-5930 44 12 4Ξ³ 4Ξ³ PROPN cana-5930 44 13 𝑐𝑑𝑑 𝑐𝑑𝑑 NOUN cana-5930 44 14 ( ( PUNCT cana-5930 44 15 𝑇2(π‘Šπ‘ 𝑇2(π‘Šπ‘ NOUN cana-5930 44 16 ) ) PUNCT cana-5930 44 17 ) ) PUNCT cana-5930 44 18 = = PUNCT cana-5930 44 19 𝑝 𝑝 NOUN cana-5930 44 20 where where SCONJ cana-5930 44 21 π‘Šπ‘ π‘Šπ‘ PROPN cana-5930 44 22 = = PUNCT cana-5930 44 23 πΆπ‘βˆ’1 πΆπ‘βˆ’1 PROPN cana-5930 44 24 + + NUM cana-5930 44 25 𝐾1 𝐾1 NOUN cana-5930 44 26 for for ADP cana-5930 44 27 ( ( PUNCT cana-5930 44 28 𝑝 𝑝 NOUN cana-5930 44 29 β‰₯ β‰₯ NOUN cana-5930 44 30 4 4 NUM cana-5930 44 31 ) ) PUNCT cana-5930 44 32 . . PUNCT cana-5930 45 1 ( ( PUNCT cana-5930 45 2 vi vi X cana-5930 45 3 ) ) PUNCT cana-5930 45 4 for for ADP cana-5930 45 5 a a DET cana-5930 45 6 corona corona NOUN cana-5930 45 7 graph graph NOUN cana-5930 45 8 𝑃𝑝 𝑃𝑝 PROPN cana-5930 45 9 ∘ ∘ PROPN cana-5930 45 10 𝐾1 𝐾1 PROPN cana-5930 45 11 , , PUNCT cana-5930 45 12 𝑝 𝑝 PRON cana-5930 45 13 β‰₯ β‰₯ NOUN cana-5930 45 14 2 2 NUM cana-5930 45 15 then then ADV cana-5930 45 16 Ξ³ Ξ³ PROPN cana-5930 45 17 𝑐𝑑𝑑 𝑐𝑑𝑑 NOUN cana-5930 45 18 ( ( PUNCT cana-5930 45 19 𝑇2(𝑃𝑝 𝑇2(𝑃𝑝 PROPN cana-5930 45 20 ∘ ∘ NOUN cana-5930 45 21 𝐾1 𝐾1 PROPN cana-5930 45 22 ) ) PUNCT cana-5930 45 23 ) ) PUNCT cana-5930 46 1 = = SYM cana-5930 46 2 2𝑝 2𝑝 NOUN cana-5930 46 3 βˆ’ βˆ’ NOUN cana-5930 46 4 1 1 X cana-5930 46 5 . . PUNCT cana-5930 46 6 ( ( PUNCT cana-5930 46 7 vii vii PROPN cana-5930 46 8 ) ) PUNCT cana-5930 46 9 for for ADP cana-5930 46 10 a a DET cana-5930 46 11 corona corona NOUN cana-5930 46 12 graph graph NOUN cana-5930 46 13 𝐢𝑝 𝐢𝑝 PROPN cana-5930 46 14 ∘ ∘ NUM cana-5930 46 15 𝐾1 𝐾1 PROPN cana-5930 46 16 , , PUNCT cana-5930 46 17 𝑝 𝑝 PRON cana-5930 46 18 β‰₯ β‰₯ NOUN cana-5930 46 19 3 3 NUM cana-5930 46 20 then then ADV cana-5930 46 21 Ξ³ Ξ³ PROPN cana-5930 46 22 𝑐𝑑𝑑 𝑐𝑑𝑑 NOUN cana-5930 46 23 ( ( PUNCT cana-5930 46 24 𝑇2(𝐢𝑝 𝑇2(𝐢𝑝 PROPN cana-5930 46 25 ∘ ∘ PROPN cana-5930 46 26 𝐾1 𝐾1 PROPN cana-5930 46 27 ) ) PUNCT cana-5930 46 28 ) ) PUNCT cana-5930 47 1 = = SYM cana-5930 47 2 2𝑝. 2𝑝. NUM cana-5930 47 3 ( ( PUNCT cana-5930 47 4 viii viii NOUN cana-5930 47 5 ) ) PUNCT cana-5930 47 6 Ξ³ Ξ³ PROPN cana-5930 47 7 𝑐𝑑𝑑 𝑐𝑑𝑑 NOUN cana-5930 47 8 ( ( PUNCT cana-5930 47 9 𝑇2(𝐾1 𝑇2(𝐾1 PROPN cana-5930 47 10 + + NOUN cana-5930 47 11 𝑃𝑛 𝑃𝑛 NOUN cana-5930 47 12 ) ) PUNCT cana-5930 47 13 ) ) PUNCT cana-5930 48 1 = = SYM cana-5930 48 2 𝑝 𝑝 PROPN cana-5930 48 3 , , PUNCT cana-5930 48 4 𝑝 𝑝 NOUN cana-5930 48 5 β‰₯ β‰₯ NOUN cana-5930 48 6 3 3 NUM cana-5930 48 7 . . PUNCT cana-5930 48 8 proposition proposition NOUN cana-5930 48 9 2.2 2.2 NUM cana-5930 48 10 . . PUNCT cana-5930 49 1 [ [ X cana-5930 49 2 6 6 NUM cana-5930 49 3 ] ] PUNCT cana-5930 49 4 let let AUX cana-5930 49 5 𝐺(𝑝 𝐺(𝑝 NUM cana-5930 49 6 , , PUNCT cana-5930 49 7 π‘ž π‘ž NOUN cana-5930 49 8 ) ) PUNCT cana-5930 49 9 be be VERB cana-5930 49 10 a a DET cana-5930 49 11 connected connected ADJ cana-5930 49 12 graph graph NOUN cana-5930 49 13 with with ADP cana-5930 49 14 Ξ΄(𝐺 Ξ΄(𝐺 PROPN cana-5930 49 15 ) ) PUNCT cana-5930 49 16 β‰₯ β‰₯ NOUN cana-5930 49 17 2 2 NUM cana-5930 49 18 then then ADV cana-5930 49 19 atleast atleast VERB cana-5930 49 20 one one NUM cana-5930 49 21 vertex vertex NOUN cana-5930 49 22 of of ADP cana-5930 49 23 g g PROPN cana-5930 49 24 is be AUX cana-5930 49 25 the the DET cana-5930 49 26 member member NOUN cana-5930 49 27 of of ADP cana-5930 49 28 ctd ctd NOUN cana-5930 49 29 - - PUNCT cana-5930 49 30 set set NOUN cana-5930 49 31 of of ADP cana-5930 49 32 𝑇2(𝐺 𝑇2(𝐺 PROPN cana-5930 49 33 ) ) PUNCT cana-5930 49 34 . . PUNCT cana-5930 50 1 proposition proposition NOUN cana-5930 50 2 2.3 2.3 NUM cana-5930 50 3 . . PUNCT cana-5930 51 1 [ [ X cana-5930 51 2 6 6 NUM cana-5930 51 3 ] ] PUNCT cana-5930 51 4 for for ADP cana-5930 51 5 any any DET cana-5930 51 6 connected connected ADJ cana-5930 51 7 graph graph NOUN cana-5930 51 8 𝐺(𝑝 𝐺(𝑝 NUM cana-5930 51 9 , , PUNCT cana-5930 51 10 π‘ž π‘ž NOUN cana-5930 51 11 ) ) PUNCT cana-5930 51 12 with with ADP cana-5930 51 13 𝑝 𝑝 PROPN cana-5930 51 14 β‰₯ β‰₯ NUM cana-5930 51 15 2,⌈ 2,⌈ PROPN cana-5930 51 16 𝑝 𝑝 NOUN cana-5930 51 17 Ξ΄(𝐺)+1 Ξ΄(𝐺)+1 NOUN cana-5930 51 18 βŒ‰ βŒ‰ SCONJ cana-5930 51 19 ≀ ≀ NUM cana-5930 51 20 γ𝑐𝑑𝑑(𝑇2(𝐺 γ𝑐𝑑𝑑(𝑇2(𝐺 NOUN cana-5930 51 21 ) ) PUNCT cana-5930 51 22 ) ) PUNCT cana-5930 52 1 ≀ ≀ PROPN cana-5930 52 2 𝑝 𝑝 ADP cana-5930 52 3 + + NUM cana-5930 52 4 π‘ž π‘ž X cana-5930 52 5 βˆ’ βˆ’ PROPN cana-5930 52 6 2 2 NUM cana-5930 52 7 . . PUNCT cana-5930 52 8 theorem theorem VERB cana-5930 52 9 2.4 2.4 NUM cana-5930 52 10 . . PUNCT cana-5930 53 1 [ [ X cana-5930 53 2 6 6 NUM cana-5930 53 3 ] ] PUNCT cana-5930 53 4 γ𝑐𝑑𝑑(𝑇2(𝐺 γ𝑐𝑑𝑑(𝑇2(𝐺 NOUN cana-5930 53 5 ) ) PUNCT cana-5930 53 6 ) ) PUNCT cana-5930 54 1 = = PUNCT cana-5930 54 2 1 1 NUM cana-5930 54 3 if if SCONJ cana-5930 54 4 and and CCONJ cana-5930 54 5 only only ADV cana-5930 54 6 if if SCONJ cana-5930 54 7 𝐺 𝐺 PROPN cana-5930 54 8 β‰… β‰… PROPN cana-5930 54 9 𝐾2 𝐾2 PROPN cana-5930 54 10 . . PUNCT cana-5930 55 1 communications communication NOUN cana-5930 55 2 on on ADP cana-5930 55 3 applied apply VERB cana-5930 55 4 nonlinear nonlinear ADJ cana-5930 55 5 analysis analysis NOUN cana-5930 55 6 issn issn NOUN cana-5930 55 7 : : PUNCT cana-5930 55 8 1074 1074 NUM cana-5930 55 9 - - PUNCT cana-5930 55 10 133x 133x NUM cana-5930 55 11 vol vol VERB cana-5930 55 12 32 32 NUM cana-5930 55 13 no no NOUN cana-5930 55 14 . . PUNCT cana-5930 56 1 10s 10 NOUN cana-5930 56 2 ( ( PUNCT cana-5930 56 3 2025 2025 NUM cana-5930 56 4 ) ) PUNCT cana-5930 56 5 3105 3105 NUM cana-5930 56 6 https://internationalpubls.com https://internationalpubls.com X cana-5930 56 7 3 3 X cana-5930 56 8 . . PUNCT cana-5930 56 9 characterisation characterisation NOUN cana-5930 56 10 of of ADP cana-5930 56 11 complementary complementary ADJ cana-5930 56 12 tree tree NOUN cana-5930 56 13 dominating dominating NOUN cana-5930 56 14 set set VERB cana-5930 56 15 in in ADP cana-5930 56 16 semi semi ADJ cana-5930 56 17 total total ADJ cana-5930 56 18 point point NOUN cana-5930 56 19 graph graph NOUN cana-5930 56 20 t2(g t2(g PRON cana-5930 56 21 ) ) PUNCT cana-5930 56 22 in in ADP cana-5930 56 23 the the DET cana-5930 56 24 following following NOUN cana-5930 56 25 , , PUNCT cana-5930 56 26 a a DET cana-5930 56 27 necessary necessary ADJ cana-5930 56 28 and and CCONJ cana-5930 56 29 sufficient sufficient ADJ cana-5930 56 30 condition condition NOUN cana-5930 56 31 for for ADP cana-5930 56 32 a a DET cana-5930 56 33 ctd ctd NOUN cana-5930 56 34 - - PUNCT cana-5930 56 35 set set NOUN cana-5930 56 36 of of ADP cana-5930 56 37 a a DET cana-5930 56 38 graph graph NOUN cana-5930 56 39 g g NOUN cana-5930 56 40 to to PART cana-5930 56 41 be be AUX cana-5930 56 42 a a DET cana-5930 56 43 ctd ctd NOUN cana-5930 56 44 - - PUNCT cana-5930 56 45 set set NOUN cana-5930 56 46 of of ADP cana-5930 56 47 its its PRON cana-5930 56 48 semi semi ADJ cana-5930 56 49 total total ADJ cana-5930 56 50 point point NOUN cana-5930 56 51 graph graph NOUN cana-5930 56 52 𝑇2(𝐺 𝑇2(𝐺 PROPN cana-5930 56 53 ) ) PUNCT cana-5930 56 54 is be AUX cana-5930 56 55 found find VERB cana-5930 56 56 . . PUNCT cana-5930 57 1 theorem theorem VERB cana-5930 57 2 3.1 3.1 NUM cana-5930 57 3 . . PUNCT cana-5930 58 1 a a DET cana-5930 58 2 ctd ctd NOUN cana-5930 58 3 - - PUNCT cana-5930 58 4 set set VERB cana-5930 58 5 d d NOUN cana-5930 58 6 of of ADP cana-5930 58 7 a a DET cana-5930 58 8 connected connected ADJ cana-5930 58 9 graph graph NOUN cana-5930 58 10 𝐺 𝐺 NOUN cana-5930 58 11 = = SYM cana-5930 58 12 ( ( PUNCT cana-5930 58 13 𝑉 𝑉 PROPN cana-5930 58 14 , , PUNCT cana-5930 58 15 𝐸 𝐸 PROPN cana-5930 58 16 ) ) PUNCT cana-5930 58 17 is be AUX cana-5930 58 18 also also ADV cana-5930 58 19 a a DET cana-5930 58 20 ctd ctd NOUN cana-5930 58 21 - - PUNCT cana-5930 58 22 set set NOUN cana-5930 58 23 of of ADP cana-5930 58 24 𝑇2(𝐺 𝑇2(𝐺 PROPN cana-5930 58 25 ) ) PUNCT cana-5930 58 26 if if SCONJ cana-5930 58 27 and and CCONJ cana-5930 58 28 only only ADV cana-5930 58 29 if if SCONJ cana-5930 58 30 ( ( PUNCT cana-5930 58 31 i i NOUN cana-5930 58 32 ) ) PUNCT cana-5930 58 33 ⟨𝐷 ⟨𝐷 ADJ cana-5930 58 34 ⟩ ⟩ NOUN cana-5930 58 35 has have VERB cana-5930 58 36 an an DET cana-5930 58 37 isolated isolated ADJ cana-5930 58 38 vertices vertex NOUN cana-5930 58 39 . . PUNCT cana-5930 59 1 ( ( PUNCT cana-5930 59 2 ii ii NOUN cana-5930 59 3 ) ) PUNCT cana-5930 59 4 for for ADP cana-5930 59 5 each each DET cana-5930 59 6 𝑣 𝑣 PROPN cana-5930 59 7 ∈ ∈ PROPN cana-5930 59 8 𝐷 𝐷 PROPN cana-5930 59 9 , , PUNCT cana-5930 59 10 𝑁(𝑣 𝑁(𝑣 NUM cana-5930 59 11 ) ) PUNCT cana-5930 59 12 ∩ ∩ NOUN cana-5930 59 13 ( ( PUNCT cana-5930 59 14 𝑉 𝑉 PROPN cana-5930 59 15 βˆ’ βˆ’ PROPN cana-5930 59 16 𝐷 𝐷 PROPN cana-5930 59 17 ) ) PUNCT cana-5930 59 18 β‰ οͺ β‰ οͺ PROPN cana-5930 59 19 and and CCONJ cana-5930 59 20 ( ( PUNCT cana-5930 59 21 iii iii NOUN cana-5930 59 22 ) ) PUNCT cana-5930 59 23 βŸ¨π‘‰ βŸ¨π‘‰ NOUN cana-5930 59 24 βˆ’ βˆ’ PROPN cana-5930 59 25 𝐷⟩ 𝐷⟩ PROPN cana-5930 59 26 β‰… β‰… X cana-5930 59 27 𝐾1 𝐾1 PROPN cana-5930 59 28 and and CCONJ cana-5930 59 29 if if SCONJ cana-5930 59 30 𝑣 𝑣 PRON cana-5930 59 31 ∈ ∈ PROPN cana-5930 59 32 𝑉 𝑉 PROPN cana-5930 59 33 βˆ’ βˆ’ PROPN cana-5930 59 34 𝐷 𝐷 PROPN cana-5930 59 35 then then ADV cana-5930 59 36 𝑁(𝑣 𝑁(𝑣 NUM cana-5930 59 37 ) ) PUNCT cana-5930 59 38 ∩ ∩ ADJ cana-5930 59 39 𝐷 𝐷 NOUN cana-5930 59 40 β‰  β‰  PROPN cana-5930 59 41 οͺ οͺ PROPN cana-5930 59 42 . . PUNCT cana-5930 60 1 proof proof NOUN cana-5930 60 2 . . PUNCT cana-5930 61 1 let let VERB cana-5930 61 2 d d PRON cana-5930 61 3 be be AUX cana-5930 61 4 a a DET cana-5930 61 5 ctd ctd NOUN cana-5930 61 6 - - PUNCT cana-5930 61 7 set set NOUN cana-5930 61 8 of of ADP cana-5930 61 9 both both DET cana-5930 61 10 g g PROPN cana-5930 61 11 and and CCONJ cana-5930 61 12 𝑇2(𝐺 𝑇2(𝐺 PROPN cana-5930 61 13 ) ) PUNCT cana-5930 61 14 . . PUNCT cana-5930 62 1 ( ( PUNCT cana-5930 62 2 i i NOUN cana-5930 62 3 ) ) PUNCT cana-5930 62 4 let let VERB cana-5930 62 5 𝑣 𝑣 PRON cana-5930 62 6 ∈ ∈ PROPN cana-5930 62 7 𝐷 𝐷 PROPN cana-5930 62 8 be be VERB cana-5930 62 9 not not PART cana-5930 62 10 an an DET cana-5930 62 11 isolated isolated ADJ cana-5930 62 12 vertex vertex NOUN cana-5930 62 13 in in ADP cana-5930 62 14 ⟨𝐷 ⟨𝐷 ADJ cana-5930 62 15 ⟩ ⟩ NOUN cana-5930 62 16 then then ADV cana-5930 62 17 its its PRON cana-5930 62 18 edge edge NOUN cana-5930 62 19 vertex vertex NOUN cana-5930 62 20 𝑣′ 𝑣′ NUM cana-5930 62 21 in in ADP cana-5930 62 22 𝑇2(𝐺 𝑇2(𝐺 PROPN cana-5930 62 23 ) ) PUNCT cana-5930 62 24 is be AUX cana-5930 62 25 isolated isolate VERB cana-5930 62 26 inβŸ¨π‘‰(t2(g inβŸ¨π‘‰(t2(g NOUN cana-5930 62 27 ) ) PUNCT cana-5930 62 28 ) ) PUNCT cana-5930 62 29 β€ˆ β€ˆ SPACE cana-5930 62 30 βˆ’ βˆ’ PROPN cana-5930 62 31 β€ˆ β€ˆ SPACE cana-5930 62 32 d d NUM cana-5930 62 33 β€ˆ β€ˆ SPACE cana-5930 62 34 ⟩ ⟩ NOUN cana-5930 62 35 which which PRON cana-5930 62 36 contradicts contradict VERB cana-5930 62 37 the the DET cana-5930 62 38 ctd ctd NOUN cana-5930 62 39 - - PUNCT cana-5930 62 40 set set NOUN cana-5930 62 41 of of ADP cana-5930 62 42 𝑇2(𝐺 𝑇2(𝐺 PROPN cana-5930 62 43 ) ) PUNCT cana-5930 62 44 . . PUNCT cana-5930 63 1 therefore therefore ADV cana-5930 63 2 , , PUNCT cana-5930 63 3 ⟨𝐷 ⟨𝐷 ADJ cana-5930 63 4 ⟩ ⟩ NOUN cana-5930 63 5 has have VERB cana-5930 63 6 an an DET cana-5930 63 7 isolated isolated ADJ cana-5930 63 8 vertices vertex NOUN cana-5930 63 9 . . PUNCT cana-5930 64 1 ( ( PUNCT cana-5930 64 2 ii ii NOUN cana-5930 64 3 ) ) PUNCT cana-5930 64 4 let let VERB cana-5930 64 5 there there PRON cana-5930 64 6 exists exist VERB cana-5930 64 7 a a DET cana-5930 64 8 vertex vertex NOUN cana-5930 64 9 𝑣 𝑣 ADP cana-5930 64 10 ∈ ∈ PROPN cana-5930 64 11 𝐷 𝐷 NOUN cana-5930 64 12 such such ADJ cana-5930 64 13 that that DET cana-5930 64 14 𝑁(𝑣 𝑁(𝑣 X cana-5930 64 15 ) ) PUNCT cana-5930 64 16 ∩ ∩ NOUN cana-5930 64 17 ( ( PUNCT cana-5930 64 18 𝑉 𝑉 PROPN cana-5930 64 19 βˆ’ βˆ’ PROPN cana-5930 64 20 𝐷 𝐷 PROPN cana-5930 64 21 ) ) PUNCT cana-5930 64 22 = = PUNCT cana-5930 65 1 οͺ οͺ PROPN cana-5930 65 2 . . PUNCT cana-5930 66 1 then then ADV cana-5930 66 2 , , PUNCT cana-5930 66 3 its its PRON cana-5930 66 4 edge edge NOUN cana-5930 66 5 vertex vertex NOUN cana-5930 66 6 𝑣′ 𝑣′ NUM cana-5930 66 7 is be AUX cana-5930 66 8 isolated isolate VERB cana-5930 66 9 in in ADP cana-5930 66 10 βŸ¨π‘‰(𝑇2(𝐺 βŸ¨π‘‰(𝑇2(𝐺 PROPN cana-5930 66 11 ) ) PUNCT cana-5930 66 12 ) ) PUNCT cana-5930 67 1 βˆ’ βˆ’ PROPN cana-5930 67 2 𝐷 𝐷 PROPN cana-5930 67 3 ⟩. ⟩. PROPN cana-5930 67 4 ( ( PUNCT cana-5930 67 5 iii iii NOUN cana-5930 67 6 ) ) PUNCT cana-5930 67 7 if if SCONJ cana-5930 67 8 𝐾2 𝐾2 NOUN cana-5930 67 9 is be AUX cana-5930 67 10 an an DET cana-5930 67 11 induced induced ADJ cana-5930 67 12 subgraph subgraph NOUN cana-5930 67 13 of of ADP cana-5930 67 14 βŸ¨π‘‰ βŸ¨π‘‰ PROPN cana-5930 67 15 βˆ’ βˆ’ PROPN cana-5930 67 16 𝐷 𝐷 PROPN cana-5930 67 17 ⟩. ⟩. NOUN cana-5930 67 18 then then ADV cana-5930 67 19 βŸ¨π‘‰(𝑇2(𝐺 βŸ¨π‘‰(𝑇2(𝐺 PROPN cana-5930 67 20 ) ) PUNCT cana-5930 67 21 ) ) PUNCT cana-5930 68 1 βˆ’ βˆ’ PROPN cana-5930 68 2 𝐷 𝐷 PROPN cana-5930 68 3 ⟩ ⟩ NOUN cana-5930 68 4 contains contain VERB cana-5930 68 5 a a DET cana-5930 68 6 cycle cycle NOUN cana-5930 68 7 . . PUNCT cana-5930 69 1 therefore therefore ADV cana-5930 69 2 , , PUNCT cana-5930 69 3 βŸ¨π‘‰ βŸ¨π‘‰ PROPN cana-5930 69 4 βˆ’ βˆ’ PROPN cana-5930 69 5 𝐷⟩ 𝐷⟩ PROPN cana-5930 69 6 β‰… β‰… PROPN cana-5930 69 7 𝐾1 𝐾1 PROPN cana-5930 69 8 . . PUNCT cana-5930 70 1 let let VERB cana-5930 70 2 𝑣 𝑣 PRON cana-5930 70 3 ∈ ∈ PROPN cana-5930 70 4 𝐾1 𝐾1 PROPN cana-5930 70 5 . . PUNCT cana-5930 71 1 since since SCONJ cana-5930 71 2 g g PROPN cana-5930 71 3 is be AUX cana-5930 71 4 connected connect VERB cana-5930 71 5 so so SCONJ cana-5930 71 6 that that SCONJ cana-5930 71 7 remaining remain VERB cana-5930 71 8 vertices vertex NOUN cana-5930 71 9 are be AUX cana-5930 71 10 adjacent adjacent ADJ cana-5930 71 11 to to ADP cana-5930 71 12 v. v. VERB cana-5930 71 13 there there PRON cana-5930 71 14 exists exist VERB cana-5930 71 15 a a DET cana-5930 71 16 edge edge NOUN cana-5930 71 17 vertices vertex NOUN cana-5930 71 18 𝑣𝑖 𝑣𝑖 ADP cana-5930 71 19 β€² β€² NUM cana-5930 71 20 ∈ ∈ NOUN cana-5930 71 21 ( ( PUNCT cana-5930 71 22 𝑣 𝑣 NOUN cana-5930 71 23 , , PUNCT cana-5930 71 24 𝑣𝑖 𝑣𝑖 NOUN cana-5930 71 25 ) ) PUNCT cana-5930 71 26 such such ADJ cana-5930 71 27 that that PRON cana-5930 71 28 βŸ¨π‘£ βŸ¨π‘£ PROPN cana-5930 71 29 , , PUNCT cana-5930 71 30 𝑣𝑖 𝑣𝑖 NOUN cana-5930 71 31 , , PUNCT cana-5930 71 32 𝑣𝑖 𝑣𝑖 ADP cana-5930 71 33 β€²βŸ© β€²βŸ© PROPN cana-5930 72 1 β‰… β‰… PROPN cana-5930 72 2 𝐢3 𝐢3 PROPN cana-5930 72 3 in in ADP cana-5930 72 4 𝑇2(𝐺).hence 𝑇2(𝐺).hence NOUN cana-5930 72 5 𝑁(𝑣 𝑁(𝑣 SYM cana-5930 72 6 ) ) PUNCT cana-5930 72 7 ∩ ∩ ADJ cana-5930 72 8 𝐷 𝐷 NOUN cana-5930 72 9 β‰  β‰  PROPN cana-5930 72 10 οͺ οͺ PROPN cana-5930 72 11 . . PUNCT cana-5930 73 1 conversely conversely ADV cana-5930 73 2 , , PUNCT cana-5930 73 3 if if SCONJ cana-5930 73 4 ( ( PUNCT cana-5930 73 5 i i NOUN cana-5930 73 6 ) ) PUNCT cana-5930 73 7 is be AUX cana-5930 73 8 true true ADJ cana-5930 73 9 , , PUNCT cana-5930 73 10 d d PROPN cana-5930 73 11 is be AUX cana-5930 73 12 dominating dominate VERB cana-5930 73 13 set set NOUN cana-5930 73 14 of of ADP cana-5930 73 15 𝑇2(𝐺 𝑇2(𝐺 PROPN cana-5930 73 16 ) ) PUNCT cana-5930 73 17 . . PUNCT cana-5930 74 1 if if SCONJ cana-5930 74 2 ( ( PUNCT cana-5930 74 3 ii ii NOUN cana-5930 74 4 ) ) PUNCT cana-5930 74 5 holds hold VERB cana-5930 74 6 , , PUNCT cana-5930 74 7 then then ADV cana-5930 74 8 βŸ¨π‘‰(𝑇2(𝐺 βŸ¨π‘‰(𝑇2(𝐺 PROPN cana-5930 74 9 ) ) PUNCT cana-5930 74 10 ) ) PUNCT cana-5930 75 1 βˆ’ βˆ’ PROPN cana-5930 75 2 𝐷 𝐷 NOUN cana-5930 75 3 ⟩ ⟩ NOUN cana-5930 75 4 is be AUX cana-5930 75 5 connected connect VERB cana-5930 75 6 and and CCONJ cana-5930 75 7 if if SCONJ cana-5930 75 8 ( ( PUNCT cana-5930 75 9 iii iii NOUN cana-5930 75 10 ) ) PUNCT cana-5930 75 11 holds hold VERB cana-5930 75 12 , , PUNCT cana-5930 75 13 then then ADV cana-5930 75 14 βŸ¨π‘‰(𝑇2(𝐺 βŸ¨π‘‰(𝑇2(𝐺 PROPN cana-5930 75 15 ) ) PUNCT cana-5930 75 16 ) ) PUNCT cana-5930 76 1 βˆ’ βˆ’ PROPN cana-5930 76 2 𝐷 𝐷 NOUN cana-5930 76 3 ⟩ ⟩ NOUN cana-5930 76 4 is be AUX cana-5930 76 5 acyclic acyclic ADJ cana-5930 76 6 . . PUNCT cana-5930 77 1 therefore therefore ADV cana-5930 77 2 , , PUNCT cana-5930 77 3 βŸ¨π‘‰(𝑇2(𝐺 βŸ¨π‘‰(𝑇2(𝐺 PROPN cana-5930 77 4 ) ) PUNCT cana-5930 77 5 ) ) PUNCT cana-5930 78 1 βˆ’ βˆ’ PROPN cana-5930 78 2 𝐷 𝐷 PROPN cana-5930 78 3 ⟩ ⟩ NOUN cana-5930 78 4 is be AUX cana-5930 78 5 a a DET cana-5930 78 6 tree tree NOUN cana-5930 78 7 . . PUNCT cana-5930 79 1 hence hence ADV cana-5930 79 2 , , PUNCT cana-5930 79 3 d d PRON cana-5930 79 4 is be AUX cana-5930 79 5 also also ADV cana-5930 79 6 a a DET cana-5930 79 7 ctd ctd NOUN cana-5930 79 8 - - PUNCT cana-5930 79 9 set set NOUN cana-5930 79 10 of of ADP cana-5930 79 11 𝑇2(𝐺 𝑇2(𝐺 PROPN cana-5930 79 12 ) ) PUNCT cana-5930 79 13 . . PUNCT cana-5930 80 1 β–‘ β–‘ PUNCT cana-5930 80 2 in in ADP cana-5930 80 3 the the DET cana-5930 80 4 following follow VERB cana-5930 80 5 , , PUNCT cana-5930 80 6 exact exact ADJ cana-5930 80 7 values value NOUN cana-5930 80 8 of of ADP cana-5930 80 9 complementary complementary ADJ cana-5930 80 10 tree tree NOUN cana-5930 80 11 domination domination NOUN cana-5930 80 12 number number NOUN cana-5930 80 13 of of ADP cana-5930 80 14 semi semi ADJ cana-5930 80 15 total total ADJ cana-5930 80 16 point point NOUN cana-5930 80 17 graph graph NOUN cana-5930 80 18 of of ADP cana-5930 80 19 some some DET cana-5930 80 20 classes class NOUN cana-5930 80 21 of of ADP cana-5930 80 22 graphs graph NOUN cana-5930 80 23 are be AUX cana-5930 80 24 given give VERB cana-5930 80 25 . . PUNCT cana-5930 81 1 proposition proposition NOUN cana-5930 81 2 3.2 3.2 NUM cana-5930 81 3 . . PUNCT cana-5930 82 1 let let VERB cana-5930 82 2 𝐢𝑝 𝐢𝑝 PROPN cana-5930 82 3 ( ( PUNCT cana-5930 82 4 𝑑 𝑑 NOUN cana-5930 82 5 ) ) PUNCT cana-5930 82 6 , , PUNCT cana-5930 82 7 𝑑 𝑑 PROPN cana-5930 82 8 β‰₯ β‰₯ NOUN cana-5930 82 9 2 2 NUM cana-5930 82 10 be be AUX cana-5930 82 11 the the DET cana-5930 82 12 one one NUM cana-5930 82 13 point point NOUN cana-5930 82 14 union union NOUN cana-5930 82 15 of of ADP cana-5930 82 16 t t PROPN cana-5930 82 17 cycles cycle NOUN cana-5930 82 18 of of ADP cana-5930 82 19 length length NOUN cana-5930 82 20 p(𝑝 p(𝑝 PROPN cana-5930 82 21 β‰₯ β‰₯ NOUN cana-5930 82 22 3 3 NUM cana-5930 82 23 ) ) PUNCT cana-5930 82 24 thenγ𝑐𝑑𝑑 thenγ𝑐𝑑𝑑 NOUN cana-5930 82 25 ( ( PUNCT cana-5930 82 26 𝑇2(𝐢𝑝 𝑇2(𝐢𝑝 PROPN cana-5930 82 27 ( ( PUNCT cana-5930 82 28 𝑑 𝑑 NOUN cana-5930 82 29 ) ) PUNCT cana-5930 82 30 ) ) PUNCT cana-5930 82 31 ) ) PUNCT cana-5930 83 1 = = PUNCT cana-5930 83 2 ( ( PUNCT cana-5930 83 3 𝑝 𝑝 PROPN cana-5930 83 4 βˆ’ βˆ’ PROPN cana-5930 83 5 1)𝑑 1)𝑑 NUM cana-5930 83 6 , , PUNCT cana-5930 83 7 𝑝 𝑝 NOUN cana-5930 83 8 β‰₯ β‰₯ NOUN cana-5930 83 9 3 3 NUM cana-5930 83 10 . . PUNCT cana-5930 84 1 proof proof NOUN cana-5930 84 2 . . PUNCT cana-5930 85 1 let let VERB cana-5930 85 2 𝐺 𝐺 PROPN cana-5930 85 3 = = SYM cana-5930 85 4 𝐢𝑝 𝐢𝑝 PROPN cana-5930 85 5 ( ( PUNCT cana-5930 85 6 𝑑 𝑑 NOUN cana-5930 85 7 ) ) PUNCT cana-5930 85 8 and and CCONJ cana-5930 85 9 u u PRON cana-5930 85 10 be be VERB cana-5930 85 11 the the DET cana-5930 85 12 point point NOUN cana-5930 85 13 of of ADP cana-5930 85 14 union union NOUN cana-5930 85 15 of of ADP cana-5930 85 16 t t PROPN cana-5930 85 17 cycles cycle NOUN cana-5930 85 18 of of ADP cana-5930 85 19 length length NOUN cana-5930 85 20 p. p. NOUN cana-5930 85 21 let let VERB cana-5930 85 22 the the DET cana-5930 85 23 vertex vertex NOUN cana-5930 85 24 set set NOUN cana-5930 85 25 of of ADP cana-5930 85 26 π‘˜π‘‘β„Ž π‘˜π‘‘β„Ž NOUN cana-5930 85 27 cycle cycle NOUN cana-5930 85 28 in in ADP cana-5930 85 29 𝐢𝑝 𝐢𝑝 PROPN cana-5930 85 30 ( ( PUNCT cana-5930 85 31 𝑑 𝑑 NOUN cana-5930 85 32 ) ) PUNCT cana-5930 85 33 be be VERB cana-5930 86 1 π‘‰π‘˜ π‘‰π‘˜ PROPN cana-5930 86 2 = = PUNCT cana-5930 86 3 { { PUNCT cana-5930 86 4 𝑒 𝑒 X cana-5930 86 5 , , PUNCT cana-5930 86 6 π‘’π‘˜1 π‘’π‘˜1 NOUN cana-5930 86 7 , , PUNCT cana-5930 86 8 π‘’π‘˜2 π‘’π‘˜2 ADV cana-5930 86 9 , , PUNCT cana-5930 86 10 … … PUNCT cana-5930 86 11 , , PUNCT cana-5930 86 12 π‘’π‘˜,π‘βˆ’1}π‘˜ π‘’π‘˜,π‘βˆ’1}π‘˜ X cana-5930 86 13 = = SYM cana-5930 86 14 1,2 1,2 NUM cana-5930 86 15 , , PUNCT cana-5930 86 16 … … PUNCT cana-5930 86 17 , , PUNCT cana-5930 86 18 𝑑(𝑑 𝑑(𝑑 X cana-5930 86 19 β‰₯ β‰₯ NOUN cana-5930 86 20 2 2 NUM cana-5930 86 21 ) ) PUNCT cana-5930 86 22 π‘‰π‘˜ π‘‰π‘˜ PROPN cana-5930 86 23 ( ( PUNCT cana-5930 86 24 𝑇2(𝐢𝑝 𝑇2(𝐢𝑝 PROPN cana-5930 86 25 ( ( PUNCT cana-5930 86 26 𝑑 𝑑 NOUN cana-5930 86 27 ) ) PUNCT cana-5930 86 28 ) ) PUNCT cana-5930 86 29 ) ) PUNCT cana-5930 87 1 = = PRON cana-5930 87 2 { { PUNCT cana-5930 87 3 𝑒 𝑒 X cana-5930 87 4 , , PUNCT cana-5930 87 5 π‘’π‘˜1 π‘’π‘˜1 NOUN cana-5930 87 6 , , PUNCT cana-5930 87 7 π‘’π‘˜2 π‘’π‘˜2 ADV cana-5930 87 8 , , PUNCT cana-5930 87 9 … … PUNCT cana-5930 87 10 , , PUNCT cana-5930 87 11 π‘’π‘˜,π‘βˆ’1 π‘’π‘˜,π‘βˆ’1 PROPN cana-5930 87 12 } } PUNCT cana-5930 87 13 βˆͺ βˆͺ X cana-5930 87 14 { { PUNCT cana-5930 87 15 π‘’π‘˜1 π‘’π‘˜1 NOUN cana-5930 87 16 β€² β€² NOUN cana-5930 87 17 , , PUNCT cana-5930 87 18 π‘’π‘˜2 π‘’π‘˜2 ADV cana-5930 87 19 β€² β€² NUM cana-5930 87 20 … … PUNCT cana-5930 87 21 π‘’π‘˜π‘ π‘’π‘˜π‘ ADJ cana-5930 87 22 β€² β€² NOUN cana-5930 87 23 } } PUNCT cana-5930 87 24 where where SCONJ cana-5930 87 25 π‘’π‘˜1 π‘’π‘˜1 NOUN cana-5930 87 26 β€² β€² VERB cana-5930 87 27 , , PUNCT cana-5930 87 28 π‘’π‘˜2 π‘’π‘˜2 ADV cana-5930 87 29 β€² β€² NUM cana-5930 87 30 , , PUNCT cana-5930 87 31 … … PUNCT cana-5930 87 32 , , PUNCT cana-5930 87 33 π‘’π‘˜π‘ π‘’π‘˜π‘ NOUN cana-5930 87 34 β€² β€² NUM cana-5930 87 35 are be AUX cana-5930 87 36 the the DET cana-5930 87 37 corresponding corresponding ADJ cana-5930 87 38 edge edge NOUN cana-5930 87 39 vertices vertex NOUN cana-5930 87 40 of of ADP cana-5930 87 41 ( ( PUNCT cana-5930 87 42 𝑒 𝑒 X cana-5930 87 43 , , PUNCT cana-5930 87 44 π‘’π‘˜1 π‘’π‘˜1 NOUN cana-5930 87 45 ) ) PUNCT cana-5930 87 46 , , PUNCT cana-5930 87 47 ( ( PUNCT cana-5930 87 48 π‘’π‘˜1 π‘’π‘˜1 NOUN cana-5930 87 49 , , PUNCT cana-5930 87 50 π‘’π‘˜2 π‘’π‘˜2 ADJ cana-5930 87 51 ) ) PUNCT cana-5930 87 52 , , PUNCT cana-5930 87 53 … … PUNCT cana-5930 87 54 , , PUNCT cana-5930 87 55 ( ( PUNCT cana-5930 87 56 𝑒 𝑒 X cana-5930 87 57 , , PUNCT cana-5930 87 58 π‘’π‘˜,π‘βˆ’1 π‘’π‘˜,π‘βˆ’1 PROPN cana-5930 87 59 ) ) PUNCT cana-5930 87 60 . . PUNCT cana-5930 88 1 let let VERB cana-5930 88 2 π·π‘˜ π·π‘˜ PROPN cana-5930 88 3 = = PRON cana-5930 88 4 { { PUNCT cana-5930 88 5 π‘’π‘˜1 π‘’π‘˜1 NOUN cana-5930 88 6 , , PUNCT cana-5930 88 7 π‘’π‘˜3 π‘’π‘˜3 ADP cana-5930 88 8 β€² β€² NUM cana-5930 88 9 , , PUNCT cana-5930 88 10 … … PUNCT cana-5930 88 11 , , PUNCT cana-5930 88 12 π‘’π‘˜π‘ π‘’π‘˜π‘ NOUN cana-5930 88 13 β€² β€² NUM cana-5930 88 14 } } PUNCT cana-5930 88 15 , , PUNCT cana-5930 88 16 π‘˜ π‘˜ X cana-5930 88 17 = = SYM cana-5930 88 18 1 1 NUM cana-5930 88 19 , , PUNCT cana-5930 88 20 2 2 NUM cana-5930 88 21 , , PUNCT cana-5930 88 22 … … PUNCT cana-5930 88 23 , , PUNCT cana-5930 88 24 𝑑 𝑑 PROPN cana-5930 88 25 𝐷 𝐷 NOUN cana-5930 88 26 = = PUNCT cana-5930 88 27 ⋃ ⋃ PROPN cana-5930 88 28 π·π‘˜ π·π‘˜ PROPN cana-5930 88 29 𝑑 𝑑 PROPN cana-5930 88 30 π‘˜=1 π‘˜=1 NOUN cana-5930 88 31 βŠ† βŠ† NUM cana-5930 88 32 𝑉(𝑇2(𝐺 𝑉(𝑇2(𝐺 NOUN cana-5930 88 33 ) ) PUNCT cana-5930 88 34 ) ) PUNCT cana-5930 88 35 . . PUNCT cana-5930 89 1 then then ADV cana-5930 89 2 βŸ¨π‘‰(𝑇2(𝐺 βŸ¨π‘‰(𝑇2(𝐺 PROPN cana-5930 89 3 ) ) PUNCT cana-5930 89 4 ) ) PUNCT cana-5930 90 1 βˆ’ βˆ’ PROPN cana-5930 90 2 𝐷 𝐷 PROPN cana-5930 90 3 ⟩ ⟩ NOUN cana-5930 90 4 is be AUX cana-5930 90 5 a a DET cana-5930 90 6 tree tree NOUN cana-5930 90 7 . . PUNCT cana-5930 91 1 hence hence ADV cana-5930 91 2 d d PROPN cana-5930 91 3 is be AUX cana-5930 91 4 a a DET cana-5930 91 5 minimum minimum ADJ cana-5930 91 6 ctd ctd NOUN cana-5930 91 7 - - PUNCT cana-5930 91 8 set set NOUN cana-5930 91 9 of of ADP cana-5930 91 10 𝑇2(𝐺 𝑇2(𝐺 PROPN cana-5930 91 11 ) ) PUNCT cana-5930 91 12 . . PUNCT cana-5930 92 1 therefore therefore ADV cana-5930 92 2 , , PUNCT cana-5930 92 3 |𝐷| |𝐷| NOUN cana-5930 92 4 = = SYM cana-5930 92 5 γ𝑐𝑑𝑑(𝑇2(𝐺 γ𝑐𝑑𝑑(𝑇2(𝐺 NOUN cana-5930 92 6 ) ) PUNCT cana-5930 92 7 ) ) PUNCT cana-5930 93 1 = = PUNCT cana-5930 93 2 ( ( PUNCT cana-5930 93 3 𝑝 𝑝 PROPN cana-5930 93 4 βˆ’ βˆ’ PROPN cana-5930 93 5 1)𝑑. 1)𝑑. PROPN cana-5930 93 6 β–‘ β–‘ PUNCT cana-5930 93 7 communications communication NOUN cana-5930 93 8 on on ADP cana-5930 93 9 applied apply VERB cana-5930 93 10 nonlinear nonlinear ADJ cana-5930 93 11 analysis analysis NOUN cana-5930 93 12 issn issn NOUN cana-5930 93 13 : : PUNCT cana-5930 93 14 1074 1074 NUM cana-5930 93 15 - - PUNCT cana-5930 93 16 133x 133x NUM cana-5930 93 17 vol vol VERB cana-5930 93 18 32 32 NUM cana-5930 93 19 no no NOUN cana-5930 93 20 . . PUNCT cana-5930 94 1 10s 10 NOUN cana-5930 94 2 ( ( PUNCT cana-5930 94 3 2025 2025 NUM cana-5930 94 4 ) ) PUNCT cana-5930 94 5 3106 3106 NUM cana-5930 94 6 https://internationalpubls.com https://internationalpubls.com X cana-5930 94 7 proposition proposition NOUN cana-5930 94 8 3.3 3.3 NUM cana-5930 94 9 . . PUNCT cana-5930 95 1 let let VERB cana-5930 95 2 g g PRON cana-5930 95 3 be be AUX cana-5930 95 4 a a DET cana-5930 95 5 unicyclic unicyclic ADJ cana-5930 95 6 graph graph NOUN cana-5930 95 7 by by ADP cana-5930 95 8 attaching attach VERB cana-5930 95 9 a a DET cana-5930 95 10 path path NOUN cana-5930 95 11 of of ADP cana-5930 95 12 length length NOUN cana-5930 95 13 ( ( PUNCT cana-5930 95 14 𝑛 𝑛 DET cana-5930 95 15 β‰₯ β‰₯ NOUN cana-5930 95 16 1 1 NUM cana-5930 95 17 ) ) PUNCT cana-5930 95 18 to to ADP cana-5930 95 19 the the DET cana-5930 95 20 𝑑(≀ 𝑑(≀ PROPN cana-5930 95 21 𝑝 𝑝 PROPN cana-5930 95 22 ) ) PUNCT cana-5930 95 23 consecutive consecutive ADJ cana-5930 95 24 vertices vertex NOUN cana-5930 95 25 of of ADP cana-5930 95 26 𝐢𝑝(𝑝 𝐢𝑝(𝑝 ADJ cana-5930 95 27 β‰₯ β‰₯ NOUN cana-5930 95 28 3 3 NUM cana-5930 95 29 ) ) PUNCT cana-5930 95 30 . . PUNCT cana-5930 96 1 then then ADV cana-5930 96 2 γ𝑐𝑑𝑑(𝑇2(𝐺 γ𝑐𝑑𝑑(𝑇2(𝐺 NOUN cana-5930 96 3 ) ) PUNCT cana-5930 96 4 ) ) PUNCT cana-5930 97 1 = = SYM cana-5930 97 2 𝑛𝑑 𝑛𝑑 PROPN cana-5930 98 1 + + CCONJ cana-5930 98 2 𝑝 𝑝 ADJ cana-5930 98 3 βˆ’ βˆ’ NOUN cana-5930 98 4 1 1 NUM cana-5930 98 5 . . PUNCT cana-5930 99 1 proof proof NOUN cana-5930 99 2 . . PUNCT cana-5930 100 1 in in ADP cana-5930 100 2 the the DET cana-5930 100 3 cycle cycle NOUN cana-5930 100 4 𝐢𝑝(𝑝 𝐢𝑝(𝑝 ADJ cana-5930 100 5 β‰₯ β‰₯ NOUN cana-5930 100 6 3 3 NUM cana-5930 100 7 ) ) PUNCT cana-5930 100 8 say say VERB cana-5930 100 9 𝑣1 𝑣1 PROPN cana-5930 100 10 , , PUNCT cana-5930 100 11 𝑣2 𝑣2 PROPN cana-5930 100 12 , , PUNCT cana-5930 100 13 … … PUNCT cana-5930 100 14 , , PUNCT cana-5930 100 15 𝑣𝑝. 𝑣𝑝. PRON cana-5930 100 16 consider consider VERB cana-5930 100 17 a a DET cana-5930 100 18 path path NOUN cana-5930 100 19 of of ADP cana-5930 100 20 length length NOUN cana-5930 100 21 π‘ƒπ‘βˆ’1 π‘ƒπ‘βˆ’1 PROPN cana-5930 100 22 in in ADP cana-5930 100 23 𝐢𝑝 𝐢𝑝 PROPN cana-5930 100 24 say say VERB cana-5930 100 25 𝑣1 𝑣1 NOUN cana-5930 100 26 , , PUNCT cana-5930 100 27 𝑣2 𝑣2 PROPN cana-5930 100 28 , , PUNCT cana-5930 100 29 … … PUNCT cana-5930 100 30 , , PUNCT cana-5930 100 31 π‘£π‘βˆ’1 π‘£π‘βˆ’1 ADV cana-5930 100 32 and and CCONJ cana-5930 100 33 attach attach VERB cana-5930 100 34 a a DET cana-5930 100 35 path path NOUN cana-5930 101 1 𝑃𝑛 𝑃𝑛 PROPN cana-5930 101 2 β€²(𝑛 β€²(𝑛 PART cana-5930 101 3 β‰₯ β‰₯ NOUN cana-5930 101 4 1 1 NUM cana-5930 101 5 ) ) PUNCT cana-5930 101 6 say say VERB cana-5930 101 7 𝑣𝑖 𝑣𝑖 ADV cana-5930 101 8 , , PUNCT cana-5930 101 9 𝑒2 𝑒2 PROPN cana-5930 101 10 , , PUNCT cana-5930 101 11 … … PUNCT cana-5930 101 12 , , PUNCT cana-5930 101 13 𝑒𝑛 𝑒𝑛 NOUN cana-5930 101 14 , , PUNCT cana-5930 101 15 𝑖 𝑖 NOUN cana-5930 101 16 = = SYM cana-5930 101 17 1 1 NUM cana-5930 101 18 , , PUNCT cana-5930 101 19 2 2 NUM cana-5930 101 20 , , PUNCT cana-5930 101 21 … … PUNCT cana-5930 101 22 , , PUNCT cana-5930 101 23 𝑑 𝑑 NOUN cana-5930 101 24 to to ADP cana-5930 101 25 the the DET cana-5930 101 26 𝑑(≀ 𝑑(≀ PROPN cana-5930 101 27 𝑝 𝑝 PROPN cana-5930 101 28 ) ) PUNCT cana-5930 101 29 consecutive consecutive ADJ cana-5930 101 30 vertices vertex NOUN cana-5930 101 31 of of ADP cana-5930 101 32 𝐢𝑝. 𝐢𝑝. PROPN cana-5930 101 33 in in ADP cana-5930 101 34 𝑇2(𝐺 𝑇2(𝐺 PROPN cana-5930 101 35 ) ) PUNCT cana-5930 101 36 , , PUNCT cana-5930 101 37 the the DET cana-5930 101 38 set set NOUN cana-5930 101 39 of of ADP cana-5930 101 40 all all DET cana-5930 101 41 edge edge NOUN cana-5930 101 42 vertices vertex NOUN cana-5930 101 43 of of ADP cana-5930 101 44 path path NOUN cana-5930 101 45 𝑃𝑛 𝑃𝑛 PROPN cana-5930 101 46 β€² β€² NUM cana-5930 101 47 of of ADP cana-5930 101 48 t t PROPN cana-5930 101 49 consecutive consecutive ADJ cana-5930 101 50 vertices vertex NOUN cana-5930 101 51 of of ADP cana-5930 101 52 𝐢𝑝 𝐢𝑝 PROPN cana-5930 101 53 , , PUNCT cana-5930 101 54 edge edge NOUN cana-5930 101 55 vertices vertex NOUN cana-5930 101 56 of of ADP cana-5930 101 57 a a DET cana-5930 101 58 path path NOUN cana-5930 101 59 π‘ƒπ‘βˆ’1 π‘ƒπ‘βˆ’1 NOUN cana-5930 101 60 and and CCONJ cana-5930 101 61 a a DET cana-5930 101 62 vertex vertex NOUN cana-5930 101 63 𝑣𝑝 𝑣𝑝 NOUN cana-5930 101 64 forms form VERB cana-5930 101 65 a a DET cana-5930 101 66 minimum minimum ADJ cana-5930 101 67 ctd ctd NOUN cana-5930 101 68 - - PUNCT cana-5930 101 69 set set NOUN cana-5930 101 70 of of ADP cana-5930 101 71 𝑇2(𝐺 𝑇2(𝐺 PROPN cana-5930 101 72 ) ) PUNCT cana-5930 101 73 . . PUNCT cana-5930 102 1 therefore therefore ADV cana-5930 102 2 , , PUNCT cana-5930 102 3 γ𝑐𝑑𝑑(𝑇2(𝐺 γ𝑐𝑑𝑑(𝑇2(𝐺 NOUN cana-5930 102 4 ) ) PUNCT cana-5930 102 5 ) ) PUNCT cana-5930 103 1 = = SYM cana-5930 103 2 𝑛𝑑 𝑛𝑑 PROPN cana-5930 104 1 + + CCONJ cana-5930 104 2 𝑝 𝑝 ADJ cana-5930 104 3 βˆ’ βˆ’ NOUN cana-5930 104 4 1 1 NUM cana-5930 104 5 . . PUNCT cana-5930 105 1 β–‘ β–‘ PUNCT cana-5930 105 2 corollary corollary ADJ cana-5930 105 3 3.4 3.4 NUM cana-5930 105 4 . . PUNCT cana-5930 106 1 g g PROPN cana-5930 106 2 be be AUX cana-5930 106 3 a a DET cana-5930 106 4 unicyclic unicyclic ADJ cana-5930 106 5 graph graph NOUN cana-5930 106 6 by by ADP cana-5930 106 7 attaching attach VERB cana-5930 106 8 one one NUM cana-5930 106 9 pendant pendant ADJ cana-5930 106 10 vertex vertex NOUN cana-5930 106 11 to to ADP cana-5930 106 12 exactly exactly ADV cana-5930 106 13 one one NUM cana-5930 106 14 vertex vertex NOUN cana-5930 106 15 of of ADP cana-5930 106 16 𝐢𝑝 𝐢𝑝 PROPN cana-5930 106 17 then then ADV cana-5930 106 18 γ𝑐𝑑𝑑(𝑇2(𝐺 γ𝑐𝑑𝑑(𝑇2(𝐺 NOUN cana-5930 106 19 ) ) PUNCT cana-5930 106 20 ) ) PUNCT cana-5930 107 1 = = PUNCT cana-5930 107 2 𝑝. 𝑝. PROPN cana-5930 107 3 corollary corollary ADJ cana-5930 107 4 3.5 3.5 NUM cana-5930 107 5 . . PUNCT cana-5930 108 1 g g PROPN cana-5930 108 2 be be AUX cana-5930 108 3 a a DET cana-5930 108 4 unicyclic unicyclic ADJ cana-5930 108 5 graph graph NOUN cana-5930 108 6 by by ADP cana-5930 108 7 attaching attach VERB cana-5930 108 8 one one NUM cana-5930 108 9 pendant pendant ADJ cana-5930 108 10 vertex vertex NOUN cana-5930 108 11 to to ADP cana-5930 108 12 𝑝 𝑝 NOUN cana-5930 108 13 βˆ’ βˆ’ NUM cana-5930 108 14 1 1 NUM cana-5930 108 15 vertices vertex NOUN cana-5930 108 16 of of ADP cana-5930 108 17 𝐢𝑝. 𝐢𝑝. PROPN cana-5930 108 18 then then ADV cana-5930 108 19 γ𝑐𝑑𝑑(𝑇2(𝐺 γ𝑐𝑑𝑑(𝑇2(𝐺 NOUN cana-5930 108 20 ) ) PUNCT cana-5930 108 21 ) ) PUNCT cana-5930 109 1 = = SYM cana-5930 109 2 2𝑝 2𝑝 NOUN cana-5930 109 3 βˆ’ βˆ’ NOUN cana-5930 109 4 2 2 X cana-5930 109 5 . . PUNCT cana-5930 109 6 corollary corollary ADJ cana-5930 109 7 3.6 3.6 NUM cana-5930 109 8 . . PUNCT cana-5930 110 1 g g PROPN cana-5930 110 2 be be AUX cana-5930 110 3 a a DET cana-5930 110 4 unicyclic unicyclic ADJ cana-5930 110 5 graph graph NOUN cana-5930 110 6 by by ADP cana-5930 110 7 attaching attach VERB cana-5930 110 8 a a DET cana-5930 110 9 path path NOUN cana-5930 110 10 of of ADP cana-5930 110 11 length length NOUN cana-5930 110 12 n n ADP cana-5930 110 13 to to ADP cana-5930 110 14 exactly exactly ADV cana-5930 110 15 one one NUM cana-5930 110 16 vertex vertex NOUN cana-5930 110 17 of of ADP cana-5930 110 18 𝐢𝑝 𝐢𝑝 PROPN cana-5930 110 19 thenγ𝑐𝑑𝑑(𝑇2(𝐺 thenγ𝑐𝑑𝑑(𝑇2(𝐺 NOUN cana-5930 110 20 ) ) PUNCT cana-5930 110 21 ) ) PUNCT cana-5930 111 1 = = PUNCT cana-5930 111 2 𝑝 𝑝 PROPN cana-5930 111 3 + + NUM cana-5930 111 4 𝑛 𝑛 PROPN cana-5930 111 5 + + NOUN cana-5930 111 6 1 1 NUM cana-5930 111 7 . . NOUN cana-5930 111 8 4 4 NUM cana-5930 111 9 . . NUM cana-5930 111 10 bounds bound NOUN cana-5930 111 11 and and CCONJ cana-5930 111 12 some some DET cana-5930 111 13 exact exact ADJ cana-5930 111 14 values value NOUN cana-5930 111 15 for for ADP cana-5930 111 16 the the DET cana-5930 111 17 complementary complementary ADJ cana-5930 111 18 tree tree NOUN cana-5930 111 19 domination domination NOUN cana-5930 111 20 number number NOUN cana-5930 111 21 of of ADP cana-5930 111 22 semi semi ADJ cana-5930 111 23 total total ADJ cana-5930 111 24 point point NOUN cana-5930 111 25 graph graph NOUN cana-5930 111 26 of of ADP cana-5930 111 27 graphs graph NOUN cana-5930 111 28 theorem theorem VERB cana-5930 111 29 4.1 4.1 NUM cana-5930 111 30 . . PUNCT cana-5930 111 31 γ𝑐𝑑𝑑(𝑇2(𝐺 γ𝑐𝑑𝑑(𝑇2(𝐺 NOUN cana-5930 111 32 ) ) PUNCT cana-5930 111 33 ) ) PUNCT cana-5930 112 1 = = SYM cana-5930 112 2 2 2 NUM cana-5930 112 3 if if SCONJ cana-5930 112 4 and and CCONJ cana-5930 112 5 only only ADV cana-5930 112 6 if if SCONJ cana-5930 112 7 𝐺 𝐺 PROPN cana-5930 112 8 β‰… β‰… PROPN cana-5930 112 9 𝐾1,2 𝐾1,2 PROPN cana-5930 112 10 or or CCONJ cana-5930 112 11 𝐢3 𝐢3 NOUN cana-5930 112 12 . . PUNCT cana-5930 113 1 proof proof NOUN cana-5930 113 2 . . PUNCT cana-5930 114 1 let let VERB cana-5930 114 2 d d PRON cana-5930 114 3 be be AUX cana-5930 114 4 a a DET cana-5930 114 5 γ𝑐𝑑𝑑-set γ𝑐𝑑𝑑-set NOUN cana-5930 114 6 of of ADP cana-5930 114 7 𝑇2(𝐺 𝑇2(𝐺 PROPN cana-5930 114 8 ) ) PUNCT cana-5930 114 9 such such ADJ cana-5930 114 10 that that DET cana-5930 114 11 |𝐷| |𝐷| NOUN cana-5930 114 12 = = SYM cana-5930 114 13 2 2 X cana-5930 114 14 . . PUNCT cana-5930 115 1 let let VERB cana-5930 115 2 𝐷 𝐷 PROPN cana-5930 115 3 = = SYM cana-5930 115 4 { { PUNCT cana-5930 115 5 𝑒1 𝑒1 PROPN cana-5930 115 6 , , PUNCT cana-5930 115 7 𝑒2 𝑒2 PROPN cana-5930 115 8 } } PUNCT cana-5930 115 9 where where SCONJ cana-5930 115 10 𝑒1 𝑒1 PROPN cana-5930 115 11 , , PUNCT cana-5930 115 12 𝑒2 𝑒2 PROPN cana-5930 115 13 ∈ ∈ PROPN cana-5930 115 14 𝑉(𝑇2(𝐺 𝑉(𝑇2(𝐺 NOUN cana-5930 115 15 ) ) PUNCT cana-5930 115 16 ) ) PUNCT cana-5930 115 17 . . PUNCT cana-5930 116 1 case case NOUN cana-5930 116 2 1 1 X cana-5930 116 3 . . PUNCT cana-5930 117 1 𝑒1 𝑒1 PROPN cana-5930 117 2 and and CCONJ cana-5930 117 3 𝑒2 𝑒2 PROPN cana-5930 117 4 are be AUX cana-5930 117 5 vertices vertex NOUN cana-5930 117 6 in in ADP cana-5930 117 7 g. g. PROPN cana-5930 117 8 then then ADV cana-5930 117 9 , , PUNCT cana-5930 117 10 d d X cana-5930 117 11 is be AUX cana-5930 117 12 also also ADV cana-5930 117 13 a a DET cana-5930 117 14 γ𝑐𝑑𝑑-set γ𝑐𝑑𝑑-set NOUN cana-5930 117 15 of of ADP cana-5930 117 16 g. g. NOUN cana-5930 117 17 by by ADP cana-5930 117 18 theorem theorem NOUN cana-5930 117 19 3.1 3.1 NUM cana-5930 117 20 , , PUNCT cana-5930 117 21 it it PRON cana-5930 117 22 can can AUX cana-5930 117 23 be be AUX cana-5930 117 24 seen see VERB cana-5930 117 25 that that SCONJ cana-5930 117 26 𝐺 𝐺 PROPN cana-5930 117 27 β‰… β‰… PROPN cana-5930 117 28 𝐾1,2 𝐾1,2 PROPN cana-5930 117 29 . . PUNCT cana-5930 118 1 case case NOUN cana-5930 118 2 2 2 NUM cana-5930 118 3 . . X cana-5930 118 4 𝑒1 𝑒1 PROPN cana-5930 118 5 , , PUNCT cana-5930 118 6 𝑒2 𝑒2 PROPN cana-5930 118 7 ∈ ∈ PROPN cana-5930 118 8 𝑉(𝑇2(𝐺 𝑉(𝑇2(𝐺 NOUN cana-5930 118 9 ) ) PUNCT cana-5930 118 10 ) ) PUNCT cana-5930 119 1 βˆ’ βˆ’ PROPN cana-5930 119 2 𝑉(𝐺 𝑉(𝐺 NOUN cana-5930 119 3 ) ) PUNCT cana-5930 119 4 . . PUNCT cana-5930 120 1 let let VERB cana-5930 120 2 𝑒1 𝑒1 PROPN cana-5930 120 3 = = SYM cana-5930 120 4 𝑒1 𝑒1 PROPN cana-5930 120 5 β€² β€² NOUN cana-5930 120 6 and and CCONJ cana-5930 120 7 𝑒2 𝑒2 PROPN cana-5930 120 8 = = PROPN cana-5930 120 9 𝑒2 𝑒2 PROPN cana-5930 120 10 β€² β€² NUM cana-5930 120 11 where where SCONJ cana-5930 120 12 ( ( PUNCT cana-5930 120 13 𝑒 𝑒 X cana-5930 120 14 , , PUNCT cana-5930 120 15 𝑒1 𝑒1 NOUN cana-5930 120 16 ) ) PUNCT cana-5930 120 17 and and CCONJ cana-5930 120 18 ( ( PUNCT cana-5930 120 19 𝑒 𝑒 X cana-5930 120 20 , , PUNCT cana-5930 120 21 𝑒2 𝑒2 PROPN cana-5930 120 22 ) ) PUNCT cana-5930 120 23 ∈ ∈ PROPN cana-5930 120 24 𝐸(𝐺 𝐸(𝐺 NOUN cana-5930 120 25 ) ) PUNCT cana-5930 120 26 . . PUNCT cana-5930 121 1 since since SCONJ cana-5930 121 2 𝑒1 𝑒1 PROPN cana-5930 121 3 β€² β€² NOUN cana-5930 121 4 and and CCONJ cana-5930 121 5 𝑒2 𝑒2 PROPN cana-5930 121 6 β€² β€² NUM cana-5930 121 7 are be AUX cana-5930 121 8 edge edge NOUN cana-5930 121 9 vertices vertex NOUN cana-5930 121 10 in in ADP cana-5930 121 11 𝑇2(𝐺 𝑇2(𝐺 PROPN cana-5930 121 12 ) ) PUNCT cana-5930 121 13 and and CCONJ cana-5930 121 14 βŸ¨π‘‰(𝑇2(𝐺 βŸ¨π‘‰(𝑇2(𝐺 PROPN cana-5930 121 15 ) ) PUNCT cana-5930 121 16 ) ) PUNCT cana-5930 122 1 βˆ’ βˆ’ PROPN cana-5930 123 1 𝐷⟩ 𝐷⟩ PROPN cana-5930 123 2 is be AUX cana-5930 123 3 connected connect VERB cana-5930 123 4 and and CCONJ cana-5930 123 5 acyclic acyclic ADJ cana-5930 123 6 . . PUNCT cana-5930 124 1 hence hence ADV cana-5930 124 2 it it PRON cana-5930 124 3 can can AUX cana-5930 124 4 be be AUX cana-5930 124 5 seen see VERB cana-5930 124 6 that that SCONJ cana-5930 124 7 𝐺 𝐺 PROPN cana-5930 124 8 β‰… β‰… PROPN cana-5930 124 9 𝐾1,2 𝐾1,2 PROPN cana-5930 124 10 . . PUNCT cana-5930 125 1 case case NOUN cana-5930 125 2 3 3 X cana-5930 125 3 . . PUNCT cana-5930 126 1 let let VERB cana-5930 126 2 𝑒1 𝑒1 PROPN cana-5930 126 3 ∈ ∈ PROPN cana-5930 126 4 𝑉(𝐺 𝑉(𝐺 NOUN cana-5930 126 5 ) ) PUNCT cana-5930 126 6 and and CCONJ cana-5930 126 7 𝑒2 𝑒2 PROPN cana-5930 126 8 = = PROPN cana-5930 126 9 𝑒2 𝑒2 PROPN cana-5930 126 10 β€² β€² NOUN cana-5930 126 11 ∈ ∈ PROPN cana-5930 126 12 𝑉(𝑇2(𝐺 𝑉(𝑇2(𝐺 NOUN cana-5930 126 13 ) ) PUNCT cana-5930 126 14 ) ) PUNCT cana-5930 127 1 βˆ’ βˆ’ ADP cana-5930 127 2 𝑉(𝐺 𝑉(𝐺 NOUN cana-5930 127 3 ) ) PUNCT cana-5930 127 4 communications communication NOUN cana-5930 127 5 on on ADP cana-5930 127 6 applied apply VERB cana-5930 127 7 nonlinear nonlinear ADJ cana-5930 127 8 analysis analysis NOUN cana-5930 127 9 issn issn NOUN cana-5930 127 10 : : PUNCT cana-5930 127 11 1074 1074 NUM cana-5930 127 12 - - PUNCT cana-5930 127 13 133x 133x NUM cana-5930 127 14 vol vol VERB cana-5930 127 15 32 32 NUM cana-5930 127 16 no no NOUN cana-5930 127 17 . . PUNCT cana-5930 128 1 10s 10 NOUN cana-5930 128 2 ( ( PUNCT cana-5930 128 3 2025 2025 NUM cana-5930 128 4 ) ) PUNCT cana-5930 128 5 3107 3107 NUM cana-5930 128 6 https://internationalpubls.com https://internationalpubls.com X cana-5930 128 7 sub sub NOUN cana-5930 128 8 case case NOUN cana-5930 128 9 3.1 3.1 NUM cana-5930 128 10 . . PUNCT cana-5930 129 1 𝑒2 𝑒2 PROPN cana-5930 129 2 = = PROPN cana-5930 129 3 𝑒2 𝑒2 PROPN cana-5930 129 4 β€² β€² NUM cana-5930 129 5 . . PUNCT cana-5930 130 1 that that PRON cana-5930 130 2 is be AUX cana-5930 130 3 𝐷 𝐷 PROPN cana-5930 130 4 = = SYM cana-5930 130 5 { { PUNCT cana-5930 130 6 𝑒1 𝑒1 PROPN cana-5930 130 7 , , PUNCT cana-5930 130 8 𝑒2 𝑒2 PROPN cana-5930 130 9 β€² β€² NUM cana-5930 130 10 } } PUNCT cana-5930 130 11 is be AUX cana-5930 130 12 a a DET cana-5930 130 13 γ𝑐𝑑𝑑-set γ𝑐𝑑𝑑-set NOUN cana-5930 130 14 of of ADP cana-5930 130 15 𝑇2(𝐺 𝑇2(𝐺 PROPN cana-5930 130 16 ) ) PUNCT cana-5930 130 17 . . PUNCT cana-5930 131 1 hence hence ADV cana-5930 131 2 𝐺 𝐺 PROPN cana-5930 131 3 β‰… β‰… PROPN cana-5930 131 4 𝐢3 𝐢3 PROPN cana-5930 131 5 . . PUNCT cana-5930 132 1 sub sub PROPN cana-5930 132 2 case case NOUN cana-5930 132 3 3.2 3.2 NUM cana-5930 132 4 . . PUNCT cana-5930 133 1 𝑒2 𝑒2 PROPN cana-5930 133 2 β‰  β‰  PROPN cana-5930 133 3 𝑒2 𝑒2 PROPN cana-5930 133 4 β€² β€² NUM cana-5930 133 5 . . PUNCT cana-5930 134 1 let let VERB cana-5930 134 2 𝑒2 𝑒2 PROPN cana-5930 134 3 = = SYM cana-5930 134 4 𝑣′ 𝑣′ PROPN cana-5930 134 5 for for ADP cana-5930 134 6 some some DET cana-5930 134 7 ( ( PUNCT cana-5930 134 8 𝑒1 𝑒1 PROPN cana-5930 134 9 , , PUNCT cana-5930 134 10 𝑣 𝑣 NOUN cana-5930 134 11 ) ) PUNCT cana-5930 134 12 ∈ ∈ PROPN cana-5930 134 13 𝐸(𝐺 𝐸(𝐺 PROPN cana-5930 134 14 ) ) PUNCT cana-5930 134 15 and and CCONJ cana-5930 134 16 𝑣′ 𝑣′ NUM cana-5930 134 17 β‰  β‰  PROPN cana-5930 134 18 𝑒2 𝑒2 PROPN cana-5930 134 19 β€² β€² NUM cana-5930 134 20 . . PUNCT cana-5930 135 1 then then ADV cana-5930 135 2 𝐷 𝐷 PROPN cana-5930 135 3 = = PUNCT cana-5930 135 4 { { PUNCT cana-5930 135 5 𝑒 𝑒 X cana-5930 135 6 , , PUNCT cana-5930 135 7 𝑣′ 𝑣′ PROPN cana-5930 135 8 } } PUNCT cana-5930 135 9 is be AUX cana-5930 135 10 a a DET cana-5930 135 11 γ𝑐𝑑𝑑-set γ𝑐𝑑𝑑-set NOUN cana-5930 135 12 of of ADP cana-5930 135 13 𝑇2(𝐺 𝑇2(𝐺 PROPN cana-5930 135 14 ) ) PUNCT cana-5930 135 15 . . PUNCT cana-5930 136 1 𝑒2 𝑒2 PROPN cana-5930 136 2 β€² β€² NUM cana-5930 136 3 β‰  β‰  PROPN cana-5930 136 4 𝑣′ 𝑣′ NUM cana-5930 136 5 implies imply VERB cana-5930 136 6 that that SCONJ cana-5930 136 7 βŸ¨π‘’2 βŸ¨π‘’2 PROPN cana-5930 136 8 β€² β€² NUM cana-5930 136 9 , , PUNCT cana-5930 136 10 𝑣 𝑣 X cana-5930 136 11 , , PUNCT cana-5930 136 12 π‘₯ π‘₯ DET cana-5930 136 13 ⟩ ⟩ NOUN cana-5930 136 14 form form NOUN cana-5930 136 15 a a DET cana-5930 136 16 cycle cycle NOUN cana-5930 136 17 in in ADP cana-5930 136 18 βŸ¨π‘‰(𝑇2(𝐺 βŸ¨π‘‰(𝑇2(𝐺 PROPN cana-5930 136 19 ) ) PUNCT cana-5930 136 20 ) ) PUNCT cana-5930 137 1 βˆ’ βˆ’ PROPN cana-5930 137 2 𝐷 𝐷 NOUN cana-5930 137 3 ⟩ ⟩ NOUN cana-5930 137 4 forms form NOUN cana-5930 137 5 either either CCONJ cana-5930 137 6 cycle cycle NOUN cana-5930 137 7 or or CCONJ cana-5930 137 8 disconnected disconnect VERB cana-5930 137 9 where where SCONJ cana-5930 137 10 π‘₯ π‘₯ NOUN cana-5930 137 11 , , PUNCT cana-5930 137 12 𝑣 𝑣 PRON cana-5930 137 13 ∈ ∈ PROPN cana-5930 137 14 𝑉(𝐺 𝑉(𝐺 NOUN cana-5930 137 15 ) ) PUNCT cana-5930 137 16 . . PUNCT cana-5930 138 1 conversely conversely ADV cana-5930 138 2 , , PUNCT cana-5930 138 3 if if SCONJ cana-5930 138 4 𝐺 𝐺 PROPN cana-5930 138 5 β‰… β‰… PROPN cana-5930 138 6 𝐾1,2 𝐾1,2 PROPN cana-5930 138 7 or or CCONJ cana-5930 138 8 𝐢3 𝐢3 PROPN cana-5930 138 9 . . PUNCT cana-5930 139 1 then then ADV cana-5930 139 2 γ𝑐𝑑𝑑(𝑇2(𝐺 γ𝑐𝑑𝑑(𝑇2(𝐺 NOUN cana-5930 139 3 ) ) PUNCT cana-5930 139 4 ) ) PUNCT cana-5930 140 1 = = SYM cana-5930 140 2 2 2 X cana-5930 140 3 . . PUNCT cana-5930 140 4 β–‘ β–‘ PUNCT cana-5930 140 5 theorem theorem ADJ cana-5930 140 6 4.2 4.2 NUM cana-5930 140 7 . . PUNCT cana-5930 140 8 γ𝑐𝑑𝑑(𝑇2(𝐺 γ𝑐𝑑𝑑(𝑇2(𝐺 NOUN cana-5930 140 9 ) ) PUNCT cana-5930 140 10 ) ) PUNCT cana-5930 141 1 = = PUNCT cana-5930 141 2 𝑝 𝑝 PROPN cana-5930 141 3 + + NUM cana-5930 141 4 π‘ž π‘ž X cana-5930 141 5 βˆ’ βˆ’ PROPN cana-5930 141 6 2 2 NUM cana-5930 141 7 if if SCONJ cana-5930 141 8 and and CCONJ cana-5930 141 9 only only ADV cana-5930 141 10 if if SCONJ cana-5930 141 11 𝐺 𝐺 PROPN cana-5930 141 12 β‰… β‰… PROPN cana-5930 141 13 𝐾2 𝐾2 PROPN cana-5930 141 14 . . PUNCT cana-5930 142 1 proof proof NOUN cana-5930 142 2 . . PUNCT cana-5930 143 1 assume assume VERB cana-5930 143 2 γ𝑐𝑑𝑑(𝑇2(𝐺 γ𝑐𝑑𝑑(𝑇2(𝐺 NOUN cana-5930 143 3 ) ) PUNCT cana-5930 143 4 ) ) PUNCT cana-5930 144 1 = = PUNCT cana-5930 144 2 𝑝 𝑝 PROPN cana-5930 144 3 + + NUM cana-5930 144 4 π‘ž π‘ž X cana-5930 144 5 βˆ’ βˆ’ PROPN cana-5930 144 6 2 2 X cana-5930 144 7 . . PUNCT cana-5930 145 1 let let VERB cana-5930 145 2 d d PRON cana-5930 145 3 be be AUX cana-5930 145 4 a a DET cana-5930 145 5 minimum minimum ADJ cana-5930 145 6 ctd ctd NOUN cana-5930 145 7 - - PUNCT cana-5930 145 8 set set NOUN cana-5930 145 9 of of ADP cana-5930 145 10 𝑇2(𝐺 𝑇2(𝐺 PROPN cana-5930 145 11 ) ) PUNCT cana-5930 145 12 having have VERB cana-5930 145 13 𝑝 𝑝 NOUN cana-5930 145 14 + + NUM cana-5930 145 15 π‘ž π‘ž X cana-5930 145 16 βˆ’ βˆ’ ADP cana-5930 145 17 2 2 NUM cana-5930 145 18 vertices vertex NOUN cana-5930 145 19 . . PUNCT cana-5930 146 1 since since SCONJ cana-5930 146 2 βŸ¨π‘‰(𝑇2(𝐺 βŸ¨π‘‰(𝑇2(𝐺 PROPN cana-5930 146 3 ) ) PUNCT cana-5930 146 4 ) ) PUNCT cana-5930 146 5 βˆ’ βˆ’ PROPN cana-5930 146 6 𝐷⟩ 𝐷⟩ PROPN cana-5930 146 7 β‰… β‰… NUM cana-5930 146 8 𝐾2 𝐾2 PROPN cana-5930 146 9 . . PUNCT cana-5930 147 1 let let VERB cana-5930 147 2 𝑉(𝑇2(𝐺 𝑉(𝑇2(𝐺 NOUN cana-5930 147 3 ) ) PUNCT cana-5930 147 4 ) ) PUNCT cana-5930 148 1 βˆ’ βˆ’ PROPN cana-5930 148 2 𝐷 𝐷 NOUN cana-5930 148 3 = = SYM cana-5930 148 4 { { PUNCT cana-5930 148 5 𝑒 𝑒 X cana-5930 148 6 , , PUNCT cana-5930 148 7 𝑣 𝑣 ADP cana-5930 148 8 } } PUNCT cana-5930 148 9 either either CCONJ cana-5930 148 10 ( ( PUNCT cana-5930 148 11 i i NOUN cana-5930 148 12 ) ) PUNCT cana-5930 148 13 𝑒 𝑒 PROPN cana-5930 148 14 , , PUNCT cana-5930 148 15 𝑣 𝑣 PRON cana-5930 148 16 ∈ ∈ PROPN cana-5930 148 17 𝑉(𝐺 𝑉(𝐺 NOUN cana-5930 148 18 ) ) PUNCT cana-5930 148 19 or or CCONJ cana-5930 148 20 ( ( PUNCT cana-5930 148 21 ii ii NOUN cana-5930 148 22 ) ) PUNCT cana-5930 148 23 𝑒 𝑒 PROPN cana-5930 148 24 ∈ ∈ PROPN cana-5930 148 25 𝑉(𝐺 𝑉(𝐺 NOUN cana-5930 148 26 ) ) PUNCT cana-5930 148 27 and and CCONJ cana-5930 148 28 𝑣 𝑣 AUX cana-5930 148 29 = = PUNCT cana-5930 148 30 𝑒′ 𝑒′ VERB cana-5930 148 31 ∈ ∈ PROPN cana-5930 148 32 𝑉(𝑇2(𝐺 𝑉(𝑇2(𝐺 NOUN cana-5930 148 33 ) ) PUNCT cana-5930 148 34 ) ) PUNCT cana-5930 149 1 βˆ’ βˆ’ PROPN cana-5930 149 2 𝑉(𝐺 𝑉(𝐺 NOUN cana-5930 149 3 ) ) PUNCT cana-5930 149 4 where where SCONJ cana-5930 149 5 𝑒′ 𝑒′ ADP cana-5930 149 6 ∈ ∈ PROPN cana-5930 149 7 ( ( PUNCT cana-5930 149 8 𝑒 𝑒 X cana-5930 149 9 , , PUNCT cana-5930 149 10 𝑣 𝑣 NOUN cana-5930 149 11 ) ) PUNCT cana-5930 149 12 . . PUNCT cana-5930 150 1 case case NOUN cana-5930 150 2 1 1 X cana-5930 150 3 . . PUNCT cana-5930 151 1 let let VERB cana-5930 151 2 𝑒′ 𝑒′ ADP cana-5930 151 3 ∈ ∈ PROPN cana-5930 151 4 𝐷 𝐷 NOUN cana-5930 151 5 where where SCONJ cana-5930 151 6 𝑒′ 𝑒′ ADJ cana-5930 151 7 be be AUX cana-5930 151 8 the the DET cana-5930 151 9 edge edge NOUN cana-5930 151 10 vertex vertex NOUN cana-5930 151 11 of of ADP cana-5930 151 12 ( ( PUNCT cana-5930 151 13 𝑒 𝑒 X cana-5930 151 14 , , PUNCT cana-5930 151 15 𝑣 𝑣 NOUN cana-5930 151 16 ) ) PUNCT cana-5930 151 17 in in ADP cana-5930 151 18 𝑇2(𝐺 𝑇2(𝐺 PROPN cana-5930 151 19 ) ) PUNCT cana-5930 151 20 . . PUNCT cana-5930 152 1 therefore therefore ADV cana-5930 152 2 , , PUNCT cana-5930 152 3 no no DET cana-5930 152 4 vertex vertex NOUN cana-5930 152 5 of of ADP cana-5930 152 6 𝑉(𝐺 𝑉(𝐺 NOUN cana-5930 152 7 ) ) PUNCT cana-5930 152 8 is be AUX cana-5930 152 9 an an DET cana-5930 152 10 element element NOUN cana-5930 152 11 of of ADP cana-5930 152 12 d d NOUN cana-5930 152 13 hence hence ADV cana-5930 152 14 𝑒′ 𝑒′ ADP cana-5930 152 15 ∈ ∈ PROPN cana-5930 152 16 𝐷. 𝐷. PROPN cana-5930 152 17 therefore therefore ADV cana-5930 152 18 𝐺 𝐺 PROPN cana-5930 152 19 β‰… β‰… PROPN cana-5930 152 20 𝐾2 𝐾2 PROPN cana-5930 152 21 . . PUNCT cana-5930 153 1 case case NOUN cana-5930 153 2 2 2 X cana-5930 153 3 . . PUNCT cana-5930 153 4 let let VERB cana-5930 153 5 𝑒 𝑒 PRON cana-5930 153 6 ∈ ∈ PROPN cana-5930 153 7 𝑉(𝐺 𝑉(𝐺 NOUN cana-5930 153 8 ) ) PUNCT cana-5930 153 9 , , PUNCT cana-5930 153 10 𝑒′ 𝑒′ ADP cana-5930 153 11 ∈ ∈ PROPN cana-5930 153 12 𝑉(𝑇2(𝐺 𝑉(𝑇2(𝐺 NOUN cana-5930 153 13 ) ) PUNCT cana-5930 153 14 ) ) PUNCT cana-5930 154 1 βˆ’ βˆ’ PROPN cana-5930 154 2 𝑉(𝐺 𝑉(𝐺 NOUN cana-5930 154 3 ) ) PUNCT cana-5930 154 4 . . PUNCT cana-5930 155 1 since since SCONJ cana-5930 155 2 βŸ¨π‘‰(𝑇2(𝐺 βŸ¨π‘‰(𝑇2(𝐺 PROPN cana-5930 155 3 ) ) PUNCT cana-5930 155 4 ) ) PUNCT cana-5930 155 5 βˆ’ βˆ’ PROPN cana-5930 155 6 𝐷⟩ 𝐷⟩ PROPN cana-5930 155 7 β‰… β‰… NUM cana-5930 155 8 𝐾2 𝐾2 PROPN cana-5930 155 9 . . PUNCT cana-5930 155 10 𝑒′ 𝑒′ PROPN cana-5930 155 11 is be AUX cana-5930 155 12 adjacent adjacent ADJ cana-5930 155 13 to to ADP cana-5930 155 14 u u NOUN cana-5930 155 15 in in ADP cana-5930 155 16 βŸ¨π‘‰(𝑇2(𝐺 βŸ¨π‘‰(𝑇2(𝐺 NOUN cana-5930 155 17 ) ) PUNCT cana-5930 155 18 ) ) PUNCT cana-5930 156 1 βˆ’ βˆ’ PROPN cana-5930 156 2 𝐷 𝐷 NOUN cana-5930 156 3 ⟩. ⟩. PROPN cana-5930 156 4 that that PRON cana-5930 156 5 is be AUX cana-5930 156 6 v v NOUN cana-5930 156 7 is be AUX cana-5930 156 8 adjacent adjacent ADJ cana-5930 156 9 to to ADP cana-5930 156 10 a a DET cana-5930 156 11 vertex vertex NOUN cana-5930 156 12 u u NOUN cana-5930 156 13 in in ADP cana-5930 156 14 g. g. PROPN cana-5930 156 15 therefore therefore ADV cana-5930 156 16 , , PUNCT cana-5930 156 17 𝐺 𝐺 PROPN cana-5930 156 18 β‰… β‰… PROPN cana-5930 156 19 𝐾2 𝐾2 PROPN cana-5930 156 20 . . PUNCT cana-5930 157 1 conversely conversely ADV cana-5930 157 2 , , PUNCT cana-5930 157 3 if if SCONJ cana-5930 157 4 𝐺 𝐺 PROPN cana-5930 157 5 β‰… β‰… NOUN cana-5930 157 6 𝐾2 𝐾2 PROPN cana-5930 157 7 then then ADV cana-5930 157 8 γ𝑐𝑑𝑑(𝑇2(𝐺 γ𝑐𝑑𝑑(𝑇2(𝐺 NOUN cana-5930 157 9 ) ) PUNCT cana-5930 157 10 ) ) PUNCT cana-5930 158 1 = = PUNCT cana-5930 158 2 𝑝 𝑝 PROPN cana-5930 158 3 + + NUM cana-5930 158 4 π‘ž π‘ž X cana-5930 158 5 βˆ’ βˆ’ PROPN cana-5930 158 6 2 2 NUM cana-5930 158 7 . . PUNCT cana-5930 158 8 β–‘ β–‘ PUNCT cana-5930 158 9 remark remark NOUN cana-5930 158 10 4.3 4.3 NUM cana-5930 158 11 . . PUNCT cana-5930 159 1 if if SCONJ cana-5930 159 2 𝑝 𝑝 PROPN cana-5930 159 3 β‰₯ β‰₯ NOUN cana-5930 159 4 3 3 NUM cana-5930 159 5 then then ADV cana-5930 159 6 γ𝑐𝑑𝑑(𝑇2(𝐺 γ𝑐𝑑𝑑(𝑇2(𝐺 NOUN cana-5930 159 7 ) ) PUNCT cana-5930 159 8 ) ) PUNCT cana-5930 160 1 ≀ ≀ PROPN cana-5930 160 2 𝑝 𝑝 ADP cana-5930 160 3 + + NUM cana-5930 160 4 π‘ž π‘ž X cana-5930 160 5 βˆ’ βˆ’ PROPN cana-5930 160 6 3 3 X cana-5930 160 7 . . X cana-5930 161 1 equality equality NOUN cana-5930 161 2 holds hold VERB cana-5930 161 3 if if SCONJ cana-5930 161 4 𝐺 𝐺 PROPN cana-5930 161 5 β‰… β‰… PROPN cana-5930 161 6 𝑃3 𝑃3 PROPN cana-5930 161 7 . . PUNCT cana-5930 161 8 theorem theorem VERB cana-5930 161 9 4.4 4.4 NUM cana-5930 161 10 . . PUNCT cana-5930 162 1 let let VERB cana-5930 162 2 𝐺(𝑝 𝐺(𝑝 NUM cana-5930 162 3 , , PUNCT cana-5930 162 4 π‘ž π‘ž NOUN cana-5930 162 5 ) ) PUNCT cana-5930 162 6 be be VERB cana-5930 162 7 a a DET cana-5930 162 8 complete complete ADJ cana-5930 162 9 graph graph NOUN cana-5930 162 10 with with ADP cana-5930 162 11 4 4 NUM cana-5930 162 12 ≀ ≀ NOUN cana-5930 162 13 𝑝 𝑝 NOUN cana-5930 162 14 ≀ ≀ NUM cana-5930 162 15 8 8 NUM cana-5930 162 16 then then ADV cana-5930 162 17 γ𝑐𝑑𝑑(𝑇2(𝐺 γ𝑐𝑑𝑑(𝑇2(𝐺 NOUN cana-5930 162 18 ) ) PUNCT cana-5930 162 19 ) ) PUNCT cana-5930 163 1 ≀ ≀ NUM cana-5930 163 2 ⌈ ⌈ X cana-5930 163 3 𝑝+π‘ž 𝑝+π‘ž NOUN cana-5930 163 4 2 2 NUM cana-5930 163 5 βŒ‰. βŒ‰. ADJ cana-5930 163 6 proof proof NOUN cana-5930 163 7 . . PUNCT cana-5930 164 1 we we PRON cana-5930 164 2 prove prove VERB cana-5930 164 3 induction induction NOUN cana-5930 164 4 on on ADP cana-5930 164 5 p. p. NOUN cana-5930 164 6 let let VERB cana-5930 164 7 𝑝 𝑝 NOUN cana-5930 164 8 = = SYM cana-5930 164 9 4 4 NUM cana-5930 164 10 , , PUNCT cana-5930 164 11 𝑒 𝑒 X cana-5930 164 12 = = PUNCT cana-5930 164 13 ( ( PUNCT cana-5930 164 14 𝑒1 𝑒1 PROPN cana-5930 164 15 , , PUNCT cana-5930 164 16 𝑒2 𝑒2 PROPN cana-5930 164 17 ) ) PUNCT cana-5930 164 18 ∈ ∈ PROPN cana-5930 164 19 𝐸(𝐺 𝐸(𝐺 PROPN cana-5930 164 20 ) ) PUNCT cana-5930 164 21 where where SCONJ cana-5930 164 22 𝑒1 𝑒1 PROPN cana-5930 164 23 , , PUNCT cana-5930 164 24 𝑒2 𝑒2 PROPN cana-5930 164 25 ∈ ∈ PROPN cana-5930 164 26 𝑉(𝐺 𝑉(𝐺 NOUN cana-5930 164 27 ) ) PUNCT cana-5930 164 28 . . PUNCT cana-5930 165 1 let let VERB cana-5930 165 2 𝐷 𝐷 NOUN cana-5930 165 3 = = SYM cana-5930 165 4 { { PUNCT cana-5930 165 5 𝑒1 𝑒1 PROPN cana-5930 165 6 , , PUNCT cana-5930 165 7 𝑒2 𝑒2 PROPN cana-5930 165 8 , , PUNCT cana-5930 165 9 𝑒12 𝑒12 NOUN cana-5930 165 10 β€² β€² NUM cana-5930 165 11 , , PUNCT cana-5930 165 12 𝑒34 𝑒34 ADP cana-5930 165 13 β€² β€² NUM cana-5930 165 14 } } PUNCT cana-5930 165 15 βŠ† βŠ† NUM cana-5930 165 16 𝑉(𝑇2(𝐺 𝑉(𝑇2(𝐺 NOUN cana-5930 165 17 ) ) PUNCT cana-5930 165 18 ) ) PUNCT cana-5930 166 1 is be AUX cana-5930 166 2 a a DET cana-5930 166 3 ctd ctd NOUN cana-5930 166 4 - - PUNCT cana-5930 166 5 set set NOUN cana-5930 166 6 of of ADP cana-5930 166 7 𝑇2(𝐺 𝑇2(𝐺 PROPN cana-5930 166 8 ) ) PUNCT cana-5930 166 9 where where SCONJ cana-5930 166 10 𝑒12 𝑒12 NOUN cana-5930 166 11 β€² β€² NUM cana-5930 166 12 , , PUNCT cana-5930 166 13 𝑒34 𝑒34 ADP cana-5930 166 14 β€² β€² NUM cana-5930 166 15 ∈ ∈ PROPN cana-5930 166 16 𝑉(𝑇2(𝐺 𝑉(𝑇2(𝐺 NOUN cana-5930 166 17 ) ) PUNCT cana-5930 166 18 ) ) PUNCT cana-5930 166 19 βˆ’ βˆ’ PROPN cana-5930 166 20 𝑉(𝐺 𝑉(𝐺 NOUN cana-5930 166 21 ) ) PUNCT cana-5930 166 22 and and CCONJ cana-5930 166 23 𝐷′ 𝐷′ NOUN cana-5930 166 24 = = SYM cana-5930 166 25 𝑉(𝑇2(𝐺 𝑉(𝑇2(𝐺 NOUN cana-5930 166 26 ) ) PUNCT cana-5930 166 27 ) ) PUNCT cana-5930 167 1 βˆ’ βˆ’ PROPN cana-5930 167 2 𝐷 𝐷 NOUN cana-5930 167 3 = = SYM cana-5930 167 4 { { PUNCT cana-5930 167 5 𝑒3 𝑒3 PROPN cana-5930 167 6 , , PUNCT cana-5930 167 7 𝑒4 𝑒4 NOUN cana-5930 167 8 , , PUNCT cana-5930 167 9 𝑒13 𝑒13 NOUN cana-5930 167 10 β€² β€² NOUN cana-5930 167 11 , , PUNCT cana-5930 167 12 𝑒23 𝑒23 ADJ cana-5930 167 13 β€² β€² NUM cana-5930 167 14 , , PUNCT cana-5930 167 15 𝑒24 𝑒24 NOUN cana-5930 167 16 β€² β€² NUM cana-5930 167 17 , , PUNCT cana-5930 167 18 𝑒41 𝑒41 NOUN cana-5930 167 19 β€² β€² NOUN cana-5930 167 20 } } PUNCT cana-5930 167 21 since since SCONJ cana-5930 167 22 𝑝 𝑝 PROPN cana-5930 167 23 β‰₯ β‰₯ NOUN cana-5930 167 24 4 4 NUM cana-5930 167 25 and and CCONJ cana-5930 167 26 Ξ΄(𝐺 Ξ΄(𝐺 NUM cana-5930 167 27 ) ) PUNCT cana-5930 167 28 β‰₯ β‰₯ NOUN cana-5930 167 29 2 2 NUM cana-5930 167 30 each each DET cana-5930 167 31 vertex vertex NOUN cana-5930 167 32 in in ADP cana-5930 167 33 βŸ¨π‘‰(𝑇2(𝐺 βŸ¨π‘‰(𝑇2(𝐺 PROPN cana-5930 167 34 ) ) PUNCT cana-5930 167 35 ) ) PUNCT cana-5930 168 1 βˆ’ βˆ’ PROPN cana-5930 168 2 𝐷′ 𝐷′ PROPN cana-5930 168 3 ⟩ ⟩ NOUN cana-5930 168 4 is be AUX cana-5930 168 5 adjacent adjacent ADJ cana-5930 168 6 to to PART cana-5930 168 7 atleast atleast VERB cana-5930 168 8 one one NUM cana-5930 168 9 vertex vertex NOUN cana-5930 168 10 in in ADP cana-5930 168 11 d d NOUN cana-5930 168 12 and and CCONJ cana-5930 168 13 βŸ¨π‘‰(𝑇2(𝐺 βŸ¨π‘‰(𝑇2(𝐺 PROPN cana-5930 168 14 ) ) PUNCT cana-5930 168 15 ) ) PUNCT cana-5930 169 1 βˆ’ βˆ’ PROPN cana-5930 170 1 𝐷⟩ 𝐷⟩ PROPN cana-5930 170 2 β‰… β‰… PROPN cana-5930 170 3 π‘†π‘š,π‘š,π‘š π‘†π‘š,π‘š,π‘š PROPN cana-5930 170 4 , , PUNCT cana-5930 170 5 π‘š π‘š X cana-5930 170 6 β‰₯ β‰₯ NUM cana-5930 170 7 2 2 NUM cana-5930 170 8 in in ADP cana-5930 170 9 𝑇2(𝐺 𝑇2(𝐺 PROPN cana-5930 170 10 ) ) PUNCT cana-5930 170 11 . . PUNCT cana-5930 171 1 therefore therefore ADV cana-5930 171 2 , , PUNCT cana-5930 171 3 |𝐷| |𝐷| X cana-5930 171 4 ≀ ≀ X cana-5930 171 5 ⌈ ⌈ NOUN cana-5930 171 6 𝑝+π‘ž 𝑝+π‘ž NOUN cana-5930 171 7 2 2 NUM cana-5930 171 8 βŒ‰. βŒ‰. ADV cana-5930 171 9 hence hence ADV cana-5930 171 10 γ𝑐𝑑𝑑(𝑇2(𝐺 γ𝑐𝑑𝑑(𝑇2(𝐺 NOUN cana-5930 171 11 ) ) PUNCT cana-5930 171 12 ) ) PUNCT cana-5930 172 1 ≀ ≀ NUM cana-5930 172 2 ⌈ ⌈ X cana-5930 172 3 𝑝+π‘ž 𝑝+π‘ž NOUN cana-5930 172 4 2 2 NUM cana-5930 172 5 βŒ‰. βŒ‰. ADJ cana-5930 172 6 equality equality NOUN cana-5930 172 7 holds hold VERB cana-5930 172 8 if if SCONJ cana-5930 172 9 𝐺 𝐺 PROPN cana-5930 172 10 β‰… β‰… PROPN cana-5930 172 11 𝐾8 𝐾8 PROPN cana-5930 172 12 . . PUNCT cana-5930 173 1 β–‘ β–‘ PUNCT cana-5930 173 2 communications communication NOUN cana-5930 173 3 on on ADP cana-5930 173 4 applied apply VERB cana-5930 173 5 nonlinear nonlinear ADJ cana-5930 173 6 analysis analysis NOUN cana-5930 173 7 issn issn NOUN cana-5930 173 8 : : PUNCT cana-5930 173 9 1074 1074 NUM cana-5930 173 10 - - PUNCT cana-5930 173 11 133x 133x NUM cana-5930 173 12 vol vol VERB cana-5930 173 13 32 32 NUM cana-5930 173 14 no no NOUN cana-5930 173 15 . . PUNCT cana-5930 174 1 10s 10 NOUN cana-5930 174 2 ( ( PUNCT cana-5930 174 3 2025 2025 NUM cana-5930 174 4 ) ) PUNCT cana-5930 174 5 3108 3108 NUM cana-5930 174 6 https://internationalpubls.com https://internationalpubls.com X cana-5930 174 7 theorem theorem VERB cana-5930 174 8 4.5 4.5 NUM cana-5930 174 9 . . PUNCT cana-5930 175 1 let let VERB cana-5930 175 2 g g PRON cana-5930 175 3 be be AUX cana-5930 175 4 a a DET cana-5930 175 5 connected connected ADJ cana-5930 175 6 graph graph NOUN cana-5930 175 7 such such ADJ cana-5930 175 8 that that SCONJ cana-5930 175 9 Ξ΄(𝐺 Ξ΄(𝐺 PROPN cana-5930 175 10 ) ) PUNCT cana-5930 175 11 β‰₯ β‰₯ NOUN cana-5930 175 12 2 2 NUM cana-5930 175 13 then then ADV cana-5930 175 14 γ𝑐𝑑𝑑(𝑇2(𝐺 γ𝑐𝑑𝑑(𝑇2(𝐺 NOUN cana-5930 175 15 ) ) PUNCT cana-5930 175 16 ) ) PUNCT cana-5930 176 1 ≀ ≀ NOUN cana-5930 176 2 π‘ž π‘ž X cana-5930 176 3 βˆ’ βˆ’ PROPN cana-5930 176 4 Ξ΄(𝐺 Ξ΄(𝐺 PROPN cana-5930 176 5 ) ) PUNCT cana-5930 177 1 + + CCONJ cana-5930 177 2 1 1 X cana-5930 177 3 . . X cana-5930 177 4 proof proof NOUN cana-5930 177 5 . . PUNCT cana-5930 178 1 let let VERB cana-5930 178 2 v v PART cana-5930 178 3 be be AUX cana-5930 178 4 a a DET cana-5930 178 5 vertex vertex NOUN cana-5930 178 6 of of ADP cana-5930 178 7 maximum maximum ADJ cana-5930 178 8 degree degree NOUN cana-5930 178 9 in in ADP cana-5930 178 10 g. g. PROPN cana-5930 178 11 let let VERB cana-5930 178 12 𝑆 𝑆 PROPN cana-5930 178 13 = = SYM cana-5930 178 14 { { PUNCT cana-5930 178 15 𝑣𝑖 𝑣𝑖 NOUN cana-5930 178 16 β€² β€² NUM cana-5930 178 17 : : PUNCT cana-5930 178 18 ( ( PUNCT cana-5930 178 19 𝑣 𝑣 NOUN cana-5930 178 20 , , PUNCT cana-5930 178 21 𝑣𝑖 𝑣𝑖 NOUN cana-5930 178 22 ) ) PUNCT cana-5930 178 23 ∈ ∈ PROPN cana-5930 178 24 𝐸(𝐺 𝐸(𝐺 NOUN cana-5930 178 25 ) ) PUNCT cana-5930 178 26 , , PUNCT cana-5930 178 27 𝑖 𝑖 NOUN cana-5930 178 28 = = SYM cana-5930 178 29 1 1 NUM cana-5930 178 30 , , PUNCT cana-5930 178 31 … … PUNCT cana-5930 178 32 , , PUNCT cana-5930 178 33 Ξ΄(𝐺 Ξ΄(𝐺 PROPN cana-5930 178 34 ) ) PUNCT cana-5930 178 35 } } PUNCT cana-5930 178 36 . . PUNCT cana-5930 179 1 any any DET cana-5930 179 2 set set NOUN cana-5930 179 3 𝐷 𝐷 NOUN cana-5930 179 4 βŠ† βŠ† NUM cana-5930 179 5 𝑉(𝑇2(𝐺 𝑉(𝑇2(𝐺 NOUN cana-5930 179 6 ) ) PUNCT cana-5930 179 7 ) ) PUNCT cana-5930 179 8 such such ADJ cana-5930 179 9 that that DET cana-5930 179 10 v(𝑇2(𝐺 v(𝑇2(𝐺 NOUN cana-5930 179 11 ) ) PUNCT cana-5930 179 12 ) ) PUNCT cana-5930 180 1 βˆ’ βˆ’ PROPN cana-5930 181 1 d d NOUN cana-5930 181 2 = = PRON cana-5930 181 3 { { PUNCT cana-5930 181 4 ⋃ ⋃ NOUN cana-5930 181 5 𝑣𝑖 𝑣𝑖 NOUN cana-5930 181 6 π‘βˆ’1 π‘βˆ’1 PROPN cana-5930 181 7 𝑖=1 𝑖=1 PUNCT cana-5930 181 8 } } PUNCT cana-5930 181 9 βˆͺ βˆͺ ADP cana-5930 181 10 𝑆. 𝑆. PROPN cana-5930 181 11 since since SCONJ cana-5930 181 12 Ξ΄(𝐺 Ξ΄(𝐺 NUM cana-5930 181 13 ) ) PUNCT cana-5930 181 14 β‰₯ β‰₯ NOUN cana-5930 181 15 2 2 NUM cana-5930 181 16 , , PUNCT cana-5930 181 17 𝑑𝑒𝑔(𝑣𝑖 𝑑𝑒𝑔(𝑣𝑖 NUM cana-5930 181 18 ) ) PUNCT cana-5930 181 19 β‰₯ β‰₯ NOUN cana-5930 181 20 2 2 NUM cana-5930 181 21 , , PUNCT cana-5930 181 22 𝑖 𝑖 NOUN cana-5930 181 23 = = SYM cana-5930 181 24 1,2 1,2 NUM cana-5930 181 25 , , PUNCT cana-5930 181 26 … … PUNCT cana-5930 181 27 , , PUNCT cana-5930 181 28 𝑝 𝑝 X cana-5930 181 29 βˆ’ βˆ’ PROPN cana-5930 181 30 1 1 NUM cana-5930 181 31 and and CCONJ cana-5930 181 32 hence hence ADV cana-5930 181 33 𝑣𝑖 𝑣𝑖 ADV cana-5930 181 34 is be AUX cana-5930 181 35 adjacent adjacent ADJ cana-5930 181 36 to to ADP cana-5930 181 37 the the DET cana-5930 181 38 vertices vertex NOUN cana-5930 181 39 of of ADP cana-5930 181 40 g g NOUN cana-5930 181 41 other other ADJ cana-5930 181 42 than than ADP cana-5930 181 43 v v NOUN cana-5930 181 44 , , PUNCT cana-5930 181 45 then then ADV cana-5930 181 46 𝑣𝑖 𝑣𝑖 SCONJ cana-5930 181 47 β€² β€² NOUN cana-5930 181 48 is be AUX cana-5930 181 49 adjacent adjacent ADJ cana-5930 181 50 to to ADP cana-5930 181 51 the the DET cana-5930 181 52 vertex vertex NOUN cana-5930 181 53 𝑣𝑖 𝑣𝑖 NOUN cana-5930 181 54 , , PUNCT cana-5930 181 55 𝑖 𝑖 NOUN cana-5930 181 56 = = SYM cana-5930 181 57 1 1 NUM cana-5930 181 58 , , PUNCT cana-5930 181 59 2 2 NUM cana-5930 181 60 , , PUNCT cana-5930 181 61 … … PUNCT cana-5930 181 62 , , PUNCT cana-5930 181 63 𝑝 𝑝 X cana-5930 181 64 βˆ’ βˆ’ PROPN cana-5930 181 65 1 1 NUM cana-5930 181 66 where where SCONJ cana-5930 181 67 𝑣𝑖 𝑣𝑖 ADV cana-5930 181 68 is be AUX cana-5930 181 69 a a DET cana-5930 181 70 vertex vertex NOUN cana-5930 181 71 in in ADP cana-5930 181 72 𝑉(𝑇2(𝐺 𝑉(𝑇2(𝐺 NOUN cana-5930 181 73 ) ) PUNCT cana-5930 181 74 ) ) PUNCT cana-5930 182 1 βˆ’ βˆ’ ADP cana-5930 182 2 𝐷. 𝐷. PROPN cana-5930 182 3 also also ADV cana-5930 182 4 , , PUNCT cana-5930 182 5 v v PRON cana-5930 182 6 is be AUX cana-5930 182 7 adjacent adjacent ADJ cana-5930 182 8 to to PART cana-5930 182 9 atleast atleast VERB cana-5930 182 10 one one NUM cana-5930 182 11 vertex vertex NOUN cana-5930 182 12 of of ADP cana-5930 182 13 g g NOUN cana-5930 182 14 and and CCONJ cana-5930 182 15 hence hence ADV cana-5930 182 16 in in ADP cana-5930 182 17 s. s. PROPN cana-5930 182 18 therefore therefore ADV cana-5930 182 19 , , PUNCT cana-5930 182 20 d d PROPN cana-5930 182 21 is be AUX cana-5930 182 22 a a DET cana-5930 182 23 dominating dominating NOUN cana-5930 182 24 set set NOUN cana-5930 182 25 of of ADP cana-5930 182 26 𝑉(𝑇2(𝐺 𝑉(𝑇2(𝐺 NOUN cana-5930 182 27 ) ) PUNCT cana-5930 182 28 ) ) PUNCT cana-5930 182 29 . . PUNCT cana-5930 183 1 moreover moreover ADV cana-5930 183 2 βŸ¨π‘‰(𝑇2(𝐺 βŸ¨π‘‰(𝑇2(𝐺 PROPN cana-5930 183 3 ) ) PUNCT cana-5930 183 4 ) ) PUNCT cana-5930 184 1 βˆ’ βˆ’ PROPN cana-5930 185 1 𝐷⟩ 𝐷⟩ PROPN cana-5930 185 2 β‰… β‰… PROPN cana-5930 185 3 𝑇 𝑇 PROPN cana-5930 185 4 ∘ ∘ PROPN cana-5930 185 5 𝐾1 𝐾1 PROPN cana-5930 185 6 and and CCONJ cana-5930 185 7 hence hence ADV cana-5930 185 8 d d PRON cana-5930 185 9 is be AUX cana-5930 185 10 a a DET cana-5930 185 11 ctd ctd NOUN cana-5930 185 12 - - PUNCT cana-5930 185 13 set set NOUN cana-5930 185 14 of of ADP cana-5930 185 15 𝑇2(𝐺 𝑇2(𝐺 PROPN cana-5930 185 16 ) ) PUNCT cana-5930 185 17 . . PUNCT cana-5930 186 1 therefore therefore ADV cana-5930 186 2 , , PUNCT cana-5930 186 3 γ𝑐𝑑𝑑(𝑇2(𝐺 γ𝑐𝑑𝑑(𝑇2(𝐺 NOUN cana-5930 186 4 ) ) PUNCT cana-5930 186 5 ) ) PUNCT cana-5930 187 1 ≀ ≀ NUM cana-5930 187 2 |𝑉(𝑇2(𝐺 |𝑉(𝑇2(𝐺 NOUN cana-5930 187 3 ) ) PUNCT cana-5930 187 4 ) ) PUNCT cana-5930 188 1 βˆ’ βˆ’ PROPN cana-5930 188 2 ( ( PUNCT cana-5930 188 3 𝑉(𝐺 𝑉(𝐺 NOUN cana-5930 188 4 ) ) PUNCT cana-5930 188 5 βˆ’ βˆ’ ADP cana-5930 189 1 𝑣 𝑣 X cana-5930 189 2 ) ) PUNCT cana-5930 189 3 βˆ’ βˆ’ NOUN cana-5930 189 4 𝑆| 𝑆| NOUN cana-5930 189 5 = = SYM cana-5930 189 6 𝑝 𝑝 PROPN cana-5930 189 7 + + CCONJ cana-5930 189 8 π‘ž π‘ž X cana-5930 189 9 βˆ’ βˆ’ PROPN cana-5930 189 10 𝑝 𝑝 PROPN cana-5930 190 1 + + CCONJ cana-5930 190 2 1 1 NUM cana-5930 190 3 βˆ’ βˆ’ PROPN cana-5930 190 4 Ξ΄(𝐺 Ξ΄(𝐺 NOUN cana-5930 190 5 ) ) PUNCT cana-5930 190 6 ≀ ≀ NOUN cana-5930 190 7 π‘ž π‘ž X cana-5930 190 8 βˆ’ βˆ’ PROPN cana-5930 190 9 Ξ΄(𝐺 Ξ΄(𝐺 PROPN cana-5930 190 10 ) ) PUNCT cana-5930 190 11 + + CCONJ cana-5930 190 12 1 1 NUM cana-5930 190 13 equality equality NOUN cana-5930 190 14 holds hold VERB cana-5930 190 15 if if SCONJ cana-5930 190 16 𝐺 𝐺 PROPN cana-5930 190 17 β‰… β‰… PROPN cana-5930 190 18 𝐾3 𝐾3 PROPN cana-5930 190 19 , , PUNCT cana-5930 190 20 𝐾4&𝐾1 𝐾4&𝐾1 PROPN cana-5930 190 21 + + CCONJ cana-5930 191 1 π‘ƒπ‘›βˆ’1 π‘ƒπ‘›βˆ’1 PROPN cana-5930 191 2 . . PUNCT cana-5930 192 1 β–‘ β–‘ PUNCT cana-5930 192 2 theorem theorem VERB cana-5930 192 3 4.6 4.6 NUM cana-5930 192 4 . . PUNCT cana-5930 193 1 let let VERB cana-5930 193 2 𝑇2(𝐺1(𝑝1 𝑇2(𝐺1(𝑝1 NOUN cana-5930 193 3 , , PUNCT cana-5930 193 4 π‘ž1 π‘ž1 NOUN cana-5930 193 5 ) ) PUNCT cana-5930 193 6 ) ) PUNCT cana-5930 193 7 and and CCONJ cana-5930 193 8 𝑇2(𝐺2(𝑝2 𝑇2(𝐺2(𝑝2 NUM cana-5930 193 9 , , PUNCT cana-5930 193 10 π‘ž2 π‘ž2 NOUN cana-5930 193 11 ) ) PUNCT cana-5930 193 12 ) ) PUNCT cana-5930 193 13 be be AUX cana-5930 193 14 two two NUM cana-5930 193 15 connected connected ADJ cana-5930 193 16 graphs graph NOUN cana-5930 193 17 of of ADP cana-5930 193 18 order order NOUN cana-5930 193 19 atleast atleast ADP cana-5930 193 20 two two NUM cana-5930 193 21 . . PUNCT cana-5930 194 1 let let VERB cana-5930 194 2 t t NOUN cana-5930 194 3 be be AUX cana-5930 194 4 an an DET cana-5930 194 5 induced induced ADJ cana-5930 194 6 sub sub NOUN cana-5930 194 7 - - NOUN cana-5930 194 8 graph graph NOUN cana-5930 194 9 of of ADP cana-5930 194 10 𝑇(𝐺1 𝑇(𝐺1 NOUN cana-5930 194 11 ) ) PUNCT cana-5930 194 12 having have VERB cana-5930 194 13 maximum maximum ADJ cana-5930 194 14 number number NOUN cana-5930 194 15 of of ADP cana-5930 194 16 vertices vertex NOUN cana-5930 194 17 such such ADJ cana-5930 194 18 that that SCONJ cana-5930 194 19 t t PROPN cana-5930 194 20 is be AUX cana-5930 194 21 a a DET cana-5930 194 22 tree tree NOUN cana-5930 194 23 . . PUNCT cana-5930 195 1 if if SCONJ cana-5930 195 2 Ξ²0 Ξ²0 PROPN cana-5930 195 3 is be AUX cana-5930 195 4 the the DET cana-5930 195 5 independence independence NOUN cana-5930 195 6 number number NOUN cana-5930 195 7 of of ADP cana-5930 195 8 𝑇2(𝐺2 𝑇2(𝐺2 NOUN cana-5930 195 9 ) ) PUNCT cana-5930 195 10 and and CCONJ cana-5930 195 11 vertices vertice VERB cana-5930 195 12 corresponding correspond VERB cana-5930 195 13 to to ADP cana-5930 195 14 the the DET cana-5930 195 15 edge edge NOUN cana-5930 195 16 joining join VERB cana-5930 195 17 from from ADP cana-5930 195 18 copies copy NOUN cana-5930 195 19 of of ADP cana-5930 195 20 𝑇(𝐺2 𝑇(𝐺2 NOUN cana-5930 195 21 ) ) PUNCT cana-5930 195 22 to to ADP cana-5930 195 23 𝑇(𝐺1 𝑇(𝐺1 NOUN cana-5930 195 24 ) ) PUNCT cana-5930 195 25 then then ADV cana-5930 195 26 γ𝑐𝑑𝑑(𝑇2(𝐺1 γ𝑐𝑑𝑑(𝑇2(𝐺1 VERB cana-5930 195 27 ∘ ∘ VERB cana-5930 195 28 𝐺2 𝐺2 NOUN cana-5930 195 29 ) ) PUNCT cana-5930 195 30 ) ) PUNCT cana-5930 195 31 ≀ ≀ NUM cana-5930 196 1 2𝑝1𝑝2 2𝑝1𝑝2 X cana-5930 196 2 + + CCONJ cana-5930 196 3 𝑝1(1 𝑝1(1 ADJ cana-5930 196 4 + + ADJ cana-5930 196 5 π‘ž2 π‘ž2 NOUN cana-5930 196 6 ) ) PUNCT cana-5930 197 1 + + CCONJ cana-5930 197 2 π‘ž1 π‘ž1 ADJ cana-5930 197 3 βˆ’ βˆ’ PROPN cana-5930 197 4 ( ( PUNCT cana-5930 197 5 𝑑 𝑑 PROPN cana-5930 197 6 βˆ’ βˆ’ PROPN cana-5930 197 7 2)Ξ²0 2)Ξ²0 ADJ cana-5930 197 8 βˆ’ βˆ’ NOUN cana-5930 197 9 𝑑. 𝑑. NOUN cana-5930 197 10 proof proof NOUN cana-5930 197 11 . . PUNCT cana-5930 198 1 let let VERB cana-5930 198 2 t t NOUN cana-5930 198 3 be be AUX cana-5930 198 4 an an DET cana-5930 198 5 induced induced ADJ cana-5930 198 6 sub sub NOUN cana-5930 198 7 - - NOUN cana-5930 198 8 graph graph NOUN cana-5930 198 9 of of ADP cana-5930 198 10 𝑇(𝐺1 𝑇(𝐺1 NOUN cana-5930 198 11 ) ) PUNCT cana-5930 198 12 having have VERB cana-5930 198 13 maximum maximum ADJ cana-5930 198 14 number number NOUN cana-5930 198 15 of of ADP cana-5930 198 16 vertices vertex NOUN cana-5930 198 17 such such ADJ cana-5930 198 18 that that SCONJ cana-5930 198 19 t t PROPN cana-5930 198 20 is be AUX cana-5930 198 21 a a DET cana-5930 198 22 tree tree NOUN cana-5930 198 23 and and CCONJ cana-5930 198 24 |𝑇| |𝑇| NOUN cana-5930 199 1 = = PUNCT cana-5930 199 2 𝑑. 𝑑. NOUN cana-5930 199 3 let let VERB cana-5930 199 4 s s PRON cana-5930 199 5 be be AUX cana-5930 199 6 a a DET cana-5930 199 7 maximum maximum ADJ cana-5930 199 8 independent independent ADJ cana-5930 199 9 set set NOUN cana-5930 199 10 of of ADP cana-5930 199 11 𝑇(𝐺2 𝑇(𝐺2 PROPN cana-5930 199 12 ) ) PUNCT cana-5930 199 13 and and CCONJ cana-5930 199 14 vertices vertice VERB cana-5930 199 15 corresponding correspond VERB cana-5930 199 16 to to ADP cana-5930 199 17 the the DET cana-5930 199 18 edge edge NOUN cana-5930 199 19 joining join VERB cana-5930 199 20 to to ADP cana-5930 199 21 the the DET cana-5930 199 22 vertices vertex NOUN cana-5930 199 23 of of ADP cana-5930 199 24 each each DET cana-5930 199 25 copies copy NOUN cana-5930 199 26 of of ADP cana-5930 199 27 𝑇(𝐺2 𝑇(𝐺2 NOUN cana-5930 199 28 ) ) PUNCT cana-5930 199 29 to to ADP cana-5930 199 30 𝑇(𝐺1 𝑇(𝐺1 NOUN cana-5930 199 31 ) ) PUNCT cana-5930 199 32 such such ADJ cana-5930 199 33 that that SCONJ cana-5930 199 34 |𝑆| |𝑆| VERB cana-5930 199 35 = = SYM cana-5930 199 36 Ξ²0 Ξ²0 NOUN cana-5930 199 37 and and CCONJ cana-5930 199 38 𝐷′ 𝐷′ NOUN cana-5930 199 39 be be AUX cana-5930 199 40 the the DET cana-5930 199 41 set set NOUN cana-5930 199 42 of of ADP cana-5930 199 43 vertices vertex NOUN cana-5930 199 44 in in ADP cana-5930 199 45 s s PRON cana-5930 199 46 in in ADP cana-5930 199 47 copies copy NOUN cana-5930 199 48 of of ADP cana-5930 199 49 𝑇(𝐺2 𝑇(𝐺2 PROPN cana-5930 199 50 ) ) PUNCT cana-5930 199 51 which which PRON cana-5930 199 52 are be AUX cana-5930 199 53 adjacent adjacent ADJ cana-5930 199 54 to to ADP cana-5930 199 55 the the DET cana-5930 199 56 vertices vertex NOUN cana-5930 199 57 of of ADP cana-5930 199 58 t t NOUN cana-5930 199 59 then then ADV cana-5930 199 60 |𝐷′| |𝐷′| X cana-5930 199 61 = = PUNCT cana-5930 199 62 ( ( PUNCT cana-5930 199 63 𝑑 𝑑 PROPN cana-5930 199 64 βˆ’ βˆ’ PROPN cana-5930 199 65 2)Ξ²0 2)Ξ²0 PROPN cana-5930 199 66 if if SCONJ cana-5930 199 67 Ξ΄(𝐺1 Ξ΄(𝐺1 PROPN cana-5930 199 68 ) ) PUNCT cana-5930 199 69 β‰₯ β‰₯ NOUN cana-5930 199 70 2 2 X cana-5930 199 71 . . PUNCT cana-5930 200 1 let let VERB cana-5930 200 2 𝐷 𝐷 NOUN cana-5930 200 3 = = PUNCT cana-5930 200 4 ( ( PUNCT cana-5930 200 5 𝑉(𝑇2(𝐺1 𝑉(𝑇2(𝐺1 PROPN cana-5930 200 6 ∘ ∘ NOUN cana-5930 200 7 𝐺2 𝐺2 NOUN cana-5930 200 8 ) ) PUNCT cana-5930 200 9 ) ) PUNCT cana-5930 200 10 ) ) PUNCT cana-5930 201 1 βˆ’ βˆ’ PROPN cana-5930 201 2 ( ( PUNCT cana-5930 201 3 𝑉(𝑇 𝑉(𝑇 NOUN cana-5930 201 4 ) ) PUNCT cana-5930 201 5 βˆͺ βˆͺ ADP cana-5930 201 6 𝐷′ 𝐷′ NOUN cana-5930 201 7 ) ) PUNCT cana-5930 201 8 then then ADV cana-5930 201 9 𝑉(𝑇2(𝐺1 𝑉(𝑇2(𝐺1 VERB cana-5930 201 10 ∘ ∘ PROPN cana-5930 201 11 𝐺2 𝐺2 NOUN cana-5930 201 12 ) ) PUNCT cana-5930 201 13 ) ) PUNCT cana-5930 202 1 βˆ’ βˆ’ PROPN cana-5930 202 2 𝐷 𝐷 NOUN cana-5930 202 3 = = PUNCT cana-5930 202 4 𝑉(𝑇 𝑉(𝑇 NOUN cana-5930 202 5 ) ) PUNCT cana-5930 202 6 βˆͺ βˆͺ NOUN cana-5930 202 7 𝐷′ 𝐷′ PROPN cana-5930 202 8 and and CCONJ cana-5930 202 9 ( ( PUNCT cana-5930 202 10 𝑑 𝑑 X cana-5930 202 11 βˆ’ βˆ’ NOUN cana-5930 202 12 2 2 NUM cana-5930 202 13 ) ) PUNCT cana-5930 202 14 vertices vertex NOUN cana-5930 202 15 of of ADP cana-5930 202 16 𝑉(𝑇 𝑉(𝑇 NOUN cana-5930 202 17 ) ) PUNCT cana-5930 202 18 are be AUX cana-5930 202 19 adjacent adjacent ADJ cana-5930 202 20 to to ADP cana-5930 202 21 ( ( PUNCT cana-5930 202 22 𝑝2 𝑝2 NOUN cana-5930 202 23 βˆ’ βˆ’ PROPN cana-5930 202 24 Ξ²0 Ξ²0 PROPN cana-5930 202 25 ) ) PUNCT cana-5930 202 26 vertices vertex NOUN cana-5930 202 27 in in ADP cana-5930 202 28 a a DET cana-5930 202 29 copy copy NOUN cana-5930 202 30 of of ADP cana-5930 202 31 𝑇2(𝐺2 𝑇2(𝐺2 NOUN cana-5930 202 32 ) ) PUNCT cana-5930 202 33 . . PUNCT cana-5930 203 1 also also ADV cana-5930 203 2 each each DET cana-5930 203 3 vertex vertex NOUN cana-5930 203 4 in in ADP cana-5930 203 5 𝐷′ 𝐷′ PROPN cana-5930 203 6 is be AUX cana-5930 203 7 adjacent adjacent ADJ cana-5930 203 8 to to PART cana-5930 203 9 atleast atleast VERB cana-5930 203 10 one one NUM cana-5930 203 11 ( ( PUNCT cana-5930 203 12 𝑝2 𝑝2 NOUN cana-5930 203 13 βˆ’ βˆ’ PROPN cana-5930 203 14 Ξ²0 Ξ²0 PROPN cana-5930 203 15 ) ) PUNCT cana-5930 203 16 vertices vertex NOUN cana-5930 203 17 in in ADP cana-5930 203 18 a a DET cana-5930 203 19 copy copy NOUN cana-5930 203 20 of of ADP cana-5930 203 21 𝑇2(𝐺2 𝑇2(𝐺2 NOUN cana-5930 203 22 ) ) PUNCT cana-5930 203 23 . . PUNCT cana-5930 204 1 therefore therefore ADV cana-5930 204 2 , , PUNCT cana-5930 204 3 d d X cana-5930 204 4 is be AUX cana-5930 204 5 a a DET cana-5930 204 6 dominating dominating NOUN cana-5930 204 7 set set NOUN cana-5930 204 8 of of ADP cana-5930 204 9 𝑇2(𝐺1 𝑇2(𝐺1 NOUN cana-5930 204 10 ∘ ∘ NOUN cana-5930 204 11 𝐺2 𝐺2 ADV cana-5930 204 12 ) ) PUNCT cana-5930 204 13 and and CCONJ cana-5930 204 14 βŸ¨π‘‰(𝑇2(𝐺1 βŸ¨π‘‰(𝑇2(𝐺1 NOUN cana-5930 204 15 ∘ ∘ NUM cana-5930 204 16 𝐺2 𝐺2 ADJ cana-5930 204 17 ) ) PUNCT cana-5930 204 18 ) ) PUNCT cana-5930 205 1 βˆ’ βˆ’ PROPN cana-5930 205 2 𝐷 𝐷 NOUN cana-5930 205 3 ⟩ ⟩ NOUN cana-5930 205 4 is be AUX cana-5930 205 5 a a DET cana-5930 205 6 tree tree NOUN cana-5930 205 7 . . PUNCT cana-5930 206 1 γ𝑐𝑑𝑑(𝑇2(𝐺1 γ𝑐𝑑𝑑(𝑇2(𝐺1 NOUN cana-5930 206 2 ∘ ∘ VERB cana-5930 206 3 𝐺2 𝐺2 NOUN cana-5930 206 4 ) ) PUNCT cana-5930 206 5 ) ) PUNCT cana-5930 207 1 ≀ ≀ NUM cana-5930 207 2 |𝐷| |𝐷| X cana-5930 207 3 ≀ ≀ NUM cana-5930 207 4 |𝑉(𝑇2(𝐺1 |𝑉(𝑇2(𝐺1 NOUN cana-5930 207 5 ∘ ∘ X cana-5930 207 6 𝐺2 𝐺2 ADJ cana-5930 207 7 ) ) PUNCT cana-5930 207 8 ) ) PUNCT cana-5930 208 1 βˆ’ βˆ’ PROPN cana-5930 208 2 ( ( PUNCT cana-5930 208 3 𝑉(𝑇 𝑉(𝑇 NOUN cana-5930 208 4 ) ) PUNCT cana-5930 208 5 βˆͺ βˆͺ ADP cana-5930 208 6 𝐷′)| 𝐷′)| NOUN cana-5930 208 7 = = SYM cana-5930 208 8 2𝑝1𝑝2 2𝑝1𝑝2 PROPN cana-5930 208 9 + + CCONJ cana-5930 208 10 𝑝1(1 𝑝1(1 ADJ cana-5930 208 11 + + ADJ cana-5930 208 12 π‘ž2 π‘ž2 NOUN cana-5930 208 13 ) ) PUNCT cana-5930 208 14 + + CCONJ cana-5930 208 15 π‘ž1 π‘ž1 ADJ cana-5930 208 16 βˆ’ βˆ’ PROPN cana-5930 209 1 𝑑(1 𝑑(1 PROPN cana-5930 209 2 + + PROPN cana-5930 209 3 Ξ²0 Ξ²0 ADJ cana-5930 209 4 ) ) PUNCT cana-5930 209 5 βˆ’ βˆ’ PROPN cana-5930 209 6 2Ξ²0 2Ξ²0 NUM cana-5930 209 7 . . PUNCT cana-5930 210 1 β–‘ β–‘ SYM cana-5930 210 2 5 5 X cana-5930 210 3 . . X cana-5930 210 4 relation relation NOUN cana-5930 210 5 between between ADP cana-5930 210 6 ctd(g ctd(g PROPN cana-5930 210 7 ) ) PUNCT cana-5930 210 8 and and CCONJ cana-5930 210 9 ctd(t2(g ctd(t2(g NOUN cana-5930 210 10 ) ) PUNCT cana-5930 210 11 ) ) PUNCT cana-5930 210 12 observation observation NOUN cana-5930 210 13 5.1 5.1 NUM cana-5930 210 14 . . PUNCT cana-5930 211 1 for for ADP cana-5930 211 2 any any DET cana-5930 211 3 connected connected ADJ cana-5930 211 4 graph graph NOUN cana-5930 211 5 𝐺(𝑝 𝐺(𝑝 NUM cana-5930 211 6 , , PUNCT cana-5930 211 7 π‘ž π‘ž NOUN cana-5930 211 8 ) ) PUNCT cana-5930 211 9 , , PUNCT cana-5930 211 10 𝑝 𝑝 PROPN cana-5930 211 11 β‰₯ β‰₯ NOUN cana-5930 211 12 2 2 NUM cana-5930 211 13 then then ADV cana-5930 211 14 γ𝑐𝑑𝑑(𝐺 γ𝑐𝑑𝑑(𝐺 NOUN cana-5930 211 15 ) ) PUNCT cana-5930 211 16 ≀ ≀ NUM cana-5930 211 17 γ𝑐𝑑𝑑(𝑇2(𝐺 γ𝑐𝑑𝑑(𝑇2(𝐺 NOUN cana-5930 211 18 ) ) PUNCT cana-5930 211 19 ) ) PUNCT cana-5930 211 20 . . PUNCT cana-5930 212 1 equality equality NOUN cana-5930 212 2 holds hold VERB cana-5930 212 3 if if SCONJ cana-5930 212 4 𝐺 𝐺 PROPN cana-5930 212 5 β‰… β‰… PROPN cana-5930 212 6 𝐾1,π‘βˆ’1 𝐾1,π‘βˆ’1 PROPN cana-5930 212 7 . . PUNCT cana-5930 213 1 communications communication NOUN cana-5930 213 2 on on ADP cana-5930 213 3 applied apply VERB cana-5930 213 4 nonlinear nonlinear ADJ cana-5930 213 5 analysis analysis NOUN cana-5930 213 6 issn issn NOUN cana-5930 213 7 : : PUNCT cana-5930 213 8 1074 1074 NUM cana-5930 213 9 - - PUNCT cana-5930 213 10 133x 133x NUM cana-5930 213 11 vol vol VERB cana-5930 213 12 32 32 NUM cana-5930 213 13 no no NOUN cana-5930 213 14 . . PUNCT cana-5930 214 1 10s 10 NOUN cana-5930 214 2 ( ( PUNCT cana-5930 214 3 2025 2025 NUM cana-5930 214 4 ) ) PUNCT cana-5930 214 5 3109 3109 NUM cana-5930 214 6 https://internationalpubls.com https://internationalpubls.com X cana-5930 214 7 theorem theorem VERB cana-5930 214 8 5.2 5.2 NUM cana-5930 214 9 . . PUNCT cana-5930 215 1 for for ADP cana-5930 215 2 any any DET cana-5930 215 3 connected connected ADJ cana-5930 215 4 graph graph NOUN cana-5930 215 5 𝐺(𝑝 𝐺(𝑝 NUM cana-5930 215 6 , , PUNCT cana-5930 215 7 π‘ž π‘ž NOUN cana-5930 215 8 ) ) PUNCT cana-5930 215 9 , , PUNCT cana-5930 215 10 𝑝 𝑝 PROPN cana-5930 215 11 β‰₯ β‰₯ NOUN cana-5930 215 12 2 2 NUM cana-5930 215 13 with with ADP cana-5930 215 14 Ξ΄(𝐺 Ξ΄(𝐺 PROPN cana-5930 215 15 ) ) PUNCT cana-5930 215 16 = = SYM cana-5930 215 17 1 1 NUM cana-5930 215 18 then then ADV cana-5930 215 19 γ𝑐𝑑𝑑(𝑇2(𝐺 γ𝑐𝑑𝑑(𝑇2(𝐺 NOUN cana-5930 215 20 ) ) PUNCT cana-5930 215 21 ) ) PUNCT cana-5930 216 1 ≀ ≀ X cana-5930 216 2 𝑝 𝑝 ADV cana-5930 216 3 βˆ’ βˆ’ PROPN cana-5930 216 4 1 1 NUM cana-5930 216 5 + + CCONJ cana-5930 216 6 γ𝑐𝑑𝑑(𝐺 γ𝑐𝑑𝑑(𝐺 NOUN cana-5930 216 7 ) ) PUNCT cana-5930 216 8 . . PUNCT cana-5930 217 1 proof proof NOUN cana-5930 217 2 . . PUNCT cana-5930 218 1 let let VERB cana-5930 218 2 d d PRON cana-5930 218 3 be be AUX cana-5930 218 4 a a DET cana-5930 218 5 minimum minimum ADJ cana-5930 218 6 ctd ctd NOUN cana-5930 218 7 - - PUNCT cana-5930 218 8 set set NOUN cana-5930 218 9 of of ADP cana-5930 218 10 g g NOUN cana-5930 218 11 and and CCONJ cana-5930 218 12 hence hence ADV cana-5930 218 13 |𝐷| |𝐷| NOUN cana-5930 218 14 = = SYM cana-5930 218 15 γ𝑐𝑑𝑑(𝐺 γ𝑐𝑑𝑑(𝐺 PROPN cana-5930 218 16 ) ) PUNCT cana-5930 218 17 . . PUNCT cana-5930 219 1 therefore therefore ADV cana-5930 219 2 , , PUNCT cana-5930 219 3 βŸ¨π‘‰(𝐺 βŸ¨π‘‰(𝐺 NOUN cana-5930 219 4 ) ) PUNCT cana-5930 219 5 βˆ’ βˆ’ PROPN cana-5930 219 6 𝐷 𝐷 PROPN cana-5930 219 7 ⟩ ⟩ NOUN cana-5930 219 8 is be AUX cana-5930 219 9 a a DET cana-5930 219 10 tree tree NOUN cana-5930 219 11 . . PUNCT cana-5930 220 1 now now ADV cana-5930 220 2 , , PUNCT cana-5930 220 3 the the DET cana-5930 220 4 set set VERB cana-5930 220 5 𝐷′ 𝐷′ NOUN cana-5930 220 6 = = NOUN cana-5930 220 7 𝐷 𝐷 PROPN cana-5930 220 8 βˆͺ βˆͺ NOUN cana-5930 220 9 ( ( PUNCT cana-5930 220 10 𝑉(𝑇2(𝐺 𝑉(𝑇2(𝐺 NOUN cana-5930 220 11 ) ) PUNCT cana-5930 220 12 ) ) PUNCT cana-5930 221 1 βˆ’ βˆ’ PROPN cana-5930 221 2 𝑉(𝐺 𝑉(𝐺 NOUN cana-5930 221 3 ) ) PUNCT cana-5930 221 4 ) ) PUNCT cana-5930 221 5 is be AUX cana-5930 221 6 a a DET cana-5930 221 7 minimum minimum ADJ cana-5930 221 8 ctd ctd NOUN cana-5930 221 9 - - PUNCT cana-5930 221 10 set set NOUN cana-5930 221 11 of of ADP cana-5930 221 12 𝑇2(𝐺 𝑇2(𝐺 PROPN cana-5930 221 13 ) ) PUNCT cana-5930 221 14 . . PUNCT cana-5930 222 1 hence hence ADV cana-5930 222 2 , , PUNCT cana-5930 222 3 γ𝑐𝑑𝑑(𝑇2(𝐺 γ𝑐𝑑𝑑(𝑇2(𝐺 NOUN cana-5930 222 4 ) ) PUNCT cana-5930 222 5 ) ) PUNCT cana-5930 223 1 ≀ ≀ NOUN cana-5930 224 1 |𝐷′| |𝐷′| X cana-5930 224 2 = = SYM cana-5930 224 3 γ𝑐𝑑𝑑(𝐺 γ𝑐𝑑𝑑(𝐺 X cana-5930 224 4 ) ) PUNCT cana-5930 225 1 + + X cana-5930 225 2 𝑝 𝑝 ADP cana-5930 225 3 βˆ’ βˆ’ NOUN cana-5930 225 4 1 1 NUM cana-5930 225 5 . . PUNCT cana-5930 225 6 β–‘ β–‘ PUNCT cana-5930 225 7 theorem theorem VERB cana-5930 225 8 5.3 5.3 NUM cana-5930 225 9 . . PUNCT cana-5930 226 1 given give VERB cana-5930 226 2 two two NUM cana-5930 226 3 integers integer NOUN cana-5930 226 4 a a PRON cana-5930 226 5 and and CCONJ cana-5930 226 6 b b NOUN cana-5930 226 7 with with ADP cana-5930 226 8 2 2 NUM cana-5930 226 9 ≀ ≀ NOUN cana-5930 226 10 π‘Ž π‘Ž PRON cana-5930 226 11 ≀ ≀ ADJ cana-5930 226 12 𝑏 𝑏 NOUN cana-5930 226 13 there there PRON cana-5930 226 14 exists exist VERB cana-5930 226 15 a a DET cana-5930 226 16 graph graph NOUN cana-5930 226 17 with with ADP cana-5930 226 18 π‘Ž π‘Ž PROPN cana-5930 226 19 + + ADP cana-5930 226 20 𝑏 𝑏 NOUN cana-5930 226 21 + + NOUN cana-5930 226 22 1 1 NUM cana-5930 226 23 vertices vertex NOUN cana-5930 226 24 such such ADJ cana-5930 226 25 that that SCONJ cana-5930 226 26 Ξ³(𝑇2(𝐺 Ξ³(𝑇2(𝐺 PROPN cana-5930 226 27 ) ) PUNCT cana-5930 226 28 ) ) PUNCT cana-5930 227 1 = = PUNCT cana-5930 228 1 π‘Ž π‘Ž X cana-5930 228 2 + + NUM cana-5930 228 3 1 1 NUM cana-5930 228 4 and and CCONJ cana-5930 228 5 γ𝑐𝑑𝑑(𝑇2(𝐺 γ𝑐𝑑𝑑(𝑇2(𝐺 NOUN cana-5930 228 6 ) ) PUNCT cana-5930 228 7 ) ) PUNCT cana-5930 229 1 = = PUNCT cana-5930 230 1 π‘Ž π‘Ž X cana-5930 230 2 + + X cana-5930 230 3 𝑏. 𝑏. VERB cana-5930 230 4 also also ADV cana-5930 230 5 γ𝑐𝑑𝑑(𝑇2(𝐺 γ𝑐𝑑𝑑(𝑇2(𝐺 NOUN cana-5930 230 6 ) ) PUNCT cana-5930 230 7 ) ) PUNCT cana-5930 231 1 ≀ ≀ NUM cana-5930 231 2 Ξ³(𝑇2(𝐺 Ξ³(𝑇2(𝐺 PROPN cana-5930 231 3 ) ) PUNCT cana-5930 231 4 ) ) PUNCT cana-5930 232 1 + + CCONJ cana-5930 232 2 𝑏. 𝑏. ADJ cana-5930 232 3 figure figure NOUN cana-5930 232 4 1 1 NUM cana-5930 232 5 . . PUNCT cana-5930 233 1 proof proof NOUN cana-5930 233 2 . . PUNCT cana-5930 234 1 in in ADP cana-5930 234 2 the the DET cana-5930 234 3 cycle cycle NOUN cana-5930 234 4 πΆπ‘Ž+2(π‘Ž πΆπ‘Ž+2(π‘Ž VERB cana-5930 234 5 β‰₯ β‰₯ NOUN cana-5930 234 6 2 2 NUM cana-5930 234 7 ) ) PUNCT cana-5930 234 8 by by ADP cana-5930 234 9 { { PUNCT cana-5930 234 10 𝑣1 𝑣1 PROPN cana-5930 234 11 , , PUNCT cana-5930 234 12 𝑣2 𝑣2 PROPN cana-5930 234 13 , , PUNCT cana-5930 234 14 … … PUNCT cana-5930 234 15 , , PUNCT cana-5930 234 16 π‘£π‘Ž+2 π‘£π‘Ž+2 X cana-5930 234 17 } } PUNCT cana-5930 234 18 of of ADP cana-5930 234 19 length length NOUN cana-5930 234 20 π‘Ž π‘Ž PRON cana-5930 234 21 + + NOUN cana-5930 234 22 2 2 NUM cana-5930 234 23 . . X cana-5930 234 24 consider consider VERB cana-5930 234 25 a a DET cana-5930 234 26 path path NOUN cana-5930 234 27 of of ADP cana-5930 234 28 length length NOUN cana-5930 234 29 a. a. NOUN cana-5930 234 30 in in ADP cana-5930 234 31 this this DET cana-5930 234 32 path path NOUN cana-5930 234 33 attach attach NOUN cana-5930 234 34 ( ( PUNCT cana-5930 234 35 𝑏 𝑏 NOUN cana-5930 234 36 βˆ’ βˆ’ PROPN cana-5930 234 37 π‘Ž π‘Ž NOUN cana-5930 234 38 ) ) PUNCT cana-5930 234 39 pendant pendant ADJ cana-5930 234 40 edges edge NOUN cana-5930 234 41 at at ADP cana-5930 234 42 exactly exactly ADV cana-5930 234 43 one one NUM cana-5930 234 44 vertex vertex NOUN cana-5930 234 45 and and CCONJ cana-5930 234 46 attach attach VERB cana-5930 234 47 one one NUM cana-5930 234 48 pendant pendant ADJ cana-5930 234 49 edge edge NOUN cana-5930 234 50 at at ADP cana-5930 234 51 each each PRON cana-5930 234 52 of of ADP cana-5930 234 53 the the DET cana-5930 234 54 remaining remain VERB cana-5930 234 55 ( ( PUNCT cana-5930 234 56 π‘Ž π‘Ž NOUN cana-5930 234 57 βˆ’ βˆ’ NUM cana-5930 234 58 1 1 NUM cana-5930 234 59 ) ) PUNCT cana-5930 234 60 vertices vertex NOUN cana-5930 234 61 . . PUNCT cana-5930 235 1 let let VERB cana-5930 235 2 the the DET cana-5930 235 3 graph graph NOUN cana-5930 235 4 thus thus ADV cana-5930 235 5 obtained obtain VERB cana-5930 235 6 be be AUX cana-5930 235 7 denoted denote VERB cana-5930 235 8 by by ADP cana-5930 235 9 g g PROPN cana-5930 235 10 and and CCONJ cana-5930 235 11 g g PROPN cana-5930 235 12 has have VERB cana-5930 235 13 π‘Ž π‘Ž PROPN cana-5930 235 14 + + NOUN cana-5930 235 15 𝑏 𝑏 NOUN cana-5930 235 16 + + CCONJ cana-5930 235 17 1 1 NUM cana-5930 235 18 vertices vertex NOUN cana-5930 235 19 . . PUNCT cana-5930 236 1 in in ADP cana-5930 236 2 𝑇2(𝐺 𝑇2(𝐺 PROPN cana-5930 236 3 ) ) PUNCT cana-5930 236 4 , , PUNCT cana-5930 236 5 edges edge NOUN cana-5930 236 6 of of ADP cana-5930 236 7 cycle cycle NOUN cana-5930 236 8 πΆπ‘Ž+2 πΆπ‘Ž+2 NOUN cana-5930 236 9 and and CCONJ cana-5930 236 10 edges edge NOUN cana-5930 236 11 of of ADP cana-5930 236 12 pendant pendant ADJ cana-5930 236 13 vertex vertex NOUN cana-5930 236 14 of of ADP cana-5930 236 15 g g PROPN cana-5930 236 16 and and CCONJ cana-5930 236 17 the the DET cana-5930 236 18 vertices vertex NOUN cana-5930 236 19 of of ADP cana-5930 236 20 g g PROPN cana-5930 236 21 are be AUX cana-5930 236 22 the the DET cana-5930 236 23 vertex vertex NOUN cana-5930 236 24 set set NOUN cana-5930 236 25 of of ADP cana-5930 236 26 𝑇2(𝐺 𝑇2(𝐺 PROPN cana-5930 236 27 ) ) PUNCT cana-5930 236 28 . . PUNCT cana-5930 236 29 therefore,|𝑉(𝑇2(𝐺))| therefore,|𝑉(𝑇2(𝐺))| PUNCT cana-5930 237 1 = = NOUN cana-5930 237 2 2π‘Ž 2π‘Ž NUM cana-5930 237 3 + + CCONJ cana-5930 237 4 2𝑏 2𝑏 NUM cana-5930 237 5 + + X cana-5930 237 6 2 2 X cana-5930 237 7 . . X cana-5930 238 1 the the DET cana-5930 238 2 set set NOUN cana-5930 238 3 { { PUNCT cana-5930 238 4 𝑣1 𝑣1 PROPN cana-5930 238 5 , , PUNCT cana-5930 238 6 𝑣2 𝑣2 PROPN cana-5930 238 7 , , PUNCT cana-5930 238 8 … … PUNCT cana-5930 238 9 , , PUNCT cana-5930 238 10 π‘£π‘Ž+1 π‘£π‘Ž+1 AUX cana-5930 238 11 } } PUNCT cana-5930 238 12 forms form VERB cana-5930 238 13 a a DET cana-5930 238 14 minimum minimum ADJ cana-5930 238 15 dominating dominating NOUN cana-5930 238 16 set set NOUN cana-5930 238 17 of of ADP cana-5930 238 18 𝑇2(𝐺 𝑇2(𝐺 PROPN cana-5930 238 19 ) ) PUNCT cana-5930 238 20 and and CCONJ cana-5930 238 21 the the DET cana-5930 238 22 set set NOUN cana-5930 238 23 consisting consist VERB cana-5930 238 24 of of ADP cana-5930 238 25 edge edge NOUN cana-5930 238 26 vertex vertex NOUN cana-5930 238 27 of of ADP cana-5930 238 28 path path NOUN cana-5930 238 29 π‘ƒπ‘Ž+1 π‘ƒπ‘Ž+1 PUNCT cana-5930 238 30 a a DET cana-5930 238 31 vertex vertex NOUN cana-5930 238 32 πΆπ‘Ž+2 πΆπ‘Ž+2 NOUN cana-5930 238 33 and and CCONJ cana-5930 238 34 all all DET cana-5930 238 35 the the DET cana-5930 238 36 edge edge NOUN cana-5930 238 37 vertex vertex NOUN cana-5930 238 38 of of ADP cana-5930 238 39 pendant pendant ADJ cana-5930 238 40 vertices vertex NOUN cana-5930 238 41 of of ADP cana-5930 238 42 g g PROPN cana-5930 238 43 forms form VERB cana-5930 238 44 a a DET cana-5930 238 45 minimum minimum ADJ cana-5930 238 46 ctd ctd NOUN cana-5930 238 47 - - PUNCT cana-5930 238 48 set set NOUN cana-5930 238 49 of of ADP cana-5930 238 50 𝑇2(𝐺 𝑇2(𝐺 PROPN cana-5930 238 51 ) ) PUNCT cana-5930 238 52 . . PUNCT cana-5930 239 1 therefore therefore ADV cana-5930 239 2 , , PUNCT cana-5930 239 3 γ𝑐𝑑𝑑(𝑇2(𝐺 γ𝑐𝑑𝑑(𝑇2(𝐺 NOUN cana-5930 239 4 ) ) PUNCT cana-5930 239 5 ) ) PUNCT cana-5930 240 1 = = PUNCT cana-5930 241 1 π‘Ž π‘Ž X cana-5930 241 2 + + X cana-5930 241 3 𝑏. 𝑏. NOUN cana-5930 241 4 if if SCONJ cana-5930 241 5 a a DET cana-5930 241 6 = = SYM cana-5930 241 7 b b NOUN cana-5930 241 8 then then ADV cana-5930 241 9 , , PUNCT cana-5930 241 10 the the DET cana-5930 241 11 equality equality NOUN cana-5930 241 12 holds hold VERB cana-5930 241 13 . . PUNCT cana-5930 242 1 β–‘ β–‘ PUNCT cana-5930 242 2 theorem theorem VERB cana-5930 242 3 5.4 5.4 NUM cana-5930 242 4 . . PUNCT cana-5930 243 1 if if SCONJ cana-5930 243 2 γ𝑐𝑑𝑑(𝐺 γ𝑐𝑑𝑑(𝐺 X cana-5930 243 3 ) ) PUNCT cana-5930 243 4 = = SYM cana-5930 243 5 1 1 NUM cana-5930 243 6 then then ADV cana-5930 243 7 γ𝑐𝑑𝑑(𝑇2(𝐺 γ𝑐𝑑𝑑(𝑇2(𝐺 NOUN cana-5930 243 8 ) ) PUNCT cana-5930 243 9 ) ) PUNCT cana-5930 244 1 = = PUNCT cana-5930 244 2 𝑝 𝑝 ADP cana-5930 244 3 βˆ’ βˆ’ NOUN cana-5930 244 4 1 1 NUM cana-5930 244 5 where where SCONJ cana-5930 244 6 𝑝 𝑝 NOUN cana-5930 244 7 β‰₯ β‰₯ NOUN cana-5930 244 8 2 2 NUM cana-5930 244 9 is be AUX cana-5930 244 10 the the DET cana-5930 244 11 number number NOUN cana-5930 244 12 of of ADP cana-5930 244 13 vertices vertex NOUN cana-5930 244 14 in in ADP cana-5930 244 15 g. g. PROPN cana-5930 244 16 proof proof NOUN cana-5930 244 17 . . PUNCT cana-5930 245 1 assume assume VERB cana-5930 245 2 γ𝑐𝑑𝑑(𝐺 γ𝑐𝑑𝑑(𝐺 X cana-5930 245 3 ) ) PUNCT cana-5930 245 4 = = SYM cana-5930 245 5 1 1 NUM cana-5930 245 6 , , PUNCT cana-5930 245 7 then then ADV cana-5930 245 8 𝐺 𝐺 PROPN cana-5930 245 9 β‰… β‰… PROPN cana-5930 245 10 𝐾1 𝐾1 PROPN cana-5930 245 11 + + CCONJ cana-5930 245 12 𝑇 𝑇 PROPN cana-5930 245 13 where where SCONJ cana-5930 245 14 t t PROPN cana-5930 245 15 is be AUX cana-5930 245 16 a a DET cana-5930 245 17 tree tree NOUN cana-5930 245 18 with with ADP cana-5930 245 19 atleast atleast ADJ cana-5930 245 20 two two NUM cana-5930 245 21 vertices vertex NOUN cana-5930 245 22 . . PUNCT cana-5930 246 1 let let VERB cana-5930 246 2 𝑉(𝐾1 𝑉(𝐾1 PRON cana-5930 246 3 ) ) PUNCT cana-5930 246 4 = = SYM cana-5930 246 5 𝑣 𝑣 NOUN cana-5930 246 6 and and CCONJ cana-5930 246 7 𝑉(𝑇 𝑉(𝑇 NOUN cana-5930 246 8 ) ) PUNCT cana-5930 247 1 = = PRON cana-5930 247 2 { { PUNCT cana-5930 247 3 𝑣1 𝑣1 PROPN cana-5930 247 4 , , PUNCT cana-5930 247 5 𝑣2 𝑣2 PROPN cana-5930 247 6 , , PUNCT cana-5930 247 7 … … PUNCT cana-5930 247 8 , , PUNCT cana-5930 247 9 π‘£π‘βˆ’1 π‘£π‘βˆ’1 PROPN cana-5930 247 10 } } PUNCT cana-5930 247 11 then then ADV cana-5930 247 12 𝑉(𝐺 𝑉(𝐺 VERB cana-5930 247 13 ) ) PUNCT cana-5930 247 14 = = SYM cana-5930 247 15 { { PUNCT cana-5930 247 16 𝑣 𝑣 NOUN cana-5930 247 17 , , PUNCT cana-5930 247 18 𝑣1 𝑣1 PROPN cana-5930 247 19 , , PUNCT cana-5930 247 20 𝑣2 𝑣2 PROPN cana-5930 247 21 , , PUNCT cana-5930 247 22 … … PUNCT cana-5930 247 23 π‘£π‘βˆ’1 π‘£π‘βˆ’1 ADP cana-5930 247 24 } } PUNCT cana-5930 247 25 and and CCONJ cana-5930 247 26 𝑉(𝑇2(𝐺 𝑉(𝑇2(𝐺 NOUN cana-5930 247 27 ) ) PUNCT cana-5930 247 28 ) ) PUNCT cana-5930 248 1 = = SYM cana-5930 248 2 𝑉(𝐺 𝑉(𝐺 NOUN cana-5930 248 3 ) ) PUNCT cana-5930 248 4 βˆͺ βˆͺ VERB cana-5930 248 5 𝑣𝑖 𝑣𝑖 ADP cana-5930 248 6 β€² β€² NUM cana-5930 248 7 βˆͺ βˆͺ ADJ cana-5930 248 8 𝑣𝑖,𝑖+1 𝑣𝑖,𝑖+1 NOUN cana-5930 248 9 β€² β€² NUM cana-5930 248 10 where where SCONJ cana-5930 248 11 𝑣𝑖 𝑣𝑖 ADV cana-5930 248 12 β€² β€² NUM cana-5930 248 13 ∈ ∈ PROPN cana-5930 248 14 ( ( PUNCT cana-5930 248 15 𝑣 𝑣 NOUN cana-5930 248 16 , , PUNCT cana-5930 248 17 𝑣𝑖 𝑣𝑖 NOUN cana-5930 248 18 ) ) PUNCT cana-5930 248 19 and and CCONJ cana-5930 248 20 𝑣𝑖,𝑖+1 𝑣𝑖,𝑖+1 NOUN cana-5930 248 21 β€² β€² NUM cana-5930 248 22 ∈ ∈ NOUN cana-5930 248 23 ( ( PUNCT cana-5930 248 24 𝑣𝑖 𝑣𝑖 NOUN cana-5930 248 25 , , PUNCT cana-5930 248 26 𝑣𝑖+1 𝑣𝑖+1 PROPN cana-5930 248 27 ) ) PUNCT cana-5930 248 28 where where SCONJ cana-5930 248 29 𝑖 𝑖 ADP cana-5930 248 30 = = SYM cana-5930 248 31 1 1 NUM cana-5930 248 32 , , PUNCT cana-5930 248 33 2 2 NUM cana-5930 248 34 , , PUNCT cana-5930 248 35 … … PUNCT cana-5930 248 36 , , PUNCT cana-5930 248 37 𝑝 𝑝 NOUN cana-5930 248 38 βˆ’ βˆ’ PROPN cana-5930 248 39 1 1 X cana-5930 248 40 . . PUNCT cana-5930 249 1 let let VERB cana-5930 249 2 𝐷 𝐷 NOUN cana-5930 249 3 = = PRON cana-5930 249 4 { { PUNCT cana-5930 249 5 𝑣𝑖,𝑖+1 𝑣𝑖,𝑖+1 NOUN cana-5930 249 6 β€² β€² NUM cana-5930 249 7 /𝑖 /𝑖 PUNCT cana-5930 250 1 = = SYM cana-5930 250 2 1,2 1,2 NUM cana-5930 250 3 , , PUNCT cana-5930 250 4 … … PUNCT cana-5930 250 5 , , PUNCT cana-5930 251 1 𝑝 𝑝 NOUN cana-5930 251 2 βˆ’ βˆ’ NOUN cana-5930 251 3 2 2 NUM cana-5930 251 4 } } PUNCT cana-5930 251 5 βˆͺ βˆͺ ADJ cana-5930 251 6 { { PUNCT cana-5930 251 7 𝑣 𝑣 NOUN cana-5930 251 8 } } PUNCT cana-5930 251 9 . . PUNCT cana-5930 252 1 then then ADV cana-5930 252 2 𝐷 𝐷 VERB cana-5930 252 3 βŠ† βŠ† NUM cana-5930 252 4 𝑉(𝑇2(𝐺 𝑉(𝑇2(𝐺 NOUN cana-5930 252 5 ) ) PUNCT cana-5930 252 6 ) ) PUNCT cana-5930 252 7 and and CCONJ cana-5930 252 8 βŸ¨π‘‰(𝑇2(𝐺 βŸ¨π‘‰(𝑇2(𝐺 PROPN cana-5930 252 9 ) ) PUNCT cana-5930 252 10 ) ) PUNCT cana-5930 253 1 βˆ’ βˆ’ PROPN cana-5930 253 2 𝐷 𝐷 PROPN cana-5930 253 3 ⟩ ⟩ NOUN cana-5930 253 4 is be AUX cana-5930 253 5 a a DET cana-5930 253 6 tree tree NOUN cana-5930 253 7 . . PUNCT cana-5930 254 1 therefore therefore ADV cana-5930 254 2 d d X cana-5930 254 3 is be AUX cana-5930 254 4 a a DET cana-5930 254 5 ctd ctd NOUN cana-5930 254 6 - - PUNCT cana-5930 254 7 set set NOUN cana-5930 254 8 of of ADP cana-5930 254 9 𝑇2(𝐺 𝑇2(𝐺 PROPN cana-5930 254 10 ) ) PUNCT cana-5930 254 11 and and CCONJ cana-5930 254 12 hence hence ADV cana-5930 254 13 , , PUNCT cana-5930 254 14 γ𝑐𝑑𝑑(𝑇2(𝐺 γ𝑐𝑑𝑑(𝑇2(𝐺 NOUN cana-5930 254 15 ) ) PUNCT cana-5930 254 16 ) ) PUNCT cana-5930 255 1 = = PUNCT cana-5930 255 2 𝑝 𝑝 ADP cana-5930 255 3 βˆ’ βˆ’ NOUN cana-5930 255 4 1 1 NUM cana-5930 255 5 . . PUNCT cana-5930 255 6 β–‘ β–‘ PUNCT cana-5930 255 7 communications communication NOUN cana-5930 255 8 on on ADP cana-5930 255 9 applied apply VERB cana-5930 255 10 nonlinear nonlinear ADJ cana-5930 255 11 analysis analysis NOUN cana-5930 255 12 issn issn NOUN cana-5930 255 13 : : PUNCT cana-5930 255 14 1074 1074 NUM cana-5930 255 15 - - PUNCT cana-5930 255 16 133x 133x NUM cana-5930 255 17 vol vol VERB cana-5930 255 18 32 32 NUM cana-5930 255 19 no no NOUN cana-5930 255 20 . . PUNCT cana-5930 256 1 10s 10 NOUN cana-5930 256 2 ( ( PUNCT cana-5930 256 3 2025 2025 NUM cana-5930 256 4 ) ) PUNCT cana-5930 256 5 3110 3110 NUM cana-5930 256 6 https://internationalpubls.com https://internationalpubls.com X cana-5930 256 7 theorem theorem VERB cana-5930 256 8 5.5 5.5 NUM cana-5930 256 9 . . PUNCT cana-5930 257 1 let let VERB cana-5930 257 2 g g PRON cana-5930 257 3 be be AUX cana-5930 257 4 a a DET cana-5930 257 5 connected connected ADJ cana-5930 257 6 graph graph NOUN cana-5930 257 7 , , PUNCT cana-5930 257 8 if if SCONJ cana-5930 257 9 γ𝑐𝑑𝑑(𝐺 γ𝑐𝑑𝑑(𝐺 X cana-5930 257 10 ) ) PUNCT cana-5930 257 11 = = SYM cana-5930 257 12 2 2 NUM cana-5930 257 13 then then ADV cana-5930 257 14 𝛾𝑐𝑑𝑑(𝑇2(𝐺 𝛾𝑐𝑑𝑑(𝑇2(𝐺 NOUN cana-5930 257 15 ) ) PUNCT cana-5930 257 16 ) ) PUNCT cana-5930 258 1 = = PRON cana-5930 258 2 { { PUNCT cana-5930 258 3 𝑝 𝑝 PROPN cana-5930 258 4 𝑖𝑓 𝑖𝑓 NUM cana-5930 258 5 𝐺 𝐺 PROPN cana-5930 258 6 β‰… β‰… PROPN cana-5930 258 7 𝐺1&𝐺2 𝐺1&𝐺2 PROPN cana-5930 258 8 𝑝 𝑝 PROPN cana-5930 258 9 βˆ’ βˆ’ PROPN cana-5930 258 10 1 1 NUM cana-5930 258 11 𝑖𝑓 𝑖𝑓 NOUN cana-5930 258 12 𝐺 𝐺 PROPN cana-5930 258 13 β‰… β‰… PROPN cana-5930 258 14 𝐺3 𝐺3 PROPN cana-5930 258 15 . . PUNCT cana-5930 259 1 ( ( PUNCT cana-5930 259 2 i i NOUN cana-5930 259 3 ) ) PUNCT cana-5930 259 4 𝐺1 𝐺1 PROPN cana-5930 259 5 is be AUX cana-5930 259 6 the the DET cana-5930 259 7 graph graph NOUN cana-5930 259 8 obtained obtain VERB cana-5930 259 9 from from ADP cana-5930 259 10 𝐾1 𝐾1 NOUN cana-5930 259 11 + + CCONJ cana-5930 259 12 𝑇 𝑇 NOUN cana-5930 259 13 with with ADP cana-5930 259 14 one one NUM cana-5930 259 15 pendant pendant ADJ cana-5930 259 16 edge edge NOUN cana-5930 259 17 attached attach VERB cana-5930 259 18 at at ADP cana-5930 259 19 the the DET cana-5930 259 20 vertex vertex NOUN cana-5930 259 21 of of ADP cana-5930 259 22 𝐾1 𝐾1 PROPN cana-5930 259 23 , , PUNCT cana-5930 259 24 where where SCONJ cana-5930 259 25 𝑇 𝑇 PROPN cana-5930 259 26 is be AUX cana-5930 259 27 any any DET cana-5930 259 28 tree tree NOUN cana-5930 259 29 with with ADP cana-5930 259 30 𝑝 𝑝 NOUN cana-5930 259 31 βˆ’ βˆ’ ADP cana-5930 259 32 2 2 NUM cana-5930 259 33 vertices vertex NOUN cana-5930 259 34 . . PUNCT cana-5930 260 1 ( ( PUNCT cana-5930 260 2 ii ii NOUN cana-5930 260 3 ) ) PUNCT cana-5930 260 4 𝐺2 𝐺2 NOUN cana-5930 260 5 is be AUX cana-5930 260 6 the the DET cana-5930 260 7 graph graph NOUN cana-5930 260 8 obtained obtain VERB cana-5930 260 9 from from ADP cana-5930 260 10 a a DET cana-5930 260 11 tree tree NOUN cana-5930 260 12 t t NOUN cana-5930 260 13 where where SCONJ cana-5930 260 14 ( ( PUNCT cana-5930 260 15 |𝑇| |𝑇| NOUN cana-5930 260 16 = = SYM cana-5930 260 17 𝑝 𝑝 PROPN cana-5930 260 18 βˆ’ βˆ’ NOUN cana-5930 260 19 2 2 NUM cana-5930 260 20 ) ) PUNCT cana-5930 260 21 by by ADP cana-5930 260 22 joining join VERB cana-5930 260 23 each each PRON cana-5930 260 24 of of ADP cana-5930 260 25 the the DET cana-5930 260 26 vertices vertex NOUN cana-5930 260 27 of of ADP cana-5930 260 28 the the DET cana-5930 260 29 tree tree NOUN cana-5930 260 30 to to ADP cana-5930 260 31 the the DET cana-5930 260 32 vertices vertex NOUN cana-5930 260 33 of of ADP cana-5930 260 34 𝐾2 𝐾2 NOUN cana-5930 260 35 such such ADJ cana-5930 260 36 that that SCONJ cana-5930 260 37 deg𝐺(𝑣 deg𝐺(𝑣 NOUN cana-5930 260 38 ) ) PUNCT cana-5930 260 39 β‰₯ β‰₯ NOUN cana-5930 260 40 2 2 NUM cana-5930 260 41 for for ADP cana-5930 260 42 all all PRON cana-5930 260 43 𝑣 𝑣 ADP cana-5930 260 44 ∈ ∈ PROPN cana-5930 260 45 𝑉(𝐾2 𝑉(𝐾2 NOUN cana-5930 260 46 ) ) PUNCT cana-5930 260 47 . . PUNCT cana-5930 261 1 ( ( PUNCT cana-5930 261 2 iii iii X cana-5930 261 3 ) ) PUNCT cana-5930 261 4 𝐺3 𝐺3 NOUN cana-5930 261 5 is be AUX cana-5930 261 6 the the DET cana-5930 261 7 graph graph NOUN cana-5930 261 8 obtained obtain VERB cana-5930 261 9 from from ADP cana-5930 261 10 a a DET cana-5930 261 11 tree tree NOUN cana-5930 261 12 by by ADP cana-5930 261 13 joining join VERB cana-5930 261 14 each each PRON cana-5930 261 15 of of ADP cana-5930 261 16 the the DET cana-5930 261 17 vertices vertex NOUN cana-5930 261 18 of of ADP cana-5930 261 19 the the DET cana-5930 261 20 tree tree NOUN cana-5930 261 21 to to ADP cana-5930 261 22 the the DET cana-5930 261 23 vertices vertex NOUN cana-5930 261 24 of of ADP cana-5930 261 25 2𝐾1 2𝐾1 NUM cana-5930 261 26 such such ADJ cana-5930 261 27 that that SCONJ cana-5930 261 28 deg𝐺(𝑣 deg𝐺(𝑣 NOUN cana-5930 261 29 ) ) PUNCT cana-5930 261 30 β‰₯ β‰₯ NOUN cana-5930 261 31 1 1 NUM cana-5930 261 32 for for ADP cana-5930 261 33 all all DET cana-5930 261 34 𝑣 𝑣 PRON cana-5930 261 35 ∈ ∈ PROPN cana-5930 261 36 𝑉(2𝐾1 𝑉(2𝐾1 PROPN cana-5930 261 37 ) ) PUNCT cana-5930 261 38 and and CCONJ cana-5930 261 39 |𝑉(𝐺)| |𝑉(𝐺)| X cana-5930 261 40 = = SYM cana-5930 261 41 𝑝. 𝑝. ADJ cana-5930 261 42 proof proof NOUN cana-5930 261 43 . . PUNCT cana-5930 262 1 assume assume VERB cana-5930 262 2 γ𝑐𝑑𝑑(𝐺 γ𝑐𝑑𝑑(𝐺 X cana-5930 262 3 ) ) PUNCT cana-5930 262 4 = = SYM cana-5930 263 1 2 2 X cana-5930 263 2 . . X cana-5930 263 3 let let VERB cana-5930 263 4 g g PRON cana-5930 263 5 be be AUX cana-5930 263 6 a a DET cana-5930 263 7 connected connected ADJ cana-5930 263 8 graph graph NOUN cana-5930 263 9 with with ADP cana-5930 263 10 𝑝 𝑝 PROPN cana-5930 263 11 β‰₯ β‰₯ NUM cana-5930 263 12 4 4 NUM cana-5930 263 13 and and CCONJ cana-5930 263 14 𝑆1 𝑆1 NOUN cana-5930 263 15 = = SYM cana-5930 263 16 { { PUNCT cana-5930 263 17 𝑒1 𝑒1 PROPN cana-5930 263 18 , , PUNCT cana-5930 263 19 𝑒2 𝑒2 PROPN cana-5930 263 20 } } PUNCT cana-5930 263 21 is be AUX cana-5930 263 22 a a DET cana-5930 263 23 minimum minimum ADJ cana-5930 263 24 ctd ctd NOUN cana-5930 263 25 - - PUNCT cana-5930 263 26 set set NOUN cana-5930 263 27 of of ADP cana-5930 263 28 g g PROPN cana-5930 263 29 then then ADV cana-5930 263 30 βŸ¨π‘‰(𝐺 βŸ¨π‘‰(𝐺 PROPN cana-5930 263 31 ) ) PUNCT cana-5930 264 1 βˆ’ βˆ’ PROPN cana-5930 264 2 𝑆1⟩ 𝑆1⟩ NOUN cana-5930 264 3 is be AUX cana-5930 264 4 a a DET cana-5930 264 5 tree tree NOUN cana-5930 264 6 𝑇. 𝑇. NOUN cana-5930 264 7 hence hence ADV cana-5930 264 8 |𝑇| |𝑇| PROPN cana-5930 264 9 = = SYM cana-5930 264 10 |𝑉(𝐺 |𝑉(𝐺 PROPN cana-5930 264 11 ) ) PUNCT cana-5930 264 12 βˆ’ βˆ’ PROPN cana-5930 265 1 𝑆1| 𝑆1| ADP cana-5930 265 2 = = SYM cana-5930 265 3 𝑝 𝑝 PROPN cana-5930 265 4 βˆ’ βˆ’ NOUN cana-5930 265 5 2 2 NUM cana-5930 265 6 . . PUNCT cana-5930 265 7 now now ADV cana-5930 265 8 construct construct VERB cana-5930 265 9 𝑇2(𝐺 𝑇2(𝐺 PROPN cana-5930 265 10 ) ) PUNCT cana-5930 265 11 , , PUNCT cana-5930 265 12 the the DET cana-5930 265 13 vertices vertex NOUN cana-5930 265 14 of of ADP cana-5930 265 15 complementary complementary ADJ cana-5930 265 16 dominating dominating NOUN cana-5930 265 17 set set NOUN cana-5930 265 18 of of ADP cana-5930 265 19 g g PROPN cana-5930 265 20 ( ( PUNCT cana-5930 265 21 tree tree NOUN cana-5930 265 22 t t PROPN cana-5930 265 23 ) ) PUNCT cana-5930 265 24 and and CCONJ cana-5930 265 25 its its PRON cana-5930 265 26 edge edge NOUN cana-5930 265 27 vertex vertex NOUN cana-5930 265 28 forms form VERB cana-5930 265 29 a a DET cana-5930 265 30 cycle cycle NOUN cana-5930 265 31 𝐢3 𝐢3 NOUN cana-5930 265 32 . . PUNCT cana-5930 266 1 let𝑆2 let𝑆2 PROPN cana-5930 266 2 = = PUNCT cana-5930 266 3 { { PUNCT cana-5930 266 4 𝑣𝑖𝑗 𝑣𝑖𝑗 PROPN cana-5930 266 5 β€² β€² NUM cana-5930 266 6 /𝑖 /𝑖 PUNCT cana-5930 267 1 β‰  β‰  PROPN cana-5930 267 2 𝑗 𝑗 PROPN cana-5930 267 3 , , PUNCT cana-5930 267 4 𝑖 𝑖 NOUN cana-5930 267 5 = = SYM cana-5930 267 6 1,2 1,2 NUM cana-5930 267 7 , , PUNCT cana-5930 267 8 … … PUNCT cana-5930 267 9 , , PUNCT cana-5930 267 10 𝑝 𝑝 PRON cana-5930 267 11 βˆ’ βˆ’ PROPN cana-5930 267 12 3 3 NUM cana-5930 267 13 , , PUNCT cana-5930 267 14 𝑗 𝑗 NOUN cana-5930 267 15 = = SYM cana-5930 267 16 1,2 1,2 NUM cana-5930 267 17 , , PUNCT cana-5930 267 18 … … PUNCT cana-5930 267 19 , , PUNCT cana-5930 267 20 𝑝 𝑝 NOUN cana-5930 267 21 βˆ’ βˆ’ NOUN cana-5930 267 22 2 2 NUM cana-5930 267 23 } } SYM cana-5930 267 24 βŠ† βŠ† NUM cana-5930 267 25 𝑉(𝑇2(𝐺))where(𝑣𝑖 𝑉(𝑇2(𝐺))where(𝑣𝑖 NUM cana-5930 267 26 , , PUNCT cana-5930 267 27 𝑣𝑗 𝑣𝑗 NOUN cana-5930 267 28 ) ) PUNCT cana-5930 267 29 ∈ ∈ NOUN cana-5930 267 30 𝐸(𝑉(𝐺 𝐸(𝑉(𝐺 NOUN cana-5930 267 31 ) ) PUNCT cana-5930 267 32 βˆ’ βˆ’ PROPN cana-5930 267 33 𝑆1 𝑆1 NOUN cana-5930 267 34 ) ) PUNCT cana-5930 267 35 . . PUNCT cana-5930 268 1 case case NOUN cana-5930 268 2 1 1 X cana-5930 268 3 . . PUNCT cana-5930 269 1 𝑒1 𝑒1 PROPN cana-5930 269 2 and and CCONJ cana-5930 269 3 𝑒2 𝑒2 PROPN cana-5930 269 4 are be AUX cana-5930 269 5 connected connect VERB cana-5930 269 6 then then ADV cana-5930 269 7 𝐺 𝐺 PROPN cana-5930 269 8 β‰… β‰… PROPN cana-5930 269 9 𝐺1 𝐺1 PROPN cana-5930 269 10 or or CCONJ cana-5930 269 11 𝐺2 𝐺2 ADJ cana-5930 269 12 . . PUNCT cana-5930 270 1 let let VERB cana-5930 270 2 𝐷 𝐷 NOUN cana-5930 270 3 = = NOUN cana-5930 270 4 𝑆1 𝑆1 NOUN cana-5930 270 5 βˆͺ βˆͺ ADJ cana-5930 270 6 𝑆2 𝑆2 PROPN cana-5930 270 7 βˆͺ βˆͺ NOUN cana-5930 270 8 { { PUNCT cana-5930 270 9 𝑒12 𝑒12 PROPN cana-5930 270 10 β€² β€² NUM cana-5930 270 11 } } PUNCT cana-5930 270 12 is be AUX cana-5930 270 13 a a DET cana-5930 270 14 minimum minimum ADJ cana-5930 270 15 ctd ctd NOUN cana-5930 270 16 - - PUNCT cana-5930 270 17 set set NOUN cana-5930 270 18 of of ADP cana-5930 270 19 𝑇2(𝐺1 𝑇2(𝐺1 NOUN cana-5930 270 20 ) ) PUNCT cana-5930 270 21 or or CCONJ cana-5930 270 22 𝑇2(𝐺2 𝑇2(𝐺2 NOUN cana-5930 270 23 ) ) PUNCT cana-5930 270 24 . . PUNCT cana-5930 271 1 hence hence ADV cana-5930 271 2 |𝐷| |𝐷| X cana-5930 271 3 = = SYM cana-5930 271 4 |𝑆1| |𝑆1| PROPN cana-5930 271 5 + + CCONJ cana-5930 271 6 |𝑆2| |𝑆2| ADP cana-5930 271 7 + + SYM cana-5930 271 8 1 1 NUM cana-5930 271 9 = = SYM cana-5930 271 10 𝑝.therefore 𝑝.therefore NOUN cana-5930 271 11 , , PUNCT cana-5930 271 12 γ𝑐𝑑𝑑(𝑇2(𝐺 γ𝑐𝑑𝑑(𝑇2(𝐺 NOUN cana-5930 271 13 ) ) PUNCT cana-5930 271 14 ) ) PUNCT cana-5930 272 1 = = PUNCT cana-5930 272 2 𝑝 𝑝 NOUN cana-5930 272 3 if if SCONJ cana-5930 272 4 𝐺 𝐺 PROPN cana-5930 272 5 β‰… β‰… PROPN cana-5930 272 6 𝐺1 𝐺1 NOUN cana-5930 272 7 or or CCONJ cana-5930 272 8 𝐺2 𝐺2 ADJ cana-5930 272 9 . . PUNCT cana-5930 273 1 case case NOUN cana-5930 273 2 2 2 NUM cana-5930 273 3 . . PUNCT cana-5930 274 1 𝑒1 𝑒1 PROPN cana-5930 274 2 and and CCONJ cana-5930 274 3 𝑒2 𝑒2 PROPN cana-5930 274 4 are be AUX cana-5930 274 5 not not PART cana-5930 274 6 connected connect VERB cana-5930 274 7 . . PUNCT cana-5930 275 1 then then ADV cana-5930 275 2 𝐺 𝐺 PROPN cana-5930 275 3 β‰… β‰… PROPN cana-5930 275 4 𝐺3 𝐺3 PROPN cana-5930 275 5 . . PUNCT cana-5930 276 1 let let VERB cana-5930 276 2 𝐷 𝐷 NOUN cana-5930 276 3 = = PUNCT cana-5930 276 4 𝑆1 𝑆1 NOUN cana-5930 276 5 βˆͺ βˆͺ ADJ cana-5930 276 6 𝑆2 𝑆2 PROPN cana-5930 276 7 is be AUX cana-5930 276 8 a a DET cana-5930 276 9 minimum minimum ADJ cana-5930 276 10 ctd ctd NOUN cana-5930 276 11 - - PUNCT cana-5930 276 12 set set NOUN cana-5930 276 13 of of ADP cana-5930 276 14 𝑇2(𝐺3 𝑇2(𝐺3 NOUN cana-5930 276 15 ) ) PUNCT cana-5930 276 16 . . PUNCT cana-5930 277 1 hence hence ADV cana-5930 277 2 |𝐷| |𝐷| X cana-5930 277 3 = = SYM cana-5930 277 4 |𝑆1| |𝑆1| PROPN cana-5930 277 5 + + NUM cana-5930 277 6 |𝑆2| |𝑆2| X cana-5930 277 7 = = SYM cana-5930 277 8 𝑝 𝑝 ADJ cana-5930 277 9 βˆ’ βˆ’ PROPN cana-5930 277 10 1 1 NUM cana-5930 277 11 . . PUNCT cana-5930 277 12 therefore therefore ADV cana-5930 277 13 , , PUNCT cana-5930 277 14 γ𝑐𝑑𝑑(𝑇2(𝐺 γ𝑐𝑑𝑑(𝑇2(𝐺 NOUN cana-5930 277 15 ) ) PUNCT cana-5930 277 16 ) ) PUNCT cana-5930 278 1 = = PUNCT cana-5930 278 2 𝑝 𝑝 NOUN cana-5930 278 3 βˆ’ βˆ’ NOUN cana-5930 278 4 1 1 NUM cana-5930 278 5 if if SCONJ cana-5930 278 6 𝐺 𝐺 PROPN cana-5930 278 7 β‰… β‰… PROPN cana-5930 278 8 𝐺3 𝐺3 PROPN cana-5930 278 9 . . PUNCT cana-5930 279 1 β–‘ β–‘ PUNCT cana-5930 279 2 theorem theorem VERB cana-5930 279 3 5.6 5.6 NUM cana-5930 279 4 . . PUNCT cana-5930 280 1 let let VERB cana-5930 280 2 g g PRON cana-5930 280 3 be be AUX cana-5930 280 4 a a DET cana-5930 280 5 connected connected ADJ cana-5930 280 6 graph graph NOUN cana-5930 280 7 with with ADP cana-5930 280 8 p p NOUN cana-5930 280 9 vertices vertex NOUN cana-5930 280 10 ( ( PUNCT cana-5930 280 11 𝑝 𝑝 NOUN cana-5930 280 12 β‰₯ β‰₯ NOUN cana-5930 280 13 3 3 NUM cana-5930 280 14 ) ) PUNCT cana-5930 280 15 , , PUNCT cana-5930 280 16 𝑉(𝑇2(𝐺 𝑉(𝑇2(𝐺 NOUN cana-5930 280 17 ) ) PUNCT cana-5930 280 18 ) ) PUNCT cana-5930 281 1 = = SYM cana-5930 281 2 𝑉(𝐺 𝑉(𝐺 NOUN cana-5930 281 3 ) ) PUNCT cana-5930 281 4 βˆͺ βˆͺ ADP cana-5930 281 5 𝑉′(𝐺 𝑉′(𝐺 NOUN cana-5930 281 6 ) ) PUNCT cana-5930 281 7 then then ADV cana-5930 281 8 , , PUNCT cana-5930 281 9 𝑉′(𝐺 𝑉′(𝐺 NOUN cana-5930 281 10 ) ) PUNCT cana-5930 281 11 = = SYM cana-5930 281 12 𝑉(𝑇2(𝐺 𝑉(𝑇2(𝐺 NUM cana-5930 281 13 ) ) PUNCT cana-5930 281 14 ) ) PUNCT cana-5930 282 1 βˆ’ βˆ’ PROPN cana-5930 282 2 𝑉(𝐺 𝑉(𝐺 NOUN cana-5930 282 3 ) ) PUNCT cana-5930 282 4 is be AUX cana-5930 282 5 a a DET cana-5930 282 6 ctd ctd NOUN cana-5930 282 7 - - PUNCT cana-5930 282 8 set set NOUN cana-5930 282 9 of of ADP cana-5930 282 10 𝑇2(𝐺 𝑇2(𝐺 PROPN cana-5930 282 11 ) ) PUNCT cana-5930 283 1 if if SCONJ cana-5930 283 2 and and CCONJ cana-5930 283 3 only only ADV cana-5930 283 4 if if SCONJ cana-5930 283 5 g g PROPN cana-5930 283 6 is be AUX cana-5930 283 7 a a DET cana-5930 283 8 tree tree NOUN cana-5930 283 9 . . PUNCT cana-5930 284 1 proof proof NOUN cana-5930 284 2 . . PUNCT cana-5930 285 1 assume assume VERB cana-5930 285 2 𝑉′(𝐺 𝑉′(𝐺 PROPN cana-5930 285 3 ) ) PUNCT cana-5930 285 4 is be AUX cana-5930 285 5 a a DET cana-5930 285 6 ctd ctd NOUN cana-5930 285 7 - - PUNCT cana-5930 285 8 set set NOUN cana-5930 285 9 of of ADP cana-5930 285 10 𝑇2(𝐺 𝑇2(𝐺 PROPN cana-5930 285 11 ) ) PUNCT cana-5930 285 12 . . PUNCT cana-5930 286 1 then then ADV cana-5930 286 2 , , PUNCT cana-5930 286 3 each each DET cana-5930 286 4 vertex vertex NOUN cana-5930 286 5 in in ADP cana-5930 286 6 𝑉(𝑇2(𝐺 𝑉(𝑇2(𝐺 NOUN cana-5930 286 7 ) ) PUNCT cana-5930 286 8 ) ) PUNCT cana-5930 287 1 βˆ’ βˆ’ PROPN cana-5930 287 2 𝑉(𝐺 𝑉(𝐺 NOUN cana-5930 287 3 ) ) PUNCT cana-5930 287 4 is be AUX cana-5930 287 5 adjacent adjacent ADJ cana-5930 287 6 to to PART cana-5930 287 7 atleast atleast VERB cana-5930 287 8 one one NUM cana-5930 287 9 vertex vertex NOUN cana-5930 287 10 in in ADP cana-5930 287 11 𝑉′(𝐺 𝑉′(𝐺 NOUN cana-5930 287 12 ) ) PUNCT cana-5930 287 13 and and CCONJ cana-5930 287 14 βŸ¨π‘‰(𝑇2(𝐺 βŸ¨π‘‰(𝑇2(𝐺 PROPN cana-5930 287 15 ) ) PUNCT cana-5930 287 16 ) ) PUNCT cana-5930 288 1 βˆ’ βˆ’ PROPN cana-5930 288 2 𝑉′(𝐺 𝑉′(𝐺 NOUN cana-5930 288 3 ) ) PUNCT cana-5930 288 4 ⟩ ⟩ NOUN cana-5930 288 5 is be AUX cana-5930 288 6 a a DET cana-5930 288 7 tree tree NOUN cana-5930 288 8 . . PUNCT cana-5930 289 1 that that PRON cana-5930 289 2 is be AUX cana-5930 289 3 βŸ¨π‘‰(𝐺)⟩ βŸ¨π‘‰(𝐺)⟩ PROPN cana-5930 289 4 is be AUX cana-5930 289 5 a a DET cana-5930 289 6 tree tree NOUN cana-5930 289 7 . . PUNCT cana-5930 290 1 conversely conversely ADV cana-5930 290 2 , , PUNCT cana-5930 290 3 assume assume VERB cana-5930 290 4 g g PROPN cana-5930 290 5 is be AUX cana-5930 290 6 a a DET cana-5930 290 7 tree.let𝐷 tree.let𝐷 NOUN cana-5930 290 8 = = SYM cana-5930 290 9 𝑉′(𝐺 𝑉′(𝐺 NOUN cana-5930 290 10 ) ) PUNCT cana-5930 290 11 that that PRON cana-5930 290 12 is be AUX cana-5930 290 13 d d NOUN cana-5930 290 14 contains contain VERB cana-5930 290 15 all all DET cana-5930 290 16 the the DET cana-5930 290 17 edge edge NOUN cana-5930 290 18 vertices vertex NOUN cana-5930 290 19 of of ADP cana-5930 290 20 g. g. PROPN cana-5930 290 21 since since SCONJ cana-5930 290 22 g g PROPN cana-5930 290 23 is be AUX cana-5930 290 24 connected connect VERB cana-5930 290 25 each each DET cana-5930 290 26 vertex vertex NOUN cana-5930 290 27 v v NOUN cana-5930 290 28 in in ADP cana-5930 290 29 𝑉(𝑇2(𝐺 𝑉(𝑇2(𝐺 NOUN cana-5930 290 30 ) ) PUNCT cana-5930 290 31 ) ) PUNCT cana-5930 291 1 βˆ’ βˆ’ PROPN cana-5930 291 2 𝐷 𝐷 NOUN cana-5930 291 3 = = PUNCT cana-5930 291 4 𝑉(𝐺 𝑉(𝐺 NOUN cana-5930 291 5 ) ) PUNCT cana-5930 291 6 forms form VERB cana-5930 291 7 a a DET cana-5930 291 8 cycle cycle NOUN cana-5930 291 9 𝐢3 𝐢3 NOUN cana-5930 291 10 in in ADP cana-5930 291 11 𝑇2(𝐺 𝑇2(𝐺 PROPN cana-5930 291 12 ) ) PUNCT cana-5930 291 13 and and CCONJ cana-5930 291 14 𝑉(𝑇2(𝐺 𝑉(𝑇2(𝐺 NOUN cana-5930 291 15 ) ) PUNCT cana-5930 291 16 ) ) PUNCT cana-5930 292 1 βˆ’ βˆ’ PROPN cana-5930 292 2 𝐷 𝐷 NOUN cana-5930 292 3 = = PUNCT cana-5930 292 4 𝑉(𝐺 𝑉(𝐺 NOUN cana-5930 292 5 ) ) PUNCT cana-5930 292 6 is be AUX cana-5930 292 7 a a DET cana-5930 292 8 tree tree NOUN cana-5930 292 9 . . PUNCT cana-5930 293 1 hence hence ADV cana-5930 293 2 , , PUNCT cana-5930 293 3 d d PROPN cana-5930 293 4 is be AUX cana-5930 293 5 a a DET cana-5930 293 6 ctd ctd NOUN cana-5930 293 7 - - PUNCT cana-5930 293 8 set set NOUN cana-5930 293 9 of of ADP cana-5930 293 10 g. g. PROPN cana-5930 293 11 β–‘ β–‘ PUNCT cana-5930 293 12 theorem theorem VERB cana-5930 293 13 5.7 5.7 NUM cana-5930 293 14 . . PUNCT cana-5930 294 1 for for ADP cana-5930 294 2 any any DET cana-5930 294 3 connected connected ADJ cana-5930 294 4 ( ( PUNCT cana-5930 294 5 𝑝 𝑝 PROPN cana-5930 294 6 , , PUNCT cana-5930 294 7 π‘ž π‘ž NOUN cana-5930 294 8 ) ) PUNCT cana-5930 294 9 graph graph NOUN cana-5930 294 10 g g NOUN cana-5930 294 11 , , PUNCT cana-5930 294 12 γ𝑐𝑑𝑑(𝑇2(𝐺 γ𝑐𝑑𝑑(𝑇2(𝐺 NOUN cana-5930 294 13 ) ) PUNCT cana-5930 294 14 ) ) PUNCT cana-5930 295 1 + + CCONJ cana-5930 295 2 Ξ΄(𝐺 Ξ΄(𝐺 X cana-5930 295 3 ) ) PUNCT cana-5930 295 4 = = SYM cana-5930 295 5 2𝑝 2𝑝 NOUN cana-5930 295 6 βˆ’ βˆ’ NOUN cana-5930 295 7 2 2 NUM cana-5930 295 8 or or CCONJ cana-5930 295 9 𝑝 𝑝 NOUN cana-5930 295 10 + + CCONJ cana-5930 295 11 π‘ž π‘ž X cana-5930 295 12 βˆ’ βˆ’ PROPN cana-5930 295 13 1 1 NUM cana-5930 295 14 if if SCONJ cana-5930 295 15 and and CCONJ cana-5930 295 16 only only ADV cana-5930 295 17 if if SCONJ cana-5930 295 18 𝐺 𝐺 PROPN cana-5930 295 19 β‰… β‰… PROPN cana-5930 295 20 𝐾1,π‘βˆ’1 𝐾1,π‘βˆ’1 PROPN cana-5930 295 21 β€ˆ β€ˆ SPACE cana-5930 295 22 ( ( PUNCT cana-5930 295 23 𝑝 𝑝 PROPN cana-5930 295 24 β‰₯ β‰₯ NOUN cana-5930 295 25 4 4 NUM cana-5930 295 26 ) ) PUNCT cana-5930 295 27 . . PUNCT cana-5930 296 1 communications communication NOUN cana-5930 296 2 on on ADP cana-5930 296 3 applied apply VERB cana-5930 296 4 nonlinear nonlinear ADJ cana-5930 296 5 analysis analysis NOUN cana-5930 296 6 issn issn NOUN cana-5930 296 7 : : PUNCT cana-5930 296 8 1074 1074 NUM cana-5930 296 9 - - PUNCT cana-5930 296 10 133x 133x NUM cana-5930 296 11 vol vol VERB cana-5930 296 12 32 32 NUM cana-5930 296 13 no no NOUN cana-5930 296 14 . . PUNCT cana-5930 297 1 10s 10 NOUN cana-5930 297 2 ( ( PUNCT cana-5930 297 3 2025 2025 NUM cana-5930 297 4 ) ) PUNCT cana-5930 297 5 3111 3111 NUM cana-5930 297 6 https://internationalpubls.com https://internationalpubls.com X cana-5930 297 7 proof proof NOUN cana-5930 297 8 . . PUNCT cana-5930 298 1 when when SCONJ cana-5930 298 2 𝐺 𝐺 PROPN cana-5930 298 3 β‰… β‰… PROPN cana-5930 298 4 𝐾1,π‘βˆ’1 𝐾1,π‘βˆ’1 PROPN cana-5930 298 5 , , PUNCT cana-5930 298 6 γ𝑐𝑑𝑑(𝑇2(𝐺 γ𝑐𝑑𝑑(𝑇2(𝐺 NOUN cana-5930 298 7 ) ) PUNCT cana-5930 298 8 ) ) PUNCT cana-5930 299 1 + + CCONJ cana-5930 299 2 Ξ΄(𝐺 Ξ΄(𝐺 X cana-5930 299 3 ) ) PUNCT cana-5930 299 4 = = SYM cana-5930 299 5 𝑝 𝑝 PROPN cana-5930 299 6 + + NUM cana-5930 299 7 π‘ž π‘ž X cana-5930 299 8 βˆ’ βˆ’ PROPN cana-5930 299 9 1 1 NUM cana-5930 299 10 . . PUNCT cana-5930 300 1 conversely conversely ADV cana-5930 300 2 , , PUNCT cana-5930 300 3 γ𝑐𝑑𝑑(𝑇2(𝐺 γ𝑐𝑑𝑑(𝑇2(𝐺 NOUN cana-5930 300 4 ) ) PUNCT cana-5930 300 5 ) ) PUNCT cana-5930 301 1 + + CCONJ cana-5930 301 2 Ξ΄(𝐺 Ξ΄(𝐺 X cana-5930 301 3 ) ) PUNCT cana-5930 301 4 = = SYM cana-5930 301 5 2𝑝 2𝑝 NOUN cana-5930 301 6 βˆ’ βˆ’ NOUN cana-5930 301 7 2 2 NUM cana-5930 301 8 is be AUX cana-5930 301 9 possible possible ADJ cana-5930 301 10 if if SCONJ cana-5930 301 11 γ𝑐𝑑𝑑(𝑇2(𝐺 γ𝑐𝑑𝑑(𝑇2(𝐺 NOUN cana-5930 301 12 ) ) PUNCT cana-5930 301 13 ) ) PUNCT cana-5930 302 1 = = PUNCT cana-5930 302 2 𝑝 𝑝 ADP cana-5930 302 3 βˆ’ βˆ’ NOUN cana-5930 302 4 1 1 NUM cana-5930 302 5 and and CCONJ cana-5930 302 6 Ξ΄(𝐺 Ξ΄(𝐺 NOUN cana-5930 302 7 ) ) PUNCT cana-5930 302 8 = = SYM cana-5930 303 1 𝑝 𝑝 NOUN cana-5930 303 2 βˆ’ βˆ’ NOUN cana-5930 303 3 1 1 NUM cana-5930 303 4 is be AUX cana-5930 303 5 possible possible ADJ cana-5930 303 6 only only ADV cana-5930 303 7 if if SCONJ cana-5930 303 8 g g PROPN cana-5930 303 9 is be AUX cana-5930 303 10 a a DET cana-5930 303 11 star star NOUN cana-5930 303 12 on on ADP cana-5930 303 13 p p NOUN cana-5930 303 14 vertices vertex NOUN cana-5930 303 15 . . PUNCT cana-5930 304 1 β–‘ β–‘ PUNCT cana-5930 304 2 theorem theorem VERB cana-5930 304 3 5.8 5.8 NUM cana-5930 304 4 . . PUNCT cana-5930 305 1 for for ADP cana-5930 305 2 any any DET cana-5930 305 3 connected connected ADJ cana-5930 305 4 ( ( PUNCT cana-5930 305 5 𝑝 𝑝 PROPN cana-5930 305 6 , , PUNCT cana-5930 305 7 π‘ž π‘ž NOUN cana-5930 305 8 ) ) PUNCT cana-5930 305 9 graph graph NOUN cana-5930 305 10 g g NOUN cana-5930 305 11 , , PUNCT cana-5930 305 12 γ𝑐𝑑𝑑(𝑇2(𝐺 γ𝑐𝑑𝑑(𝑇2(𝐺 NOUN cana-5930 305 13 ) ) PUNCT cana-5930 305 14 ) ) PUNCT cana-5930 306 1 + + CCONJ cana-5930 306 2 Ξ΄(𝐺 Ξ΄(𝐺 X cana-5930 306 3 ) ) PUNCT cana-5930 306 4 = = SYM cana-5930 306 5 2𝑝 2𝑝 NOUN cana-5930 306 6 βˆ’ βˆ’ PROPN cana-5930 306 7 𝑛 𝑛 PROPN cana-5930 306 8 ( ( PUNCT cana-5930 306 9 𝑝 𝑝 NOUN cana-5930 306 10 β‰₯ β‰₯ NOUN cana-5930 306 11 4 4 NUM cana-5930 306 12 ) ) PUNCT cana-5930 306 13 where where SCONJ cana-5930 306 14 𝑛 𝑛 ADV cana-5930 306 15 = = PUNCT cana-5930 306 16 π‘‘π‘–π‘Žπ‘š(𝐺 π‘‘π‘–π‘Žπ‘š(𝐺 X cana-5930 306 17 ) ) PUNCT cana-5930 306 18 , , PUNCT cana-5930 306 19 𝑛 𝑛 PRON cana-5930 306 20 β‰₯ β‰₯ NUM cana-5930 306 21 2 2 NUM cana-5930 306 22 if if SCONJ cana-5930 306 23 𝐺 𝐺 PROPN cana-5930 306 24 β‰… β‰… PROPN cana-5930 306 25 broom broom NOUN cana-5930 306 26 graph graph NOUN cana-5930 306 27 . . PUNCT cana-5930 307 1 proof proof NOUN cana-5930 307 2 . . PUNCT cana-5930 308 1 for for ADP cana-5930 308 2 the the DET cana-5930 308 3 graphs graph NOUN cana-5930 308 4 given give VERB cana-5930 308 5 in in ADP cana-5930 308 6 the the DET cana-5930 308 7 theorem theorem ADJ cana-5930 308 8 Ξ΄(𝐺 Ξ΄(𝐺 PROPN cana-5930 308 9 ) ) PUNCT cana-5930 308 10 = = SYM cana-5930 308 11 𝑝 𝑝 PROPN cana-5930 308 12 βˆ’ βˆ’ PROPN cana-5930 308 13 𝑛 𝑛 PROPN cana-5930 308 14 , , PUNCT cana-5930 308 15 γ𝑐𝑑𝑑(𝑇2(𝐺 γ𝑐𝑑𝑑(𝑇2(𝐺 NOUN cana-5930 308 16 ) ) PUNCT cana-5930 308 17 ) ) PUNCT cana-5930 309 1 = = PUNCT cana-5930 309 2 𝑝 𝑝 PROPN cana-5930 309 3 then then ADV cana-5930 309 4 γ𝑐𝑑𝑑(𝑇2(𝐺 γ𝑐𝑑𝑑(𝑇2(𝐺 NOUN cana-5930 309 5 ) ) PUNCT cana-5930 309 6 ) ) PUNCT cana-5930 310 1 + + CCONJ cana-5930 310 2 Ξ΄(𝐺 Ξ΄(𝐺 X cana-5930 310 3 ) ) PUNCT cana-5930 310 4 = = SYM cana-5930 310 5 2𝑝 2𝑝 NOUN cana-5930 310 6 βˆ’ βˆ’ NOUN cana-5930 310 7 𝑛. 𝑛. NOUN cana-5930 310 8 conversely conversely ADV cana-5930 310 9 , , PUNCT cana-5930 310 10 γ𝑐𝑑𝑑(𝑇2(𝐺 γ𝑐𝑑𝑑(𝑇2(𝐺 NOUN cana-5930 310 11 ) ) PUNCT cana-5930 310 12 ) ) PUNCT cana-5930 311 1 + + CCONJ cana-5930 311 2 Ξ΄(𝐺 Ξ΄(𝐺 X cana-5930 311 3 ) ) PUNCT cana-5930 311 4 = = SYM cana-5930 311 5 2𝑝 2𝑝 NOUN cana-5930 311 6 βˆ’ βˆ’ NOUN cana-5930 311 7 𝑛 𝑛 PRON cana-5930 311 8 only only ADV cana-5930 311 9 possible possible ADJ cana-5930 311 10 if if SCONJ cana-5930 311 11 ( ( PUNCT cana-5930 311 12 i i NOUN cana-5930 311 13 ) ) PUNCT cana-5930 311 14 γ𝑐𝑑𝑑(𝑇2(𝐺 γ𝑐𝑑𝑑(𝑇2(𝐺 NOUN cana-5930 311 15 ) ) PUNCT cana-5930 311 16 ) ) PUNCT cana-5930 312 1 = = PUNCT cana-5930 312 2 𝑝 𝑝 ADP cana-5930 312 3 βˆ’ βˆ’ NOUN cana-5930 312 4 1 1 NUM cana-5930 312 5 and and CCONJ cana-5930 312 6 Ξ΄(𝐺 Ξ΄(𝐺 NOUN cana-5930 312 7 ) ) PUNCT cana-5930 312 8 = = SYM cana-5930 312 9 𝑝 𝑝 PROPN cana-5930 312 10 βˆ’ βˆ’ PROPN cana-5930 312 11 ( ( PUNCT cana-5930 312 12 𝑛 𝑛 PROPN cana-5930 312 13 βˆ’ βˆ’ PROPN cana-5930 312 14 1 1 NUM cana-5930 312 15 ) ) PUNCT cana-5930 312 16 in in ADP cana-5930 312 17 this this DET cana-5930 312 18 case case NOUN cana-5930 312 19 γ𝑐𝑑𝑑(𝑇2(𝐺 γ𝑐𝑑𝑑(𝑇2(𝐺 NOUN cana-5930 312 20 ) ) PUNCT cana-5930 312 21 ) ) PUNCT cana-5930 313 1 = = PUNCT cana-5930 313 2 𝑝 𝑝 NOUN cana-5930 313 3 βˆ’ βˆ’ NOUN cana-5930 313 4 1 1 NUM cana-5930 313 5 if if SCONJ cana-5930 313 6 and and CCONJ cana-5930 313 7 only only ADV cana-5930 313 8 if if SCONJ cana-5930 313 9 g g PROPN cana-5930 313 10 is be AUX cana-5930 313 11 a a DET cana-5930 313 12 tree tree NOUN cana-5930 313 13 on on ADP cana-5930 313 14 p p NOUN cana-5930 313 15 vertices vertex NOUN cana-5930 313 16 . . PUNCT cana-5930 314 1 but but CCONJ cana-5930 314 2 for for ADP cana-5930 314 3 a a DET cana-5930 314 4 star star NOUN cana-5930 314 5 Ξ΄(𝐺 Ξ΄(𝐺 PROPN cana-5930 314 6 ) ) PUNCT cana-5930 314 7 = = SYM cana-5930 314 8 𝑝 𝑝 ADP cana-5930 314 9 βˆ’ βˆ’ PROPN cana-5930 314 10 1 1 NUM cana-5930 314 11 and and CCONJ cana-5930 314 12 π‘‘π‘–π‘Žπ‘š(𝐺 π‘‘π‘–π‘Žπ‘š(𝐺 PRON cana-5930 314 13 ) ) PUNCT cana-5930 314 14 = = SYM cana-5930 314 15 2 2 X cana-5930 314 16 . . PUNCT cana-5930 314 17 ( ( PUNCT cana-5930 314 18 ii ii NOUN cana-5930 314 19 ) ) PUNCT cana-5930 314 20 if if SCONJ cana-5930 314 21 g g PROPN cana-5930 314 22 is be AUX cana-5930 314 23 a a DET cana-5930 314 24 broom broom NOUN cana-5930 314 25 graph graph NOUN cana-5930 314 26 on on ADP cana-5930 314 27 p p NOUN cana-5930 314 28 vertices vertex NOUN cana-5930 314 29 with with ADP cana-5930 314 30 path path NOUN cana-5930 314 31 of of ADP cana-5930 314 32 length length NOUN cana-5930 314 33 2 2 NUM cana-5930 314 34 , , PUNCT cana-5930 314 35 Ξ΄(𝐺 Ξ΄(𝐺 ADJ cana-5930 314 36 ) ) PUNCT cana-5930 314 37 = = SYM cana-5930 314 38 𝑝 𝑝 NOUN cana-5930 314 39 βˆ’ βˆ’ ADP cana-5930 314 40 2 2 NUM cana-5930 314 41 and and CCONJ cana-5930 314 42 π‘‘π‘–π‘Žπ‘š(𝐺 π‘‘π‘–π‘Žπ‘š(𝐺 PRON cana-5930 314 43 ) ) PUNCT cana-5930 314 44 = = SYM cana-5930 314 45 3 3 X cana-5930 314 46 . . PUNCT cana-5930 314 47 γ𝑐𝑑𝑑(𝑇2(𝐺 γ𝑐𝑑𝑑(𝑇2(𝐺 NOUN cana-5930 314 48 ) ) PUNCT cana-5930 314 49 ) ) PUNCT cana-5930 315 1 + + CCONJ cana-5930 316 1 Ξ΄(𝐺 Ξ΄(𝐺 X cana-5930 316 2 ) ) PUNCT cana-5930 316 3 = = SYM cana-5930 316 4 𝑝 𝑝 ADP cana-5930 316 5 βˆ’ βˆ’ NOUN cana-5930 316 6 1 1 NUM cana-5930 317 1 + + X cana-5930 317 2 𝑝 𝑝 NOUN cana-5930 317 3 βˆ’ βˆ’ PROPN cana-5930 317 4 2 2 NUM cana-5930 317 5 = = SYM cana-5930 317 6 2𝑝 2𝑝 NOUN cana-5930 317 7 βˆ’ βˆ’ NOUN cana-5930 317 8 3 3 NUM cana-5930 317 9 therefore therefore ADV cana-5930 317 10 g g PROPN cana-5930 317 11 is be AUX cana-5930 317 12 a a DET cana-5930 317 13 broom broom NOUN cana-5930 317 14 graph graph NOUN cana-5930 317 15 of of ADP cana-5930 317 16 length length NOUN cana-5930 317 17 𝑛 𝑛 PRON cana-5930 317 18 βˆ’ βˆ’ PROPN cana-5930 317 19 1.i.e 1.i.e NUM cana-5930 317 20 . . PUNCT cana-5930 317 21 , , PUNCT cana-5930 317 22 𝑛 𝑛 PROPN cana-5930 317 23 = = PUNCT cana-5930 317 24 π‘‘π‘–π‘Žπ‘š(𝐺)(𝑛 π‘‘π‘–π‘Žπ‘š(𝐺)(𝑛 NOUN cana-5930 317 25 β‰₯ β‰₯ NOUN cana-5930 317 26 2 2 NUM cana-5930 317 27 ) ) PUNCT cana-5930 317 28 . . PUNCT cana-5930 318 1 β–‘ β–‘ PUNCT cana-5930 318 2 theorem theorem VERB cana-5930 318 3 5.9 5.9 NUM cana-5930 318 4 . . PUNCT cana-5930 319 1 let let VERB cana-5930 319 2 𝑑1 𝑑1 NOUN cana-5930 319 3 and and CCONJ cana-5930 319 4 𝑑2 𝑑2 NOUN cana-5930 319 5 be be AUX cana-5930 319 6 two two NUM cana-5930 319 7 trees tree NOUN cana-5930 319 8 with with ADP cana-5930 319 9 order order NOUN cana-5930 319 10 𝑝1 𝑝1 NOUN cana-5930 319 11 β‰₯ β‰₯ NOUN cana-5930 319 12 2 2 NUM cana-5930 319 13 and and CCONJ cana-5930 319 14 𝑝2 𝑝2 PROPN cana-5930 319 15 β‰₯ β‰₯ NOUN cana-5930 319 16 2 2 NUM cana-5930 319 17 respectively respectively ADV cana-5930 319 18 . . PUNCT cana-5930 320 1 then then ADV cana-5930 320 2 γ𝑐𝑑𝑑(𝑇2(𝑑1 γ𝑐𝑑𝑑(𝑇2(𝑑1 VERB cana-5930 320 3 ∘ ∘ PROPN cana-5930 320 4 𝑑2 𝑑2 PROPN cana-5930 320 5 ) ) PUNCT cana-5930 320 6 ) ) PUNCT cana-5930 320 7 ≀ ≀ NUM cana-5930 321 1 2γ𝑐𝑑𝑑(𝑑1 2γ𝑐𝑑𝑑(𝑑1 NUM cana-5930 321 2 ∘ ∘ NUM cana-5930 321 3 𝑑2 𝑑2 PROPN cana-5930 321 4 ) ) PUNCT cana-5930 321 5 + + CCONJ cana-5930 321 6 𝑝1(1 𝑝1(1 ADJ cana-5930 321 7 + + NUM cana-5930 321 8 𝑝2 𝑝2 NOUN cana-5930 321 9 ) ) PUNCT cana-5930 321 10 βˆ’ βˆ’ PROPN cana-5930 322 1 1 1 X cana-5930 322 2 . . PUNCT cana-5930 323 1 proof proof NOUN cana-5930 323 2 . . PUNCT cana-5930 324 1 we we PRON cana-5930 324 2 have have VERB cana-5930 324 3 γ𝑐𝑑𝑑(𝐺 γ𝑐𝑑𝑑(𝐺 X cana-5930 324 4 ) ) PUNCT cana-5930 324 5 ≀ ≀ NOUN cana-5930 324 6 𝑝1(𝑝2 𝑝1(𝑝2 ADP cana-5930 324 7 βˆ’ βˆ’ NOUN cana-5930 324 8 1)[4 1)[4 NUM cana-5930 324 9 ] ] PUNCT cana-5930 324 10 . . PUNCT cana-5930 325 1 let let VERB cana-5930 325 2 d d PRON cana-5930 325 3 be be AUX cana-5930 325 4 the the DET cana-5930 325 5 minimum minimum ADJ cana-5930 325 6 ctd ctd NOUN cana-5930 325 7 - - PUNCT cana-5930 325 8 set set NOUN cana-5930 325 9 of of ADP cana-5930 325 10 𝑑1 𝑑1 NOUN cana-5930 325 11 ∘ ∘ PROPN cana-5930 325 12 𝑑2 𝑑2 PROPN cana-5930 325 13 . . PUNCT cana-5930 326 1 hence hence ADV cana-5930 326 2 |𝐷| |𝐷| VERB cana-5930 326 3 ≀ ≀ PROPN cana-5930 326 4 𝑝1(𝑝2 𝑝1(𝑝2 ADP cana-5930 326 5 βˆ’ βˆ’ PROPN cana-5930 326 6 1 1 NUM cana-5930 326 7 ) ) PUNCT cana-5930 326 8 . . PUNCT cana-5930 327 1 let let VERB cana-5930 327 2 𝐷′ 𝐷′ PROPN cana-5930 327 3 be be AUX cana-5930 327 4 the the DET cana-5930 327 5 number number NOUN cana-5930 327 6 of of ADP cana-5930 327 7 edge edge NOUN cana-5930 327 8 vertices vertex NOUN cana-5930 327 9 of of ADP cana-5930 327 10 𝑇2(𝑑1 𝑇2(𝑑1 PROPN cana-5930 327 11 ∘ ∘ PROPN cana-5930 327 12 𝑑2 𝑑2 PROPN cana-5930 327 13 ) ) PUNCT cana-5930 327 14 . . PUNCT cana-5930 328 1 then then ADV cana-5930 328 2 𝐷 𝐷 PROPN cana-5930 328 3 βˆͺ βˆͺ VERB cana-5930 328 4 𝐷′ 𝐷′ NOUN cana-5930 328 5 βŠ† βŠ† NUM cana-5930 328 6 𝑉(𝑇2(𝐺 𝑉(𝑇2(𝐺 NOUN cana-5930 328 7 ) ) PUNCT cana-5930 328 8 ) ) PUNCT cana-5930 328 9 is be AUX cana-5930 328 10 a a DET cana-5930 328 11 minimum minimum ADJ cana-5930 328 12 ctdset ctdset NOUN cana-5930 328 13 of of ADP cana-5930 328 14 𝑇2(𝐺 𝑇2(𝐺 PROPN cana-5930 328 15 ) ) PUNCT cana-5930 328 16 . . PUNCT cana-5930 329 1 therefore therefore ADV cana-5930 329 2 , , PUNCT cana-5930 329 3 γ𝑐𝑑𝑑(𝑇2(𝐺 γ𝑐𝑑𝑑(𝑇2(𝐺 NOUN cana-5930 329 4 ) ) PUNCT cana-5930 329 5 ) ) PUNCT cana-5930 330 1 ≀ ≀ NOUN cana-5930 330 2 |𝐷 |𝐷 NOUN cana-5930 330 3 βˆͺ βˆͺ ADP cana-5930 330 4 𝐷′| 𝐷′| PROPN cana-5930 330 5 ≀ ≀ NOUN cana-5930 330 6 2𝑝1(𝑝2 2𝑝1(𝑝2 NUM cana-5930 330 7 βˆ’ βˆ’ NUM cana-5930 330 8 1 1 NUM cana-5930 330 9 ) ) PUNCT cana-5930 330 10 + + CCONJ cana-5930 330 11 𝑝1(1 𝑝1(1 ADJ cana-5930 330 12 + + NUM cana-5930 330 13 𝑝2 𝑝2 NOUN cana-5930 330 14 ) ) PUNCT cana-5930 330 15 βˆ’ βˆ’ PROPN cana-5930 330 16 1 1 NUM cana-5930 330 17 ≀ ≀ NOUN cana-5930 330 18 2γ𝑐𝑑𝑑(𝐺 2γ𝑐𝑑𝑑(𝐺 NUM cana-5930 330 19 ) ) PUNCT cana-5930 330 20 + + CCONJ cana-5930 330 21 𝑝1(1 𝑝1(1 ADJ cana-5930 330 22 + + NUM cana-5930 330 23 𝑝2 𝑝2 NOUN cana-5930 330 24 ) ) PUNCT cana-5930 330 25 βˆ’ βˆ’ PROPN cana-5930 331 1 1 1 NUM cana-5930 331 2 . . PUNCT cana-5930 331 3 β–‘ β–‘ PUNCT cana-5930 331 4 theorem theorem VERB cana-5930 331 5 5.10 5.10 NUM cana-5930 331 6 . . PUNCT cana-5930 332 1 let let VERB cana-5930 332 2 g g PRON cana-5930 332 3 be be AUX cana-5930 332 4 a a DET cana-5930 332 5 ( ( PUNCT cana-5930 332 6 𝑝 𝑝 PROPN cana-5930 332 7 , , PUNCT cana-5930 332 8 π‘ž π‘ž NOUN cana-5930 332 9 ) ) PUNCT cana-5930 332 10 , , PUNCT cana-5930 332 11 𝑝 𝑝 PROPN cana-5930 332 12 β‰₯ β‰₯ NOUN cana-5930 332 13 5 5 NUM cana-5930 332 14 , , PUNCT cana-5930 332 15 graph graph VERB cana-5930 332 16 such such ADJ cana-5930 332 17 that that SCONJ cana-5930 332 18 both both CCONJ cana-5930 332 19 g g PROPN cana-5930 332 20 and and CCONJ cana-5930 332 21 𝐺 𝐺 PROPN cana-5930 332 22 are be AUX cana-5930 332 23 connected connect VERB cana-5930 332 24 then then ADV cana-5930 332 25 ( ( PUNCT cana-5930 332 26 i i NOUN cana-5930 332 27 ) ) PUNCT cana-5930 332 28 8 8 NUM cana-5930 332 29 ≀ ≀ NUM cana-5930 332 30 γ𝑐𝑑𝑑(𝑇2(𝐺 γ𝑐𝑑𝑑(𝑇2(𝐺 NOUN cana-5930 332 31 ) ) PUNCT cana-5930 332 32 ) ) PUNCT cana-5930 333 1 + + CCONJ cana-5930 333 2 γ𝑐𝑑𝑑 γ𝑐𝑑𝑑 NOUN cana-5930 333 3 ( ( PUNCT cana-5930 333 4 𝑇2(𝐺 𝑇2(𝐺 NOUN cana-5930 333 5 ) ) PUNCT cana-5930 333 6 ) ) PUNCT cana-5930 333 7 ≀ ≀ ADV cana-5930 333 8 2(𝑝 2(𝑝 NUM cana-5930 333 9 + + CCONJ cana-5930 333 10 π‘ž π‘ž X cana-5930 333 11 βˆ’ βˆ’ PROPN cana-5930 333 12 4 4 NUM cana-5930 333 13 ) ) PUNCT cana-5930 333 14 ( ( PUNCT cana-5930 333 15 ii ii NOUN cana-5930 333 16 ) ) PUNCT cana-5930 333 17 4 4 NUM cana-5930 333 18 ≀ ≀ NOUN cana-5930 333 19 γ𝑐𝑑𝑑(𝑇2(𝐺 γ𝑐𝑑𝑑(𝑇2(𝐺 NOUN cana-5930 333 20 ) ) PUNCT cana-5930 333 21 ) ) PUNCT cana-5930 333 22 . . PUNCT cana-5930 334 1 γ𝑐𝑑𝑑 γ𝑐𝑑𝑑 PROPN cana-5930 334 2 ( ( PUNCT cana-5930 334 3 𝑇2(𝐺 𝑇2(𝐺 NOUN cana-5930 334 4 ) ) PUNCT cana-5930 334 5 ) ) PUNCT cana-5930 334 6 ≀ ≀ NOUN cana-5930 334 7 ( ( PUNCT cana-5930 334 8 𝑝 𝑝 AUX cana-5930 334 9 + + NUM cana-5930 334 10 π‘ž π‘ž X cana-5930 334 11 βˆ’ βˆ’ PROPN cana-5930 334 12 4)2 4)2 PROPN cana-5930 334 13 communications communication NOUN cana-5930 334 14 on on ADP cana-5930 334 15 applied apply VERB cana-5930 334 16 nonlinear nonlinear ADJ cana-5930 334 17 analysis analysis NOUN cana-5930 334 18 issn issn NOUN cana-5930 334 19 : : PUNCT cana-5930 334 20 1074 1074 NUM cana-5930 334 21 - - PUNCT cana-5930 334 22 133x 133x NUM cana-5930 334 23 vol vol VERB cana-5930 334 24 32 32 NUM cana-5930 334 25 no no NOUN cana-5930 334 26 . . PUNCT cana-5930 335 1 10s 10 NOUN cana-5930 335 2 ( ( PUNCT cana-5930 335 3 2025 2025 NUM cana-5930 335 4 ) ) PUNCT cana-5930 335 5 3112 3112 NUM cana-5930 335 6 https://internationalpubls.com https://internationalpubls.com X cana-5930 335 7 proof proof NOUN cana-5930 335 8 . . PUNCT cana-5930 336 1 by by ADP cana-5930 336 2 theorem theorem ADJ cana-5930 336 3 3.4 3.4 NUM cana-5930 336 4 γ𝑐𝑑𝑑(𝑇2(𝐺 γ𝑐𝑑𝑑(𝑇2(𝐺 NOUN cana-5930 336 5 ) ) PUNCT cana-5930 336 6 ) ) PUNCT cana-5930 337 1 = = PUNCT cana-5930 337 2 𝑝 𝑝 PROPN cana-5930 337 3 + + NUM cana-5930 337 4 π‘ž π‘ž X cana-5930 337 5 βˆ’ βˆ’ PROPN cana-5930 337 6 2 2 NUM cana-5930 337 7 if if SCONJ cana-5930 337 8 and and CCONJ cana-5930 337 9 only only ADV cana-5930 337 10 if if SCONJ cana-5930 337 11 g g PROPN cana-5930 337 12 is be AUX cana-5930 337 13 a a DET cana-5930 337 14 graph graph NOUN cana-5930 337 15 𝐾2 𝐾2 NOUN cana-5930 337 16 . . PUNCT cana-5930 338 1 but but CCONJ cana-5930 338 2 in in ADP cana-5930 338 3 this this DET cana-5930 338 4 case case NOUN cana-5930 338 5 𝐺 𝐺 NOUN cana-5930 338 6 is be AUX cana-5930 338 7 disconnected disconnect VERB cana-5930 338 8 . . PUNCT cana-5930 339 1 γ𝑐𝑑𝑑(𝑇2(𝐺 γ𝑐𝑑𝑑(𝑇2(𝐺 NOUN cana-5930 339 2 ) ) PUNCT cana-5930 339 3 ) ) PUNCT cana-5930 340 1 = = PUNCT cana-5930 340 2 𝑝 𝑝 PROPN cana-5930 340 3 + + NUM cana-5930 340 4 π‘ž π‘ž X cana-5930 340 5 βˆ’ βˆ’ PROPN cana-5930 340 6 3 3 NUM cana-5930 340 7 if if SCONJ cana-5930 340 8 and and CCONJ cana-5930 340 9 only only ADV cana-5930 340 10 if if SCONJ cana-5930 340 11 g g PROPN cana-5930 340 12 is be AUX cana-5930 340 13 the the DET cana-5930 340 14 graph graph NOUN cana-5930 340 15 𝑃3&𝐢4 𝑃3&𝐢4 PROPN cana-5930 340 16 . . PUNCT cana-5930 341 1 but but CCONJ cana-5930 341 2 in in ADP cana-5930 341 3 this this DET cana-5930 341 4 case case NOUN cana-5930 341 5 𝐺 𝐺 NOUN cana-5930 341 6 is be AUX cana-5930 341 7 disconnected disconnect VERB cana-5930 341 8 . . PUNCT cana-5930 342 1 therefore therefore ADV cana-5930 342 2 , , PUNCT cana-5930 342 3 γ𝑐𝑑𝑑(𝑇2(𝐺 γ𝑐𝑑𝑑(𝑇2(𝐺 NOUN cana-5930 342 4 ) ) PUNCT cana-5930 342 5 ) ) PUNCT cana-5930 343 1 ≀ ≀ PROPN cana-5930 343 2 𝑝 𝑝 ADP cana-5930 343 3 + + NUM cana-5930 343 4 π‘ž π‘ž X cana-5930 343 5 βˆ’ βˆ’ PROPN cana-5930 343 6 4 4 NUM cana-5930 343 7 . . PUNCT cana-5930 343 8 hence hence ADV cana-5930 343 9 , , PUNCT cana-5930 343 10 γ𝑐𝑑𝑑(𝑇2(𝐺 γ𝑐𝑑𝑑(𝑇2(𝐺 NOUN cana-5930 343 11 ) ) PUNCT cana-5930 343 12 ) ) PUNCT cana-5930 344 1 + + CCONJ cana-5930 344 2 γ𝑐𝑑𝑑 γ𝑐𝑑𝑑 NOUN cana-5930 344 3 ( ( PUNCT cana-5930 344 4 𝑇2(𝐺 𝑇2(𝐺 NOUN cana-5930 344 5 ) ) PUNCT cana-5930 344 6 ) ) PUNCT cana-5930 344 7 ≀ ≀ ADV cana-5930 344 8 2(𝑝 2(𝑝 NUM cana-5930 344 9 + + CCONJ cana-5930 344 10 π‘ž π‘ž X cana-5930 344 11 βˆ’ βˆ’ PROPN cana-5930 344 12 4 4 NUM cana-5930 344 13 ) ) PUNCT cana-5930 344 14 . . PUNCT cana-5930 345 1 for for ADP cana-5930 345 2 lower low ADJ cana-5930 345 3 bound bound ADJ cana-5930 345 4 γ𝑐𝑑𝑑(𝑇2(𝐺 γ𝑐𝑑𝑑(𝑇2(𝐺 NOUN cana-5930 345 5 ) ) PUNCT cana-5930 345 6 ) ) PUNCT cana-5930 346 1 = = PUNCT cana-5930 346 2 4 4 NUM cana-5930 346 3 if if SCONJ cana-5930 346 4 and and CCONJ cana-5930 346 5 only only ADV cana-5930 346 6 if if SCONJ cana-5930 346 7 𝐺 𝐺 PROPN cana-5930 346 8 β‰… β‰… PROPN cana-5930 346 9 𝑃4 𝑃4 PROPN cana-5930 346 10 and and CCONJ cana-5930 346 11 𝐢5 𝐢5 NOUN cana-5930 346 12 . . PUNCT cana-5930 347 1 in in ADP cana-5930 347 2 this this DET cana-5930 347 3 case case NOUN cana-5930 347 4 𝐺 𝐺 NOUN cana-5930 347 5 is be AUX cana-5930 347 6 connected connect VERB cana-5930 347 7 . . PUNCT cana-5930 348 1 hence hence ADV cana-5930 348 2 γ𝑐𝑑𝑑(𝐺 γ𝑐𝑑𝑑(𝐺 PROPN cana-5930 348 3 ) ) PUNCT cana-5930 349 1 + + SYM cana-5930 349 2 γ𝑐𝑑𝑑(𝐺 γ𝑐𝑑𝑑(𝐺 X cana-5930 349 3 ) ) PUNCT cana-5930 349 4 β‰₯ β‰₯ NOUN cana-5930 349 5 4 4 NUM cana-5930 349 6 . . PUNCT cana-5930 349 7 ( ( PUNCT cana-5930 349 8 ii ii NOUN cana-5930 349 9 ) ) PUNCT cana-5930 349 10 follows follow VERB cana-5930 349 11 similarly similarly ADV cana-5930 349 12 . . PUNCT cana-5930 350 1 β–‘ β–‘ PUNCT cana-5930 350 2 refrences refrence NOUN cana-5930 350 3 [ [ X cana-5930 350 4 1 1 X cana-5930 350 5 ] ] X cana-5930 350 6 b. b. PROPN cana-5930 350 7 basavanagoud basavanagoud PROPN cana-5930 350 8 , , PUNCT cana-5930 350 9 s.m s.m PROPN cana-5930 350 10 . . PROPN cana-5930 350 11 hosamani hosamani PROPN cana-5930 350 12 , , PUNCT cana-5930 350 13 and and CCONJ cana-5930 350 14 s.h s.h PROPN cana-5930 350 15 . . PROPN cana-5930 350 16 malghan malghan PROPN cana-5930 350 17 . . PUNCT cana-5930 351 1 β€œ " PUNCT cana-5930 351 2 domination domination NOUN cana-5930 351 3 insemi insemi NOUN cana-5930 351 4 total total ADJ cana-5930 351 5 - - PUNCT cana-5930 351 6 point point NOUN cana-5930 351 7 graph graph NOUN cana-5930 351 8 ” " PUNCT cana-5930 351 9 . . PUNCT cana-5930 352 1 j. j. PROPN cana-5930 352 2 comp comp PROPN cana-5930 352 3 . . PROPN cana-5930 352 4 & & CCONJ cana-5930 352 5 math math PROPN cana-5930 352 6 . . PUNCT cana-5930 353 1 sci sci PROPN cana-5930 353 2 . . PROPN cana-5930 353 3 , , PUNCT cana-5930 353 4 1(5):598–605 1(5):598–605 PROPN cana-5930 353 5 , , PUNCT cana-5930 353 6 2010 2010 NUM cana-5930 353 7 . . PUNCT cana-5930 354 1 [ [ X cana-5930 354 2 2 2 NUM cana-5930 354 3 ] ] PUNCT cana-5930 354 4 f. f. PROPN cana-5930 354 5 harary harary PROPN cana-5930 354 6 . . PUNCT cana-5930 355 1 β€œ " PUNCT cana-5930 355 2 graph graph NOUN cana-5930 355 3 theory theory NOUN cana-5930 355 4 ” " PUNCT cana-5930 355 5 , , PUNCT cana-5930 355 6 addison addison PROPN cana-5930 355 7 - - PUNCT cana-5930 355 8 wesley wesley PROPN cana-5930 355 9 , , PUNCT cana-5930 355 10 reading read VERB cana-5930 355 11 mass mass PROPN cana-5930 355 12 . . PUNCT cana-5930 355 13 1972 1972 NUM cana-5930 355 14 . . PUNCT cana-5930 356 1 [ [ X cana-5930 356 2 3 3 X cana-5930 356 3 ] ] PUNCT cana-5930 356 4 s. s. PROPN cana-5930 356 5 muthammai muthammai PROPN cana-5930 356 6 , , PUNCT cana-5930 356 7 m. m. NOUN cana-5930 356 8 bhanumathi bhanumathi PROPN cana-5930 356 9 and and CCONJ cana-5930 356 10 p. p. PROPN cana-5930 356 11 vidhya vidhya PROPN cana-5930 356 12 , , PUNCT cana-5930 356 13 β€œ " PUNCT cana-5930 356 14 complementary complementary ADJ cana-5930 356 15 tree tree NOUN cana-5930 356 16 domination domination NOUN cana-5930 356 17 number number NOUN cana-5930 356 18 of of ADP cana-5930 356 19 a a DET cana-5930 356 20 graph graph NOUN cana-5930 356 21 ” " PUNCT cana-5930 356 22 . . PUNCT cana-5930 357 1 international international PROPN cana-5930 357 2 mathematical mathematical PROPN cana-5930 357 3 forum forum PROPN cana-5930 357 4 , , PUNCT cana-5930 357 5 6(26):1273βˆ’1282 6(26):1273βˆ’1282 NUM cana-5930 357 6 , , PUNCT cana-5930 357 7 2011 2011 NUM cana-5930 357 8 . . PUNCT cana-5930 358 1 [ [ X cana-5930 358 2 4 4 X cana-5930 358 3 ] ] PUNCT cana-5930 358 4 s. s. PROPN cana-5930 358 5 muthammai muthammai PROPN cana-5930 358 6 and and CCONJ cana-5930 358 7 p. p. PROPN cana-5930 358 8 vidhya vidhya PROPN cana-5930 358 9 , , PUNCT cana-5930 358 10 β€œ " PUNCT cana-5930 358 11 more more ADJ cana-5930 358 12 results result NOUN cana-5930 358 13 on on ADP cana-5930 358 14 complementary complementary ADJ cana-5930 358 15 tree tree NOUN cana-5930 358 16 domination domination NOUN cana-5930 358 17 number number NOUN cana-5930 358 18 of of ADP cana-5930 358 19 graphs”,international graphs”,international ADJ cana-5930 358 20 journal journal NOUN cana-5930 358 21 of of ADP cana-5930 358 22 mathematics mathematic NOUN cana-5930 358 23 and and CCONJ cana-5930 358 24 its its PRON cana-5930 358 25 applications application NOUN cana-5930 358 26 , , PUNCT cana-5930 358 27 vol vol NOUN cana-5930 358 28 . . PROPN cana-5930 358 29 4 4 NUM cana-5930 358 30 , , PUNCT cana-5930 358 31 no no INTJ cana-5930 358 32 . . NOUN cana-5930 358 33 1 1 NUM cana-5930 358 34 - - SYM cana-5930 358 35 d d PROPN cana-5930 358 36 , , PUNCT cana-5930 358 37 pp pp ADJ cana-5930 358 38 . . PUNCT cana-5930 359 1 17 17 NUM cana-5930 359 2 - - SYM cana-5930 359 3 20 20 NUM cana-5930 359 4 , , PUNCT cana-5930 359 5 2016 2016 NUM cana-5930 359 6 . . PUNCT cana-5930 360 1 [ [ X cana-5930 360 2 5 5 X cana-5930 360 3 ] ] PUNCT cana-5930 360 4 s. s. PROPN cana-5930 360 5 muthammai muthammai PROPN cana-5930 360 6 and and CCONJ cana-5930 360 7 p. p. NOUN cana-5930 360 8 vidhya vidhya PROPN cana-5930 360 9 β€œ " PUNCT cana-5930 360 10 complementary complementary ADJ cana-5930 360 11 tree tree NOUN cana-5930 360 12 domination domination NOUN cana-5930 360 13 in in ADP cana-5930 360 14 splitting splitting NOUN cana-5930 360 15 graphs graph NOUN cana-5930 360 16 of of ADP cana-5930 360 17 graphs”,international graphs”,international ADJ cana-5930 360 18 journal journal NOUN cana-5930 360 19 of of ADP cana-5930 360 20 mathematics mathematics NOUN cana-5930 360 21 trends trend NOUN cana-5930 360 22 and and CCONJ cana-5930 360 23 technology technology NOUN cana-5930 360 24 vol vol NOUN cana-5930 360 25 . . PROPN cana-5930 361 1 31 31 NUM cana-5930 362 1 no.2,pp no.2,pp ADJ cana-5930 362 2 . . PUNCT cana-5930 363 1 53 53 NUM cana-5930 363 2 - - SYM cana-5930 363 3 56 56 NUM cana-5930 363 4 , , PUNCT cana-5930 363 5 2016 2016 NUM cana-5930 363 6 . . PUNCT cana-5930 364 1 [ [ X cana-5930 364 2 6 6 NUM cana-5930 364 3 ] ] PUNCT cana-5930 364 4 o. o. ADJ cana-5930 364 5 ore ore PROPN cana-5930 364 6 . . PUNCT cana-5930 365 1 β€œ " PUNCT cana-5930 365 2 theory theory NOUN cana-5930 365 3 of of ADP cana-5930 365 4 graphs graph NOUN cana-5930 365 5 ” " PUNCT cana-5930 365 6 , , PUNCT cana-5930 365 7 amer amer PROPN cana-5930 365 8 . . PROPN cana-5930 365 9 math math PROPN cana-5930 365 10 soc soc PROPN cana-5930 365 11 . . PUNCT cana-5930 366 1 colloq colloq PROPN cana-5930 366 2 . . PUNCT cana-5930 367 1 publ publ PROPN cana-5930 367 2 . . PUNCT cana-5930 367 3 , , PUNCT cana-5930 367 4 providence.38 providence.38 PROPN cana-5930 367 5 , , PUNCT cana-5930 367 6 1962 1962 NUM cana-5930 367 7 . . PUNCT cana-5930 368 1 [ [ X cana-5930 368 2 7 7 X cana-5930 368 3 ] ] X cana-5930 368 4 e. e. PROPN cana-5930 368 5 sampath sampath PROPN cana-5930 368 6 kumar kumar PROPN cana-5930 368 7 and and CCONJ cana-5930 368 8 s.b s.b PROPN cana-5930 368 9 . . PROPN cana-5930 368 10 chikkodimath chikkodimath NOUN cana-5930 368 11 . . PUNCT cana-5930 369 1 β€œ " PUNCT cana-5930 369 2 semi semi ADJ cana-5930 369 3 - - ADJ cana-5930 369 4 total total ADJ cana-5930 369 5 graphs graph NOUN cana-5930 369 6 of of ADP cana-5930 369 7 a a DET cana-5930 369 8 graph graph NOUN cana-5930 369 9 i i NOUN cana-5930 369 10 ” " PUNCT cana-5930 369 11 . . PUNCT cana-5930 370 1 journal journal NOUN cana-5930 370 2 of of ADP cana-5930 370 3 the the DET cana-5930 370 4 karnatak karnatak PROPN cana-5930 370 5 university university NOUN cana-5930 370 6 - - PUNCT cana-5930 370 7 science science NOUN cana-5930 370 8 , , PUNCT cana-5930 370 9 xviii:274–280 xviii:274–280 NUM cana-5930 370 10 , , PUNCT cana-5930 370 11 1973 1973 NUM cana-5930 370 12 . . PUNCT cana-5930 371 1 [ [ X cana-5930 371 2 8 8 NUM cana-5930 371 3 ] ] X cana-5930 371 4 p. p. NOUN cana-5930 371 5 vidhya vidhya PROPN cana-5930 371 6 , , PUNCT cana-5930 371 7 s. s. PROPN cana-5930 371 8 jayalakshmi jayalakshmi PROPN cana-5930 371 9 , , PUNCT cana-5930 371 10 and and CCONJ cana-5930 371 11 g. g. PROPN cana-5930 371 12 mahalakshmi mahalakshmi PROPN cana-5930 371 13 . . PUNCT cana-5930 372 1 β€œ " PUNCT cana-5930 372 2 complementary complementary ADJ cana-5930 372 3 tree tree NOUN cana-5930 372 4 domination domination NOUN cana-5930 372 5 number number NOUN cana-5930 372 6 of of ADP cana-5930 372 7 semi semi ADJ cana-5930 372 8 total total ADJ cana-5930 372 9 point point NOUN cana-5930 372 10 graph graph NOUN cana-5930 372 11 ” " PUNCT cana-5930 372 12 . . PUNCT cana-5930 373 1 the the DET cana-5930 373 2 international international ADJ cana-5930 373 3 journal journal NOUN cana-5930 373 4 of of ADP cana-5930 373 5 analytical analytical ADJ cana-5930 373 6 and and CCONJ cana-5930 373 7 experimental experimental ADJ cana-5930 373 8 modal modal ADJ cana-5930 373 9 analysis analysis NOUN cana-5930 373 10 , , PUNCT cana-5930 373 11 xii:682–689 xii:682–689 NOUN cana-5930 373 12 , , PUNCT cana-5930 373 13 2020 2020 NUM cana-5930 373 14 . . PUNCT