id sid tid token lemma pos cana-3627 1 1 communications communication NOUN cana-3627 1 2 on on ADP cana-3627 1 3 applied apply VERB cana-3627 1 4 nonlinear nonlinear ADJ cana-3627 1 5 analysis analysis NOUN cana-3627 1 6 issn issn NOUN cana-3627 1 7 : : PUNCT cana-3627 1 8 1074 1074 NUM cana-3627 1 9 - - PUNCT cana-3627 1 10 133x 133x NUM cana-3627 1 11 vol vol NOUN cana-3627 1 12 32 32 NUM cana-3627 1 13 no no NOUN cana-3627 1 14 . . PUNCT cana-3627 2 1 8s 8s PROPN cana-3627 2 2 ( ( PUNCT cana-3627 2 3 2025 2025 NUM cana-3627 2 4 ) ) PUNCT cana-3627 2 5 182 182 NUM cana-3627 2 6 https://internationalpubls.com https://internationalpubls.com X cana-3627 2 7 effect effect NOUN cana-3627 2 8 of of ADP cana-3627 2 9 edge edge NOUN cana-3627 2 10 deletion deletion NOUN cana-3627 2 11 on on ADP cana-3627 2 12 the the DET cana-3627 2 13 weak weak ADJ cana-3627 2 14 roman roman ADJ cana-3627 2 15 domination domination NOUN cana-3627 2 16 number number NOUN cana-3627 2 17 of of ADP cana-3627 2 18 a a DET cana-3627 2 19 graph graph NOUN cana-3627 2 20 p. p. NOUN cana-3627 2 21 roushini roushini NOUN cana-3627 2 22 leely leely ADV cana-3627 2 23 pushpam1 pushpam1 PROPN cana-3627 2 24 , , PUNCT cana-3627 2 25 m. m. NOUN cana-3627 2 26 kamalam2 kamalam2 PROPN cana-3627 2 27 1 1 NUM cana-3627 2 28 department department NOUN cana-3627 2 29 of of ADP cana-3627 2 30 mathematics mathematics PROPN cana-3627 2 31 , , PUNCT cana-3627 2 32 d. d. PROPN cana-3627 2 33 b. b. PROPN cana-3627 2 34 jain jain PROPN cana-3627 2 35 college college PROPN cana-3627 2 36 , , PUNCT cana-3627 2 37 chennai chennai PROPN cana-3627 2 38 600097 600097 NUM cana-3627 2 39 , , PUNCT cana-3627 2 40 tamil tamil PROPN cana-3627 2 41 nadu nadu PROPN cana-3627 2 42 , , PUNCT cana-3627 2 43 india india PROPN cana-3627 2 44 . . PUNCT cana-3627 3 1 e e X cana-3627 3 2 - - NOUN cana-3627 3 3 mail mail NOUN cana-3627 3 4 : : PUNCT cana-3627 3 5 roushinip@yahoo.com roushinip@yahoo.com X cana-3627 3 6 2 2 NUM cana-3627 3 7 shri shri PROPN cana-3627 3 8 s. s. PROPN cana-3627 3 9 s. s. PROPN cana-3627 3 10 shasun shasun PROPN cana-3627 3 11 jain jain PROPN cana-3627 3 12 college college PROPN cana-3627 3 13 for for ADP cana-3627 3 14 women woman NOUN cana-3627 3 15 , , PUNCT cana-3627 3 16 chennai chennai PROPN cana-3627 3 17 600017 600017 NUM cana-3627 3 18 , , PUNCT cana-3627 3 19 tamil tamil PROPN cana-3627 3 20 nadu nadu PROPN cana-3627 3 21 , , PUNCT cana-3627 3 22 india india PROPN cana-3627 3 23 . . PUNCT cana-3627 4 1 e e X cana-3627 4 2 - - NOUN cana-3627 4 3 mail mail NOUN cana-3627 4 4 : : PUNCT cana-3627 4 5 divinegrace27@gmail.com divinegrace27@gmail.com X cana-3627 4 6 article article NOUN cana-3627 4 7 history history NOUN cana-3627 4 8 : : PUNCT cana-3627 4 9 received receive VERB cana-3627 4 10 : : PUNCT cana-3627 4 11 26 26 NUM cana-3627 4 12 - - SYM cana-3627 4 13 10 10 NUM cana-3627 4 14 - - PUNCT cana-3627 4 15 2024 2024 NUM cana-3627 4 16 revised revise VERB cana-3627 4 17 : : PUNCT cana-3627 4 18 25 25 NUM cana-3627 4 19 - - SYM cana-3627 4 20 11 11 NUM cana-3627 4 21 - - PUNCT cana-3627 4 22 2024 2024 NUM cana-3627 4 23 accepted accept VERB cana-3627 4 24 : : PUNCT cana-3627 4 25 22 22 NUM cana-3627 4 26 - - SYM cana-3627 4 27 12 12 NUM cana-3627 4 28 - - PUNCT cana-3627 4 29 2024 2024 NUM cana-3627 4 30 abstract abstract NOUN cana-3627 4 31 : : PUNCT cana-3627 4 32 consider consider VERB cana-3627 4 33 a a DET cana-3627 4 34 graph graph NOUN cana-3627 4 35 Ξ³ Ξ³ X cana-3627 4 36 with with ADP cana-3627 4 37 vertex vertex NOUN cana-3627 4 38 set set VERB cana-3627 4 39 𝑉 𝑉 PROPN cana-3627 4 40 and and CCONJ cana-3627 4 41 edge edge NOUN cana-3627 4 42 set set VERB cana-3627 4 43 𝐸. 𝐸. PROPN cana-3627 4 44 for for SCONJ cana-3627 4 45 a a DET cana-3627 4 46 function function NOUN cana-3627 4 47 𝑔 𝑔 NOUN cana-3627 4 48 defined define VERB cana-3627 4 49 on on ADP cana-3627 4 50 the the DET cana-3627 4 51 vertex vertex NOUN cana-3627 4 52 set set NOUN cana-3627 4 53 having have VERB cana-3627 4 54 values value NOUN cana-3627 4 55 in in ADP cana-3627 4 56 the the DET cana-3627 4 57 set set NOUN cana-3627 4 58 { { PUNCT cana-3627 4 59 0 0 NUM cana-3627 4 60 , , PUNCT cana-3627 4 61 1,2 1,2 NUM cana-3627 4 62 } } PUNCT cana-3627 4 63 , , PUNCT cana-3627 4 64 the the DET cana-3627 4 65 weight weight NOUN cana-3627 4 66 of of ADP cana-3627 4 67 a a DET cana-3627 4 68 vertex vertex NOUN cana-3627 4 69 π‘₯ π‘₯ NOUN cana-3627 4 70 is be AUX cana-3627 4 71 𝑔(π‘₯ 𝑔(π‘₯ PROPN cana-3627 4 72 ) ) PUNCT cana-3627 4 73 . . PUNCT cana-3627 5 1 the the DET cana-3627 5 2 weight weight NOUN cana-3627 5 3 of of ADP cana-3627 5 4 a a DET cana-3627 5 5 subset subset ADJ cana-3627 5 6 𝑋 𝑋 NOUN cana-3627 5 7 of of ADP cana-3627 5 8 𝑉 𝑉 PROPN cana-3627 5 9 is be AUX cana-3627 5 10 denoted denote VERB cana-3627 5 11 by by ADP cana-3627 5 12 𝑔(𝑋 𝑔(𝑋 PROPN cana-3627 5 13 ) ) PUNCT cana-3627 5 14 and and CCONJ cana-3627 5 15 is be AUX cana-3627 5 16 defined define VERB cana-3627 5 17 as as ADP cana-3627 5 18 the the DET cana-3627 5 19 sum sum NOUN cana-3627 5 20 of of ADP cana-3627 5 21 the the DET cana-3627 5 22 weights weight NOUN cana-3627 5 23 of of ADP cana-3627 5 24 all all DET cana-3627 5 25 the the DET cana-3627 5 26 vertices vertex NOUN cana-3627 5 27 in in ADP cana-3627 5 28 𝑋. 𝑋. PROPN cana-3627 5 29 an an DET cana-3627 5 30 undefended undefended ADJ cana-3627 5 31 vertex vertex NOUN cana-3627 5 32 under under ADP cana-3627 5 33 𝑔 𝑔 PROPN cana-3627 5 34 is be AUX cana-3627 5 35 a a DET cana-3627 5 36 vertex vertex NOUN cana-3627 5 37 π‘Ž π‘Ž NOUN cana-3627 5 38 in in ADP cana-3627 5 39 Ξ³ Ξ³ NOUN cana-3627 5 40 with with ADP cana-3627 5 41 a a DET cana-3627 5 42 neighbourhood neighbourhood NOUN cana-3627 5 43 with with ADP cana-3627 5 44 weight weight NOUN cana-3627 5 45 0 0 NUM cana-3627 5 46 . . PUNCT cana-3627 6 1 a a DET cana-3627 6 2 weak weak ADJ cana-3627 6 3 roman roman ADJ cana-3627 6 4 dominating dominating NOUN cana-3627 6 5 function function NOUN cana-3627 6 6 𝑔 𝑔 PROPN cana-3627 6 7 ( ( PUNCT cana-3627 6 8 π‘Šπ‘…π·πΉ π‘Šπ‘…π·πΉ NOUN cana-3627 6 9 ) ) PUNCT cana-3627 6 10 is be AUX cana-3627 6 11 a a DET cana-3627 6 12 function function NOUN cana-3627 6 13 such such ADJ cana-3627 6 14 that that PRON cana-3627 6 15 for for ADP cana-3627 6 16 any any DET cana-3627 6 17 vertex vertex NOUN cana-3627 7 1 π‘₯ π‘₯ NOUN cana-3627 7 2 having have VERB cana-3627 7 3 weight weight NOUN cana-3627 7 4 0 0 NUM cana-3627 8 1 , , PUNCT cana-3627 8 2 there there PRON cana-3627 8 3 is be VERB cana-3627 8 4 a a DET cana-3627 8 5 vertex vertex NOUN cana-3627 8 6 𝑦 𝑦 NOUN cana-3627 8 7 in in ADP cana-3627 8 8 the the DET cana-3627 8 9 open open ADJ cana-3627 8 10 neighbourhood neighbourhood NOUN cana-3627 8 11 of of ADP cana-3627 8 12 π‘₯ π‘₯ NOUN cana-3627 8 13 having have VERB cana-3627 8 14 positive positive ADJ cana-3627 8 15 weight weight NOUN cana-3627 8 16 such such ADJ cana-3627 8 17 that that SCONJ cana-3627 8 18 , , PUNCT cana-3627 8 19 under under ADP cana-3627 8 20 the the DET cana-3627 8 21 function function NOUN cana-3627 8 22 β„Ž β„Ž PROPN cana-3627 8 23 defined define VERB cana-3627 8 24 on on ADP cana-3627 8 25 𝑉 𝑉 PROPN cana-3627 8 26 having have VERB cana-3627 8 27 values value NOUN cana-3627 8 28 in in ADP cana-3627 8 29 the the DET cana-3627 8 30 set set NOUN cana-3627 8 31 { { PUNCT cana-3627 8 32 0,1,2 0,1,2 NOUN cana-3627 8 33 } } PUNCT cana-3627 8 34 defined define VERB cana-3627 8 35 by by ADP cana-3627 8 36 β„Ž(π‘₯ β„Ž(π‘₯ NOUN cana-3627 8 37 ) ) PUNCT cana-3627 8 38 = = SYM cana-3627 8 39 1 1 NUM cana-3627 8 40 , , PUNCT cana-3627 8 41 β„Ž(𝑦 β„Ž(𝑦 NOUN cana-3627 8 42 ) ) PUNCT cana-3627 8 43 = = SYM cana-3627 8 44 𝑔(𝑦 𝑔(𝑦 PROPN cana-3627 8 45 ) ) PUNCT cana-3627 8 46 βˆ’ βˆ’ PROPN cana-3627 8 47 1 1 NUM cana-3627 8 48 , , PUNCT cana-3627 8 49 β„Ž(𝑧 β„Ž(𝑧 PROPN cana-3627 8 50 ) ) PUNCT cana-3627 8 51 = = SYM cana-3627 8 52 𝑔(𝑧 𝑔(𝑧 PROPN cana-3627 8 53 ) ) PUNCT cana-3627 8 54 , , PUNCT cana-3627 8 55 if if SCONJ cana-3627 8 56 𝑧 𝑧 PRON cana-3627 8 57 ∈ ∈ PROPN cana-3627 8 58 𝑉 𝑉 PROPN cana-3627 8 59 βˆ’ βˆ’ PROPN cana-3627 8 60 { { PUNCT cana-3627 8 61 π‘₯ π‘₯ NOUN cana-3627 8 62 , , PUNCT cana-3627 8 63 𝑦 𝑦 NOUN cana-3627 8 64 } } PUNCT cana-3627 8 65 , , PUNCT cana-3627 8 66 there there PRON cana-3627 8 67 are be VERB cana-3627 8 68 no no DET cana-3627 8 69 undefended undefended ADJ cana-3627 8 70 vertices vertex NOUN cana-3627 8 71 in in ADP cana-3627 8 72 Ξ³ Ξ³ PROPN cana-3627 8 73 . . PUNCT cana-3627 9 1 the the DET cana-3627 9 2 number number NOUN cana-3627 9 3 π›Ύπ‘Ÿ(Ξ³ π›Ύπ‘Ÿ(Ξ³ NUM cana-3627 9 4 ) ) PUNCT cana-3627 9 5 , , PUNCT cana-3627 9 6 the the DET cana-3627 9 7 minimum minimum NOUN cana-3627 9 8 of of ADP cana-3627 9 9 the the DET cana-3627 9 10 weights weight NOUN cana-3627 9 11 of of ADP cana-3627 9 12 all all DET cana-3627 9 13 the the DET cana-3627 9 14 π‘Šπ‘…π·πΉπ‘  π‘Šπ‘…π·πΉπ‘  NOUN cana-3627 9 15 defined define VERB cana-3627 9 16 on on ADP cana-3627 9 17 Ξ³ Ξ³ X cana-3627 9 18 is be AUX cana-3627 9 19 called call VERB cana-3627 9 20 the the DET cana-3627 9 21 weak weak ADJ cana-3627 9 22 roman roman ADJ cana-3627 9 23 domination domination NOUN cana-3627 9 24 number number NOUN cana-3627 9 25 of of ADP cana-3627 9 26 Ξ³ Ξ³ PROPN cana-3627 9 27 . . PUNCT cana-3627 10 1 the the DET cana-3627 10 2 π‘Šπ‘…π·πΉ π‘Šπ‘…π·πΉ NOUN cana-3627 10 3 whose whose DET cana-3627 10 4 weight weight NOUN cana-3627 10 5 is be AUX cana-3627 10 6 π›Ύπ‘Ÿ(Ξ³ π›Ύπ‘Ÿ(Ξ³ NUM cana-3627 10 7 ) ) PUNCT cana-3627 10 8 is be AUX cana-3627 10 9 called call VERB cana-3627 10 10 a a DET cana-3627 10 11 π›Ύπ‘Ÿ π›Ύπ‘Ÿ ADJ cana-3627 10 12 βˆ’ βˆ’ PROPN cana-3627 10 13 function function NOUN cana-3627 10 14 and and CCONJ cana-3627 10 15 π›Ύπ‘Ÿ(Ξ³ π›Ύπ‘Ÿ(Ξ³ NUM cana-3627 10 16 ) ) PUNCT cana-3627 10 17 is be AUX cana-3627 10 18 called call VERB cana-3627 10 19 the the DET cana-3627 10 20 π›Ύπ‘Ÿ π›Ύπ‘Ÿ ADJ cana-3627 10 21 βˆ’ βˆ’ PROPN cana-3627 10 22 value value NOUN cana-3627 10 23 of of ADP cana-3627 10 24 Ξ³ Ξ³ PROPN cana-3627 10 25 . . PROPN cana-3627 11 1 with with ADP cana-3627 11 2 respect respect NOUN cana-3627 11 3 to to ADP cana-3627 11 4 a a DET cana-3627 11 5 graph graph NOUN cana-3627 11 6 Ξ³ Ξ³ X cana-3627 11 7 , , PUNCT cana-3627 11 8 we we PRON cana-3627 11 9 say say VERB cana-3627 11 10 that that SCONJ cana-3627 11 11 an an DET cana-3627 11 12 edge edge NOUN cana-3627 11 13 π‘₯ π‘₯ PRON cana-3627 11 14 ∈ ∈ NOUN cana-3627 11 15 πΈβˆ’ πΈβˆ’ VERB cana-3627 11 16 if if SCONJ cana-3627 11 17 and and CCONJ cana-3627 11 18 only only ADV cana-3627 11 19 if if SCONJ cana-3627 11 20 its its PRON cana-3627 11 21 removal removal NOUN cana-3627 11 22 will will AUX cana-3627 11 23 reduce reduce VERB cana-3627 11 24 the the DET cana-3627 11 25 π›Ύπ‘Ÿ π›Ύπ‘Ÿ ADJ cana-3627 11 26 βˆ’ βˆ’ PROPN cana-3627 11 27 value value NOUN cana-3627 11 28 of of ADP cana-3627 11 29 Ξ³ Ξ³ PROPN cana-3627 11 30 . . PROPN cana-3627 11 31 similarly similarly ADV cana-3627 11 32 , , PUNCT cana-3627 11 33 we we PRON cana-3627 11 34 say say VERB cana-3627 11 35 that that SCONJ cana-3627 11 36 an an DET cana-3627 11 37 edge edge NOUN cana-3627 11 38 π‘₯ π‘₯ X cana-3627 11 39 ∈ ∈ NOUN cana-3627 11 40 𝐸+ 𝐸+ PUNCT cana-3627 11 41 if if SCONJ cana-3627 11 42 and and CCONJ cana-3627 11 43 only only ADV cana-3627 11 44 if if SCONJ cana-3627 11 45 its its PRON cana-3627 11 46 removal removal NOUN cana-3627 11 47 will will AUX cana-3627 11 48 increase increase VERB cana-3627 11 49 the the DET cana-3627 11 50 π›Ύπ‘Ÿ π›Ύπ‘Ÿ ADJ cana-3627 11 51 βˆ’ βˆ’ PROPN cana-3627 11 52 value value NOUN cana-3627 11 53 of of ADP cana-3627 11 54 Ξ³ Ξ³ NOUN cana-3627 11 55 and and CCONJ cana-3627 11 56 an an DET cana-3627 11 57 edge edge NOUN cana-3627 11 58 π‘₯ π‘₯ DET cana-3627 11 59 ∈ ∈ NOUN cana-3627 11 60 𝐸0 𝐸0 NOUN cana-3627 12 1 if if SCONJ cana-3627 12 2 and and CCONJ cana-3627 12 3 only only ADV cana-3627 12 4 if if SCONJ cana-3627 12 5 its its PRON cana-3627 12 6 removal removal NOUN cana-3627 12 7 will will AUX cana-3627 12 8 leave leave VERB cana-3627 12 9 unaltered unaltere VERB cana-3627 12 10 the the DET cana-3627 12 11 π›Ύπ‘Ÿ π›Ύπ‘Ÿ ADJ cana-3627 12 12 βˆ’ βˆ’ PROPN cana-3627 12 13 value value NOUN cana-3627 12 14 of of ADP cana-3627 12 15 Ξ³ Ξ³ PROPN cana-3627 12 16 . . PROPN cana-3627 12 17 in in ADP cana-3627 12 18 this this DET cana-3627 12 19 paper paper NOUN cana-3627 12 20 , , PUNCT cana-3627 12 21 we we PRON cana-3627 12 22 categorize categorize VERB cana-3627 12 23 edges edge NOUN cana-3627 12 24 of of ADP cana-3627 12 25 a a DET cana-3627 12 26 graph graph NOUN cana-3627 12 27 as as ADP cana-3627 12 28 belonging belong VERB cana-3627 12 29 to to ADP cana-3627 12 30 πΈβˆ’ πΈβˆ’ ADJ cana-3627 12 31 , , PUNCT cana-3627 12 32 𝐸+ 𝐸+ X cana-3627 12 33 or or CCONJ cana-3627 12 34 𝐸0 𝐸0 NOUN cana-3627 12 35 . . PUNCT cana-3627 13 1 keywords keyword NOUN cana-3627 13 2 : : PUNCT cana-3627 13 3 weak weak ADJ cana-3627 13 4 roman roman ADJ cana-3627 13 5 dominating dominating NOUN cana-3627 13 6 number number NOUN cana-3627 13 7 , , PUNCT cana-3627 13 8 changing change VERB cana-3627 13 9 and and CCONJ cana-3627 13 10 unchanging unchanging ADJ cana-3627 13 11 edges edge NOUN cana-3627 13 12 , , PUNCT cana-3627 13 13 edge edge NOUN cana-3627 13 14 deletion deletion NOUN cana-3627 13 15 . . PUNCT cana-3627 14 1 1 1 X cana-3627 14 2 . . X cana-3627 14 3 introduction introduction NOUN cana-3627 14 4 for for ADP cana-3627 14 5 a a DET cana-3627 14 6 graph graph NOUN cana-3627 15 1 Ξ³ Ξ³ NOUN cana-3627 15 2 having have VERB cana-3627 15 3 vertex vertex NOUN cana-3627 15 4 set set VERB cana-3627 15 5 𝑉 𝑉 PROPN cana-3627 15 6 and and CCONJ cana-3627 15 7 edge edge NOUN cana-3627 15 8 set set VERB cana-3627 15 9 𝐸 𝐸 PROPN cana-3627 15 10 , , PUNCT cana-3627 15 11 the the DET cana-3627 15 12 open open ADJ cana-3627 15 13 neighbourhood neighbourhood NOUN cana-3627 15 14 of of ADP cana-3627 15 15 the the DET cana-3627 15 16 vertex vertex NOUN cana-3627 15 17 π‘₯ π‘₯ PROPN cana-3627 15 18 ( ( PUNCT cana-3627 15 19 denoted denote VERB cana-3627 15 20 by by ADP cana-3627 15 21 𝑁(π‘₯ 𝑁(π‘₯ NOUN cana-3627 15 22 ) ) PUNCT cana-3627 15 23 ) ) PUNCT cana-3627 15 24 is be AUX cana-3627 15 25 the the DET cana-3627 15 26 set set NOUN cana-3627 15 27 of of ADP cana-3627 15 28 all all DET cana-3627 15 29 those those DET cana-3627 15 30 vertices vertex NOUN cana-3627 15 31 that that PRON cana-3627 15 32 are be AUX cana-3627 15 33 adjacent adjacent ADJ cana-3627 15 34 to to ADP cana-3627 15 35 π‘₯ π‘₯ PROPN cana-3627 15 36 and and CCONJ cana-3627 15 37 the the DET cana-3627 15 38 closed closed ADJ cana-3627 15 39 neighbourhood neighbourhood NOUN cana-3627 15 40 of of ADP cana-3627 15 41 the the DET cana-3627 15 42 vertex vertex NOUN cana-3627 15 43 π‘₯ π‘₯ PROPN cana-3627 15 44 ( ( PUNCT cana-3627 15 45 denoted denote VERB cana-3627 15 46 by by ADP cana-3627 15 47 𝑁[π‘₯ 𝑁[π‘₯ NOUN cana-3627 15 48 ] ] PUNCT cana-3627 15 49 ) ) PUNCT cana-3627 15 50 is be AUX cana-3627 15 51 the the DET cana-3627 15 52 union union NOUN cana-3627 15 53 of of ADP cana-3627 15 54 the the DET cana-3627 15 55 open open ADJ cana-3627 15 56 neighbourhood neighbourhood NOUN cana-3627 15 57 of of ADP cana-3627 15 58 π‘₯ π‘₯ PROPN cana-3627 15 59 and and CCONJ cana-3627 15 60 { { PUNCT cana-3627 15 61 π‘₯ π‘₯ NOUN cana-3627 15 62 } } PUNCT cana-3627 15 63 . . PUNCT cana-3627 16 1 for for ADP cana-3627 16 2 a a DET cana-3627 16 3 function function NOUN cana-3627 16 4 𝑔 𝑔 NOUN cana-3627 16 5 defined define VERB cana-3627 16 6 on on ADP cana-3627 16 7 the the DET cana-3627 16 8 vertex vertex NOUN cana-3627 16 9 set set VERB cana-3627 16 10 𝑉 𝑉 PROPN cana-3627 16 11 and and CCONJ cana-3627 16 12 having have VERB cana-3627 16 13 values value NOUN cana-3627 16 14 in in ADP cana-3627 16 15 the the DET cana-3627 16 16 set set NOUN cana-3627 16 17 { { PUNCT cana-3627 16 18 0 0 NUM cana-3627 16 19 , , PUNCT cana-3627 16 20 1,2 1,2 NUM cana-3627 16 21 } } PUNCT cana-3627 16 22 , , PUNCT cana-3627 16 23 the the DET cana-3627 16 24 weight weight NOUN cana-3627 16 25 of of ADP cana-3627 16 26 a a DET cana-3627 16 27 vertex vertex NOUN cana-3627 16 28 π‘₯ π‘₯ NOUN cana-3627 16 29 is be AUX cana-3627 16 30 𝑔(π‘₯ 𝑔(π‘₯ PROPN cana-3627 16 31 ) ) PUNCT cana-3627 16 32 . . PUNCT cana-3627 17 1 the the DET cana-3627 17 2 weight weight NOUN cana-3627 17 3 of of ADP cana-3627 17 4 a a DET cana-3627 17 5 subset subset ADJ cana-3627 17 6 𝑋 𝑋 NOUN cana-3627 17 7 of of ADP cana-3627 17 8 𝑉 𝑉 PROPN cana-3627 17 9 is be AUX cana-3627 17 10 denoted denote VERB cana-3627 17 11 by by ADP cana-3627 17 12 𝑔(𝑋 𝑔(𝑋 PROPN cana-3627 17 13 ) ) PUNCT cana-3627 17 14 and and CCONJ cana-3627 17 15 is be AUX cana-3627 17 16 defined define VERB cana-3627 17 17 as as ADP cana-3627 17 18 the the DET cana-3627 17 19 sum sum NOUN cana-3627 17 20 of of ADP cana-3627 17 21 the the DET cana-3627 17 22 weights weight NOUN cana-3627 17 23 of of ADP cana-3627 17 24 all all DET cana-3627 17 25 the the DET cana-3627 17 26 vertices vertex NOUN cana-3627 17 27 in in ADP cana-3627 17 28 𝑋. 𝑋. PROPN cana-3627 17 29 for for ADP cana-3627 17 30 𝑙 𝑙 PROPN cana-3627 17 31 ∈ ∈ PROPN cana-3627 17 32 { { PUNCT cana-3627 17 33 0,1,2 0,1,2 NOUN cana-3627 17 34 } } PUNCT cana-3627 17 35 , , PUNCT cana-3627 17 36 we we PRON cana-3627 17 37 say say VERB cana-3627 17 38 that that SCONJ cana-3627 17 39 a a DET cana-3627 17 40 vertex vertex NOUN cana-3627 17 41 π‘₯ π‘₯ X cana-3627 17 42 ∈ ∈ NOUN cana-3627 18 1 𝑉𝑙 𝑉𝑙 ADV cana-3627 18 2 if if SCONJ cana-3627 18 3 and and CCONJ cana-3627 18 4 only only ADV cana-3627 18 5 if if SCONJ cana-3627 18 6 the the DET cana-3627 18 7 weight weight NOUN cana-3627 18 8 of of ADP cana-3627 18 9 π‘₯ π‘₯ PROPN cana-3627 18 10 under under ADP cana-3627 18 11 𝑔 𝑔 PROPN cana-3627 18 12 is be AUX cana-3627 18 13 𝑙. 𝑙. ADJ cana-3627 18 14 in in ADP cana-3627 18 15 view view NOUN cana-3627 18 16 of of ADP cana-3627 18 17 the the DET cana-3627 18 18 one one NUM cana-3627 18 19 - - PUNCT cana-3627 18 20 to to ADP cana-3627 18 21 - - PUNCT cana-3627 18 22 one one NUM cana-3627 18 23 correspondence correspondence NOUN cana-3627 18 24 between between ADP cana-3627 18 25 the the DET cana-3627 18 26 functions function NOUN cana-3627 19 1 𝑔 𝑔 PROPN cana-3627 19 2 defined define VERB cana-3627 19 3 on on ADP cana-3627 19 4 𝑉 𝑉 PROPN cana-3627 19 5 and and CCONJ cana-3627 19 6 having have VERB cana-3627 19 7 values value NOUN cana-3627 19 8 in in ADP cana-3627 19 9 the the DET cana-3627 19 10 set set NOUN cana-3627 19 11 { { PUNCT cana-3627 19 12 0,1,2 0,1,2 NOUN cana-3627 19 13 } } PUNCT cana-3627 19 14 and and CCONJ cana-3627 19 15 the the DET cana-3627 19 16 sets set NOUN cana-3627 19 17 ( ( PUNCT cana-3627 19 18 { { PUNCT cana-3627 19 19 𝑉𝑙 𝑉𝑙 NOUN cana-3627 19 20 } } PUNCT cana-3627 19 21 : : PUNCT cana-3627 20 1 𝑙 𝑙 X cana-3627 20 2 = = SYM cana-3627 20 3 0,1,2 0,1,2 NUM cana-3627 20 4 ) ) PUNCT cana-3627 20 5 , , PUNCT cana-3627 20 6 we we PRON cana-3627 20 7 can can AUX cana-3627 20 8 write write VERB cana-3627 20 9 𝑔 𝑔 PROPN cana-3627 20 10 = = SYM cana-3627 20 11 ( ( PUNCT cana-3627 20 12 𝑉0 𝑉0 NOUN cana-3627 20 13 , , PUNCT cana-3627 20 14 𝑉1 𝑉1 PROPN cana-3627 20 15 , , PUNCT cana-3627 20 16 𝑉2 𝑉2 PROPN cana-3627 20 17 ) ) PUNCT cana-3627 20 18 . . PUNCT cana-3627 21 1 communications communication NOUN cana-3627 21 2 on on ADP cana-3627 21 3 applied apply VERB cana-3627 21 4 nonlinear nonlinear ADJ cana-3627 21 5 analysis analysis NOUN cana-3627 21 6 issn issn NOUN cana-3627 21 7 : : PUNCT cana-3627 21 8 1074 1074 NUM cana-3627 21 9 - - PUNCT cana-3627 21 10 133x 133x NUM cana-3627 21 11 vol vol NOUN cana-3627 21 12 32 32 NUM cana-3627 21 13 no no NOUN cana-3627 21 14 . . PUNCT cana-3627 22 1 8s 8s PROPN cana-3627 22 2 ( ( PUNCT cana-3627 22 3 2025 2025 NUM cana-3627 22 4 ) ) PUNCT cana-3627 22 5 183 183 NUM cana-3627 22 6 https://internationalpubls.com https://internationalpubls.com X cana-3627 23 1 the the DET cana-3627 23 2 function function NOUN cana-3627 23 3 𝑔 𝑔 PROPN cana-3627 23 4 is be AUX cana-3627 23 5 called call VERB cana-3627 23 6 a a DET cana-3627 23 7 roman roman ADJ cana-3627 23 8 dominating dominating NOUN cana-3627 23 9 function function NOUN cana-3627 23 10 ( ( PUNCT cana-3627 23 11 𝑅𝐷𝐹 𝑅𝐷𝐹 PROPN cana-3627 23 12 ) ) PUNCT cana-3627 24 1 [ [ X cana-3627 24 2 2 2 NUM cana-3627 24 3 ] ] PUNCT cana-3627 24 4 , , PUNCT cana-3627 24 5 any any DET cana-3627 24 6 vertex vertex NOUN cana-3627 24 7 π‘Ž π‘Ž NOUN cana-3627 24 8 of of ADP cana-3627 24 9 Ξ³ Ξ³ NOUN cana-3627 24 10 having have VERB cana-3627 24 11 weight weight NOUN cana-3627 24 12 0 0 NUM cana-3627 24 13 has have VERB cana-3627 24 14 at at ADV cana-3627 24 15 least least ADV cana-3627 24 16 one one NUM cana-3627 24 17 vertex vertex NOUN cana-3627 24 18 with with ADP cana-3627 24 19 weight weight NOUN cana-3627 24 20 2 2 NUM cana-3627 24 21 in in ADP cana-3627 24 22 its its PRON cana-3627 24 23 open open ADJ cana-3627 24 24 neighbourhood neighbourhood NOUN cana-3627 24 25 . . PUNCT cana-3627 25 1 the the DET cana-3627 25 2 number number NOUN cana-3627 25 3 𝛾𝑅(Ξ³ 𝛾𝑅(Ξ³ PROPN cana-3627 25 4 ) ) PUNCT cana-3627 25 5 is be AUX cana-3627 25 6 the the DET cana-3627 25 7 minimum minimum NOUN cana-3627 25 8 of of ADP cana-3627 25 9 the the DET cana-3627 25 10 weights weight NOUN cana-3627 25 11 of of ADP cana-3627 25 12 all all DET cana-3627 25 13 𝑅𝐷𝐹𝑠 𝑅𝐷𝐹𝑠 NOUN cana-3627 25 14 defined define VERB cana-3627 25 15 on on ADP cana-3627 25 16 Ξ³ Ξ³ PROPN cana-3627 25 17 . . PUNCT cana-3627 26 1 the the DET cana-3627 26 2 𝑅𝐷𝐹 𝑅𝐷𝐹 PROPN cana-3627 26 3 𝑔 𝑔 VERB cana-3627 26 4 with with ADP cana-3627 26 5 weight weight NOUN cana-3627 26 6 𝛾𝑅(Ξ³ 𝛾𝑅(Ξ³ PROPN cana-3627 26 7 ) ) PUNCT cana-3627 26 8 is be AUX cana-3627 26 9 called call VERB cana-3627 26 10 a a DET cana-3627 26 11 𝛾𝑅 𝛾𝑅 PROPN cana-3627 26 12 βˆ’ βˆ’ PROPN cana-3627 26 13 function function NOUN cana-3627 26 14 on on ADP cana-3627 26 15 Ξ³ Ξ³ NOUN cana-3627 26 16 and and CCONJ cana-3627 26 17 the the DET cana-3627 26 18 number number NOUN cana-3627 26 19 𝛾𝑅(Ξ³ 𝛾𝑅(Ξ³ PROPN cana-3627 26 20 ) ) PUNCT cana-3627 26 21 itself itself PRON cana-3627 26 22 is be AUX cana-3627 26 23 called call VERB cana-3627 26 24 the the DET cana-3627 26 25 𝛾𝑅 𝛾𝑅 PROPN cana-3627 26 26 βˆ’ βˆ’ PROPN cana-3627 26 27 value value NOUN cana-3627 26 28 of of ADP cana-3627 26 29 Ξ³ Ξ³ PROPN cana-3627 26 30 . . PUNCT cana-3627 27 1 many many ADJ cana-3627 27 2 researchers researcher NOUN cana-3627 27 3 have have AUX cana-3627 27 4 investigated investigate VERB cana-3627 27 5 the the DET cana-3627 27 6 roman roman ADJ cana-3627 27 7 dominating dominating NOUN cana-3627 27 8 number number NOUN cana-3627 27 9 of of ADP cana-3627 27 10 graphs graph NOUN cana-3627 27 11 [ [ X cana-3627 27 12 1 1 NUM cana-3627 27 13 , , PUNCT cana-3627 27 14 4 4 NUM cana-3627 27 15 , , PUNCT cana-3627 27 16 5 5 NUM cana-3627 27 17 , , PUNCT cana-3627 27 18 7 7 NUM cana-3627 27 19 , , PUNCT cana-3627 27 20 8 8 NUM cana-3627 27 21 , , PUNCT cana-3627 27 22 9 9 NUM cana-3627 27 23 ] ] PUNCT cana-3627 27 24 . . PUNCT cana-3627 28 1 a a DET cana-3627 28 2 weak weak ADJ cana-3627 28 3 roman roman ADJ cana-3627 28 4 dominating dominating NOUN cana-3627 28 5 function function NOUN cana-3627 28 6 𝑔 𝑔 PROPN cana-3627 28 7 ( ( PUNCT cana-3627 28 8 π‘Šπ‘…π·πΉ π‘Šπ‘…π·πΉ NOUN cana-3627 28 9 ) ) PUNCT cana-3627 29 1 [ [ X cana-3627 29 2 3 3 X cana-3627 29 3 ] ] PUNCT cana-3627 29 4 is be AUX cana-3627 29 5 a a DET cana-3627 29 6 function function NOUN cana-3627 29 7 such such ADJ cana-3627 29 8 that that PRON cana-3627 29 9 for for ADP cana-3627 29 10 any any DET cana-3627 29 11 vertex vertex NOUN cana-3627 29 12 π‘₯ π‘₯ ADP cana-3627 29 13 having have VERB cana-3627 29 14 positive positive ADJ cana-3627 29 15 weight weight NOUN cana-3627 29 16 , , PUNCT cana-3627 29 17 there there PRON cana-3627 29 18 is be VERB cana-3627 29 19 a a DET cana-3627 29 20 vertex vertex NOUN cana-3627 29 21 𝑦 𝑦 NOUN cana-3627 29 22 in in ADP cana-3627 29 23 the the DET cana-3627 29 24 open open ADJ cana-3627 29 25 neighbourhood neighbourhood NOUN cana-3627 29 26 of of ADP cana-3627 29 27 π‘₯ π‘₯ NOUN cana-3627 29 28 having have VERB cana-3627 29 29 positive positive ADJ cana-3627 29 30 weight weight NOUN cana-3627 29 31 and and CCONJ cana-3627 29 32 under under ADP cana-3627 29 33 the the DET cana-3627 29 34 function function NOUN cana-3627 29 35 β„Ž β„Ž PROPN cana-3627 29 36 defined define VERB cana-3627 29 37 on on ADP cana-3627 29 38 𝑉 𝑉 PROPN cana-3627 29 39 having have VERB cana-3627 29 40 values value NOUN cana-3627 29 41 in in ADP cana-3627 29 42 the the DET cana-3627 29 43 set set NOUN cana-3627 29 44 { { PUNCT cana-3627 29 45 0,1,2 0,1,2 NOUN cana-3627 29 46 } } PUNCT cana-3627 29 47 defined define VERB cana-3627 29 48 by by ADP cana-3627 29 49 β„Ž(π‘₯ β„Ž(π‘₯ NOUN cana-3627 29 50 ) ) PUNCT cana-3627 29 51 = = SYM cana-3627 29 52 1 1 NUM cana-3627 29 53 , , PUNCT cana-3627 29 54 β„Ž(𝑦 β„Ž(𝑦 NOUN cana-3627 29 55 ) ) PUNCT cana-3627 29 56 = = SYM cana-3627 29 57 𝑔(𝑦 𝑔(𝑦 PROPN cana-3627 29 58 ) ) PUNCT cana-3627 29 59 βˆ’ βˆ’ PROPN cana-3627 29 60 1 1 NUM cana-3627 29 61 , , PUNCT cana-3627 29 62 β„Ž(𝑧 β„Ž(𝑧 PROPN cana-3627 29 63 ) ) PUNCT cana-3627 29 64 = = SYM cana-3627 29 65 𝑔(𝑧 𝑔(𝑧 PROPN cana-3627 29 66 ) ) PUNCT cana-3627 29 67 , , PUNCT cana-3627 29 68 𝑧 𝑧 PROPN cana-3627 29 69 ∈ ∈ PROPN cana-3627 29 70 𝑉 𝑉 PROPN cana-3627 29 71 βˆ’ βˆ’ PROPN cana-3627 29 72 { { PUNCT cana-3627 29 73 π‘₯ π‘₯ NOUN cana-3627 29 74 , , PUNCT cana-3627 29 75 𝑦 𝑦 NOUN cana-3627 29 76 } } PUNCT cana-3627 29 77 , , PUNCT cana-3627 29 78 there there PRON cana-3627 29 79 are be VERB cana-3627 29 80 no no DET cana-3627 29 81 undefended undefended ADJ cana-3627 29 82 vertices vertex NOUN cana-3627 29 83 in in ADP cana-3627 29 84 Ξ³ Ξ³ PROPN cana-3627 29 85 . . PUNCT cana-3627 30 1 the the DET cana-3627 30 2 number number NOUN cana-3627 30 3 π›Ύπ‘Ÿ(Ξ³ π›Ύπ‘Ÿ(Ξ³ NUM cana-3627 30 4 ) ) PUNCT cana-3627 30 5 , , PUNCT cana-3627 30 6 the the DET cana-3627 30 7 minimum minimum NOUN cana-3627 30 8 of of ADP cana-3627 30 9 the the DET cana-3627 30 10 weights weight NOUN cana-3627 30 11 of of ADP cana-3627 30 12 all all DET cana-3627 30 13 the the DET cana-3627 30 14 π‘Šπ‘…π·πΉπ‘  π‘Šπ‘…π·πΉπ‘  NOUN cana-3627 30 15 defined define VERB cana-3627 30 16 on on ADP cana-3627 30 17 Ξ³ Ξ³ X cana-3627 30 18 is be AUX cana-3627 30 19 called call VERB cana-3627 30 20 the the DET cana-3627 30 21 weak weak ADJ cana-3627 30 22 roman roman ADJ cana-3627 30 23 domination domination NOUN cana-3627 30 24 number number NOUN cana-3627 30 25 of of ADP cana-3627 30 26 Ξ³ Ξ³ PROPN cana-3627 30 27 . . PUNCT cana-3627 31 1 the the DET cana-3627 31 2 π‘Šπ‘…π·πΉ π‘Šπ‘…π·πΉ NOUN cana-3627 31 3 whose whose DET cana-3627 31 4 weight weight NOUN cana-3627 31 5 is be AUX cana-3627 31 6 π›Ύπ‘Ÿ(Ξ³ π›Ύπ‘Ÿ(Ξ³ NUM cana-3627 31 7 ) ) PUNCT cana-3627 31 8 is be AUX cana-3627 31 9 called call VERB cana-3627 31 10 a a DET cana-3627 31 11 π›Ύπ‘Ÿ π›Ύπ‘Ÿ ADJ cana-3627 31 12 βˆ’ βˆ’ NOUN cana-3627 31 13 function function NOUN cana-3627 31 14 defined define VERB cana-3627 31 15 on on ADP cana-3627 31 16 Ξ³ Ξ³ NOUN cana-3627 31 17 and and CCONJ cana-3627 31 18 π›Ύπ‘Ÿ(Ξ³ π›Ύπ‘Ÿ(Ξ³ NUM cana-3627 31 19 ) ) PUNCT cana-3627 31 20 is be AUX cana-3627 31 21 called call VERB cana-3627 31 22 the the DET cana-3627 31 23 π›Ύπ‘Ÿ π›Ύπ‘Ÿ ADJ cana-3627 31 24 βˆ’ βˆ’ PROPN cana-3627 31 25 value value NOUN cana-3627 31 26 of of ADP cana-3627 31 27 Ξ³ Ξ³ PROPN cana-3627 31 28 . . PUNCT cana-3627 32 1 π‘Šπ‘…π·πΉπ‘  π‘Šπ‘…π·πΉπ‘  NOUN cana-3627 32 2 are be AUX cana-3627 32 3 marginally marginally ADV cana-3627 32 4 compromising compromise VERB cana-3627 32 5 but but CCONJ cana-3627 32 6 more more ADJ cana-3627 32 7 accommodative accommodative ADJ cana-3627 32 8 versions version NOUN cana-3627 32 9 of of ADP cana-3627 32 10 𝑅𝐷𝐹𝑠. 𝑅𝐷𝐹𝑠. PRON cana-3627 32 11 researchers researcher NOUN cana-3627 32 12 have have AUX cana-3627 32 13 extensively extensively ADV cana-3627 32 14 worked work VERB cana-3627 32 15 on on ADP cana-3627 32 16 the the DET cana-3627 32 17 parameter parameter NOUN cana-3627 32 18 weak weak ADJ cana-3627 32 19 roman roman ADJ cana-3627 32 20 domination domination NOUN cana-3627 32 21 number number NOUN cana-3627 32 22 of of ADP cana-3627 32 23 graphs graph NOUN cana-3627 32 24 [ [ X cana-3627 32 25 6 6 NUM cana-3627 32 26 , , PUNCT cana-3627 32 27 10 10 NUM cana-3627 32 28 , , PUNCT cana-3627 32 29 11 11 NUM cana-3627 32 30 , , PUNCT cana-3627 32 31 12 12 NUM cana-3627 32 32 ] ] PUNCT cana-3627 32 33 . . PUNCT cana-3627 33 1 with with ADP cana-3627 33 2 respect respect NOUN cana-3627 33 3 to to ADP cana-3627 33 4 a a DET cana-3627 33 5 graph graph NOUN cana-3627 33 6 Ξ³ Ξ³ X cana-3627 33 7 , , PUNCT cana-3627 33 8 we we PRON cana-3627 33 9 say say VERB cana-3627 33 10 that that SCONJ cana-3627 33 11 an an DET cana-3627 33 12 edge edge NOUN cana-3627 33 13 π‘₯ π‘₯ PRON cana-3627 33 14 ∈ ∈ NOUN cana-3627 33 15 πΈβˆ’ πΈβˆ’ VERB cana-3627 33 16 if if SCONJ cana-3627 33 17 and and CCONJ cana-3627 33 18 only only ADV cana-3627 33 19 if if SCONJ cana-3627 33 20 its its PRON cana-3627 33 21 removal removal NOUN cana-3627 33 22 will will AUX cana-3627 33 23 reduce reduce VERB cana-3627 33 24 the the DET cana-3627 33 25 π›Ύπ‘Ÿ π›Ύπ‘Ÿ ADJ cana-3627 33 26 βˆ’ βˆ’ PROPN cana-3627 33 27 value value NOUN cana-3627 33 28 of of ADP cana-3627 33 29 Ξ³ Ξ³ PROPN cana-3627 33 30 . . PROPN cana-3627 33 31 similarly similarly ADV cana-3627 33 32 , , PUNCT cana-3627 33 33 we we PRON cana-3627 33 34 say say VERB cana-3627 33 35 that that SCONJ cana-3627 33 36 an an DET cana-3627 33 37 edge edge NOUN cana-3627 33 38 π‘₯ π‘₯ X cana-3627 33 39 ∈ ∈ NOUN cana-3627 33 40 𝐸+ 𝐸+ PUNCT cana-3627 33 41 if if SCONJ cana-3627 33 42 and and CCONJ cana-3627 33 43 only only ADV cana-3627 33 44 if if SCONJ cana-3627 33 45 its its PRON cana-3627 33 46 removal removal NOUN cana-3627 33 47 will will AUX cana-3627 33 48 increase increase VERB cana-3627 33 49 the the DET cana-3627 33 50 π›Ύπ‘Ÿ π›Ύπ‘Ÿ ADJ cana-3627 33 51 βˆ’ βˆ’ PROPN cana-3627 33 52 value value NOUN cana-3627 33 53 of of ADP cana-3627 33 54 Ξ³ Ξ³ NOUN cana-3627 33 55 and and CCONJ cana-3627 33 56 an an DET cana-3627 33 57 edge edge NOUN cana-3627 33 58 π‘₯ π‘₯ DET cana-3627 33 59 ∈ ∈ NOUN cana-3627 33 60 𝐸0 𝐸0 NOUN cana-3627 34 1 if if SCONJ cana-3627 34 2 and and CCONJ cana-3627 34 3 only only ADV cana-3627 34 4 if if SCONJ cana-3627 34 5 its its PRON cana-3627 34 6 removal removal NOUN cana-3627 34 7 will will AUX cana-3627 34 8 leave leave VERB cana-3627 34 9 unaltered unaltere VERB cana-3627 34 10 the the DET cana-3627 34 11 π›Ύπ‘Ÿ π›Ύπ‘Ÿ ADJ cana-3627 34 12 βˆ’ βˆ’ PROPN cana-3627 34 13 value value NOUN cana-3627 34 14 of of ADP cana-3627 34 15 Ξ³ Ξ³ PROPN cana-3627 34 16 . . PUNCT cana-3627 35 1 the the DET cana-3627 35 2 behaviour behaviour NOUN cana-3627 35 3 of of ADP cana-3627 35 4 an an DET cana-3627 35 5 edge edge NOUN cana-3627 35 6 to to PART cana-3627 35 7 belong belong VERB cana-3627 35 8 to to ADP cana-3627 35 9 πΈβˆ’ πΈβˆ’ ADJ cana-3627 35 10 , , PUNCT cana-3627 35 11 𝐸+ 𝐸+ X cana-3627 35 12 or or CCONJ cana-3627 35 13 𝐸0 𝐸0 NOUN cana-3627 35 14 is be AUX cana-3627 35 15 called call VERB cana-3627 35 16 the the DET cana-3627 35 17 changing change VERB cana-3627 35 18 and and CCONJ cana-3627 35 19 unchanging unchanging ADJ cana-3627 35 20 behaviour behaviour NOUN cana-3627 35 21 of of ADP cana-3627 35 22 the the DET cana-3627 35 23 edge edge NOUN cana-3627 35 24 . . PUNCT cana-3627 36 1 this this DET cana-3627 36 2 paper paper NOUN cana-3627 36 3 is be AUX cana-3627 36 4 dedicated dedicate VERB cana-3627 36 5 to to PART cana-3627 36 6 study study VERB cana-3627 36 7 the the DET cana-3627 36 8 changing change VERB cana-3627 36 9 and and CCONJ cana-3627 36 10 unchanging unchanging ADJ cana-3627 36 11 behaviour behaviour NOUN cana-3627 36 12 of of ADP cana-3627 36 13 an an DET cana-3627 36 14 edge edge NOUN cana-3627 36 15 with with ADP cana-3627 36 16 respect respect NOUN cana-3627 36 17 to to ADP cana-3627 36 18 a a DET cana-3627 36 19 π‘Šπ‘…π·πΉ. π‘Šπ‘…π·πΉ. ADJ cana-3627 36 20 definition definition NOUN cana-3627 36 21 1.1 1.1 NUM cana-3627 36 22 . . PUNCT cana-3627 37 1 [ [ X cana-3627 37 2 12 12 NUM cana-3627 37 3 ] ] PUNCT cana-3627 37 4 under under ADP cana-3627 37 5 a a DET cana-3627 37 6 π‘Šπ‘…π·πΉ π‘Šπ‘…π·πΉ NOUN cana-3627 37 7 , , PUNCT cana-3627 37 8 a a DET cana-3627 37 9 vertex vertex NOUN cana-3627 37 10 π‘Ž π‘Ž X cana-3627 37 11 ∈ ∈ PROPN cana-3627 37 12 𝑉 𝑉 PROPN cana-3627 37 13 with with ADP cana-3627 37 14 weight weight NOUN cana-3627 37 15 0 0 NUM cana-3627 37 16 is be AUX cana-3627 37 17 said say VERB cana-3627 37 18 to to PART cana-3627 37 19 be be AUX cana-3627 37 20 dependent dependent ADJ cana-3627 37 21 on on ADP cana-3627 37 22 another another DET cana-3627 37 23 vertex vertex NOUN cana-3627 37 24 𝑏 𝑏 PROPN cana-3627 37 25 ∈ ∈ PROPN cana-3627 37 26 𝑉 𝑉 PROPN cana-3627 37 27 of of ADP cana-3627 37 28 positive positive ADJ cana-3627 37 29 weight weight NOUN cana-3627 37 30 , , PUNCT cana-3627 37 31 if if SCONJ cana-3627 37 32 the the DET cana-3627 37 33 increase increase NOUN cana-3627 37 34 in in ADP cana-3627 37 35 the the DET cana-3627 37 36 weight weight NOUN cana-3627 37 37 of of ADP cana-3627 37 38 π‘Ž π‘Ž NOUN cana-3627 37 39 by by ADP cana-3627 37 40 1 1 NUM cana-3627 37 41 and and CCONJ cana-3627 37 42 the the DET cana-3627 37 43 simultaneous simultaneous ADJ cana-3627 37 44 decrease decrease NOUN cana-3627 37 45 of of ADP cana-3627 37 46 the the DET cana-3627 37 47 weight weight NOUN cana-3627 37 48 of of ADP cana-3627 37 49 𝑏 𝑏 NUM cana-3627 37 50 by by ADP cana-3627 37 51 1 1 NUM cana-3627 37 52 , , PUNCT cana-3627 37 53 will will AUX cana-3627 37 54 not not PART cana-3627 37 55 create create VERB cana-3627 37 56 an an DET cana-3627 37 57 undefended undefended ADJ cana-3627 37 58 vertex vertex NOUN cana-3627 37 59 in in ADP cana-3627 37 60 𝑉. 𝑉. NOUN cana-3627 37 61 we we PRON cana-3627 37 62 then then ADV cana-3627 37 63 write write VERB cana-3627 37 64 , , PUNCT cana-3627 37 65 π‘Ž π‘Ž PROPN cana-3627 37 66 ∈ ∈ PROPN cana-3627 37 67 𝐷𝐺(𝑏 𝐷𝐺(𝑏 NOUN cana-3627 37 68 ) ) PUNCT cana-3627 37 69 . . PUNCT cana-3627 38 1 notation notation NOUN cana-3627 38 2 1.1 1.1 NUM cana-3627 38 3 . . PUNCT cana-3627 39 1 [ [ X cana-3627 39 2 12 12 NUM cana-3627 39 3 ] ] X cana-3627 39 4 𝐻π‘₯ 𝐻π‘₯ PROPN cana-3627 39 5 = = SYM cana-3627 39 6 ( ( PUNCT cana-3627 39 7 𝑉1 𝑉1 PROPN cana-3627 39 8 βˆͺ βˆͺ X cana-3627 39 9 𝑉2 𝑉2 PROPN cana-3627 39 10 ) ) PUNCT cana-3627 39 11 βˆ’ βˆ’ PROPN cana-3627 39 12 { { PUNCT cana-3627 39 13 π‘₯ π‘₯ NOUN cana-3627 39 14 } } PUNCT cana-3627 39 15 , , PUNCT cana-3627 39 16 οΏ½ οΏ½ PROPN cana-3627 39 17 Μ… Μ… NOUN cana-3627 39 18 οΏ½ οΏ½ NOUN cana-3627 39 19 (𝑒 (𝑒 NUM cana-3627 39 20 ) ) PUNCT cana-3627 39 21 = = PUNCT cana-3627 39 22 ( ( PUNCT cana-3627 39 23 𝑁(𝑒 𝑁(𝑒 NOUN cana-3627 39 24 ) ) PUNCT cana-3627 39 25 ∩ ∩ ADJ cana-3627 39 26 𝑉0 𝑉0 NOUN cana-3627 39 27 ) ) PUNCT cana-3627 40 1 βˆ’ βˆ’ PROPN cana-3627 40 2 ⋃ ⋃ PROPN cana-3627 40 3 𝐷γ(π‘₯)π‘₯∈𝐻 𝐷γ(π‘₯)π‘₯∈𝐻 PROPN cana-3627 40 4 . . PUNCT cana-3627 41 1 definition definition NOUN cana-3627 41 2 1.2 1.2 NUM cana-3627 41 3 . . PUNCT cana-3627 42 1 [ [ X cana-3627 42 2 12 12 NUM cana-3627 42 3 ] ] PUNCT cana-3627 42 4 let let VERB cana-3627 42 5 𝑒 𝑒 PRON cana-3627 42 6 ∈ ∈ PROPN cana-3627 42 7 𝑉. 𝑉. NOUN cana-3627 42 8 a a PRON cana-3627 42 9 ( ( PUNCT cana-3627 42 10 𝑒 𝑒 NOUN cana-3627 42 11 : : SYM cana-3627 42 12 1 1 NUM cana-3627 42 13 ) ) PUNCT cana-3627 42 14 βˆ’ βˆ’ NOUN cana-3627 42 15 set set VERB cana-3627 42 16 in in ADP cana-3627 42 17 Ξ³ Ξ³ PROPN cana-3627 42 18 is be AUX cana-3627 42 19 a a DET cana-3627 42 20 set set NOUN cana-3627 42 21 𝑆 𝑆 PROPN cana-3627 42 22 βŠ† βŠ† NUM cana-3627 42 23 𝑁(𝑒 𝑁(𝑒 NUM cana-3627 42 24 ) ) PUNCT cana-3627 42 25 such such ADJ cana-3627 42 26 that that PRON cana-3627 42 27 for for ADP cana-3627 42 28 any any DET cana-3627 42 29 π‘₯ π‘₯ PRON cana-3627 42 30 ∈ ∈ PROPN cana-3627 42 31 𝑆 𝑆 PROPN cana-3627 42 32 , , PUNCT cana-3627 42 33 𝑆 𝑆 PROPN cana-3627 42 34 βˆ’ βˆ’ PROPN cana-3627 42 35 𝑁(𝐻𝑒 𝑁(𝐻𝑒 NUM cana-3627 42 36 ) ) PUNCT cana-3627 42 37 βŠ† βŠ† NUM cana-3627 42 38 𝑁[π‘₯ 𝑁[π‘₯ NUM cana-3627 42 39 ] ] PUNCT cana-3627 42 40 . . PUNCT cana-3627 43 1 definition definition NOUN cana-3627 43 2 1.3 1.3 NUM cana-3627 43 3 . . PUNCT cana-3627 44 1 [ [ X cana-3627 44 2 12 12 NUM cana-3627 44 3 ] ] PUNCT cana-3627 44 4 let let VERB cana-3627 44 5 𝑒 𝑒 PRON cana-3627 44 6 ∈ ∈ PROPN cana-3627 44 7 𝑉. 𝑉. NOUN cana-3627 44 8 a a PRON cana-3627 44 9 ( ( PUNCT cana-3627 44 10 𝑒 𝑒 NOUN cana-3627 44 11 : : SYM cana-3627 44 12 1 1 NUM cana-3627 44 13 ) ) PUNCT cana-3627 44 14 βˆ’ βˆ’ NOUN cana-3627 44 15 set set VERB cana-3627 44 16 in in ADP cana-3627 44 17 Ξ³ Ξ³ PROPN cana-3627 44 18 is be AUX cana-3627 44 19 a a DET cana-3627 44 20 set set NOUN cana-3627 44 21 𝑆 𝑆 PROPN cana-3627 44 22 βŠ† βŠ† NUM cana-3627 44 23 𝑁(𝑒 𝑁(𝑒 NOUN cana-3627 44 24 ) ) PUNCT cana-3627 44 25 that that PRON cana-3627 44 26 is be AUX cana-3627 44 27 not not PART cana-3627 44 28 a a DET cana-3627 44 29 ( ( PUNCT cana-3627 44 30 𝑒 𝑒 NOUN cana-3627 44 31 : : SYM cana-3627 44 32 1 1 NUM cana-3627 44 33 ) ) PUNCT cana-3627 44 34 βˆ’ βˆ’ NOUN cana-3627 44 35 set set NOUN cana-3627 44 36 . . PUNCT cana-3627 45 1 2 2 X cana-3627 45 2 . . X cana-3627 45 3 categorizing categorize VERB cana-3627 45 4 an an DET cana-3627 45 5 edge edge NOUN cana-3627 45 6 for for ADP cana-3627 45 7 membership membership NOUN cana-3627 45 8 in in ADP cana-3627 45 9 π‘¬βˆ’ π‘¬βˆ’ PROPN cana-3627 45 10 , , PUNCT cana-3627 45 11 𝑬+ 𝑬+ X cana-3627 45 12 and and CCONJ cana-3627 45 13 π‘¬πŸŽ π‘¬πŸŽ NOUN cana-3627 45 14 we we PRON cana-3627 45 15 start start VERB cana-3627 45 16 by by ADP cana-3627 45 17 stating state VERB cana-3627 45 18 two two NUM cana-3627 45 19 trivial trivial ADJ cana-3627 45 20 observations observation NOUN cana-3627 45 21 . . PUNCT cana-3627 46 1 observation observation NOUN cana-3627 46 2 2.1 2.1 NUM cana-3627 46 3 . . PUNCT cana-3627 47 1 πΈβˆ’ πΈβˆ’ VERB cana-3627 47 2 = = PROPN cana-3627 47 3 βˆ…. βˆ…. ADP cana-3627 47 4 observation observation NOUN cana-3627 47 5 2.2 2.2 NUM cana-3627 47 6 . . PUNCT cana-3627 48 1 if if SCONJ cana-3627 48 2 π‘₯𝑦 π‘₯𝑦 NOUN cana-3627 48 3 is be AUX cana-3627 48 4 an an DET cana-3627 48 5 edge edge NOUN cana-3627 48 6 of of ADP cana-3627 48 7 Ξ³ Ξ³ NOUN cana-3627 48 8 and and CCONJ cana-3627 48 9 if if SCONJ cana-3627 48 10 { { PUNCT cana-3627 48 11 π‘₯ π‘₯ NOUN cana-3627 48 12 , , PUNCT cana-3627 48 13 𝑦 𝑦 NOUN cana-3627 48 14 } } PUNCT cana-3627 48 15 βŠ† βŠ† NUM cana-3627 48 16 𝑉0 𝑉0 NOUN cana-3627 48 17 or or CCONJ cana-3627 48 18 if if SCONJ cana-3627 48 19 { { PUNCT cana-3627 48 20 π‘₯ π‘₯ NOUN cana-3627 48 21 , , PUNCT cana-3627 48 22 𝑦 𝑦 NOUN cana-3627 48 23 } } PUNCT cana-3627 48 24 βŠ† βŠ† NUM cana-3627 48 25 𝑉1 𝑉1 NOUN cana-3627 48 26 βˆͺ βˆͺ NOUN cana-3627 48 27 𝑉2 𝑉2 NOUN cana-3627 48 28 , , PUNCT cana-3627 48 29 then then ADV cana-3627 48 30 π‘₯𝑦 π‘₯𝑦 PROPN cana-3627 48 31 ∈ ∈ PROPN cana-3627 48 32 𝐸0 𝐸0 NOUN cana-3627 48 33 . . PUNCT cana-3627 49 1 theorem theorem VERB cana-3627 49 2 2.1 2.1 NUM cana-3627 49 3 . . PUNCT cana-3627 50 1 let let VERB cana-3627 50 2 π‘₯𝑦 π‘₯𝑦 NOUN cana-3627 50 3 be be AUX cana-3627 50 4 an an DET cana-3627 50 5 edge edge NOUN cana-3627 50 6 of of ADP cana-3627 50 7 the the DET cana-3627 50 8 graph graph NOUN cana-3627 50 9 Ξ³ Ξ³ PROPN cana-3627 50 10 such such ADJ cana-3627 50 11 that that SCONJ cana-3627 50 12 π‘₯ π‘₯ PROPN cana-3627 50 13 has have VERB cana-3627 50 14 positive positive ADJ cana-3627 50 15 weight weight NOUN cana-3627 50 16 and and CCONJ cana-3627 50 17 𝑦 𝑦 NOUN cana-3627 50 18 has have VERB cana-3627 50 19 weight weight NOUN cana-3627 50 20 0 0 NUM cana-3627 50 21 under under ADP cana-3627 50 22 a a DET cana-3627 50 23 π‘Šπ‘…π·πΉ π‘Šπ‘…π·πΉ NOUN cana-3627 50 24 𝑔. 𝑔. NOUN cana-3627 50 25 then then ADV cana-3627 50 26 , , PUNCT cana-3627 50 27 the the DET cana-3627 50 28 following follow VERB cana-3627 50 29 conditions condition NOUN cana-3627 50 30 imply imply VERB cana-3627 50 31 and and CCONJ cana-3627 50 32 are be AUX cana-3627 50 33 implied imply VERB cana-3627 50 34 by by ADP cana-3627 50 35 the the DET cana-3627 50 36 membership membership NOUN cana-3627 50 37 of of ADP cana-3627 50 38 π‘₯ π‘₯ PROPN cana-3627 50 39 in in ADP cana-3627 50 40 𝐸0 𝐸0 NOUN cana-3627 50 41 . . PUNCT cana-3627 51 1 1 1 X cana-3627 51 2 . . X cana-3627 51 3 𝑦 𝑦 PROPN cana-3627 51 4 ∈ ∈ PROPN cana-3627 51 5 οΏ½ οΏ½ PROPN cana-3627 51 6 Μ… Μ… NOUN cana-3627 51 7 οΏ½ οΏ½ NOUN cana-3627 51 8 (π‘₯ (π‘₯ NOUN cana-3627 51 9 ) ) PUNCT cana-3627 51 10 and and CCONJ cana-3627 51 11 a. a. NOUN cana-3627 51 12 π‘₯ π‘₯ PRON cana-3627 51 13 ∈ ∈ PROPN cana-3627 51 14 𝑉2 𝑉2 NOUN cana-3627 51 15 and and CCONJ cana-3627 51 16 οΏ½ οΏ½ PROPN cana-3627 51 17 Μ… Μ… NOUN cana-3627 51 18 οΏ½ οΏ½ NOUN cana-3627 51 19 (π‘₯ (π‘₯ NOUN cana-3627 51 20 ) ) PUNCT cana-3627 51 21 βˆ’ βˆ’ PROPN cana-3627 51 22 { { PUNCT cana-3627 51 23 𝑦 𝑦 NOUN cana-3627 51 24 } } PUNCT cana-3627 51 25 is be AUX cana-3627 51 26 a a DET cana-3627 51 27 ( ( PUNCT cana-3627 51 28 π‘₯ π‘₯ NOUN cana-3627 51 29 : : SYM cana-3627 51 30 1 1 NUM cana-3627 51 31 ) ) PUNCT cana-3627 51 32 βˆ’ βˆ’ NOUN cana-3627 51 33 set set VERB cana-3627 51 34 in in ADP cana-3627 51 35 Ξ³ Ξ³ NOUN cana-3627 51 36 or or CCONJ cana-3627 51 37 communications communication NOUN cana-3627 51 38 on on ADP cana-3627 51 39 applied apply VERB cana-3627 51 40 nonlinear nonlinear ADJ cana-3627 51 41 analysis analysis NOUN cana-3627 51 42 issn issn NOUN cana-3627 51 43 : : PUNCT cana-3627 51 44 1074 1074 NUM cana-3627 51 45 - - PUNCT cana-3627 51 46 133x 133x NUM cana-3627 51 47 vol vol NOUN cana-3627 51 48 32 32 NUM cana-3627 51 49 no no NOUN cana-3627 51 50 . . PUNCT cana-3627 52 1 8s 8s PROPN cana-3627 52 2 ( ( PUNCT cana-3627 52 3 2025 2025 NUM cana-3627 52 4 ) ) PUNCT cana-3627 52 5 184 184 NUM cana-3627 52 6 https://internationalpubls.com https://internationalpubls.com X cana-3627 52 7 b. b. PROPN cana-3627 52 8 π‘₯ π‘₯ X cana-3627 52 9 ∈ ∈ PROPN cana-3627 52 10 𝑉1 𝑉1 NOUN cana-3627 52 11 , , PUNCT cana-3627 52 12 π‘₯ π‘₯ DET cana-3627 52 13 ∈ ∈ PROPN cana-3627 52 14 𝐷γ(πœ” 𝐷γ(πœ” NOUN cana-3627 52 15 ) ) PUNCT cana-3627 52 16 , , PUNCT cana-3627 52 17 for for ADP cana-3627 52 18 some some DET cana-3627 52 19 πœ” πœ” PRON cana-3627 52 20 ∈ ∈ NOUN cana-3627 52 21 𝑉1 𝑉1 NOUN cana-3627 52 22 βˆͺ βˆͺ NOUN cana-3627 52 23 𝑉2 𝑉2 NOUN cana-3627 52 24 and and CCONJ cana-3627 52 25 for for ADP cana-3627 52 26 no no DET cana-3627 52 27 vertex vertex NOUN cana-3627 52 28 𝑧 𝑧 PRON cana-3627 52 29 ∈ ∈ PROPN cana-3627 52 30 𝑁(π‘₯ 𝑁(π‘₯ NUM cana-3627 52 31 ) ) PUNCT cana-3627 52 32 ∩ ∩ NOUN cana-3627 52 33 𝑉0 𝑉0 NOUN cana-3627 53 1 having have VERB cana-3627 53 2 𝑁(𝑧 𝑁(𝑧 NUM cana-3627 53 3 ) ) PUNCT cana-3627 53 4 ∩ ∩ NOUN cana-3627 53 5 𝑉2 𝑉2 NOUN cana-3627 53 6 = = SYM cana-3627 53 7 βˆ… βˆ… NOUN cana-3627 53 8 and and CCONJ cana-3627 53 9 for for ADP cana-3627 53 10 every every DET cana-3627 53 11 𝑑 𝑑 PROPN cana-3627 53 12 ∈ ∈ PROPN cana-3627 53 13 ( ( PUNCT cana-3627 53 14 𝑁(𝑧 𝑁(𝑧 NOUN cana-3627 53 15 ) ) PUNCT cana-3627 53 16 βˆ’ βˆ’ NOUN cana-3627 53 17 { { PUNCT cana-3627 53 18 π‘₯ π‘₯ NOUN cana-3627 53 19 } } PUNCT cana-3627 53 20 ) ) PUNCT cana-3627 53 21 ∩ ∩ PROPN cana-3627 53 22 𝑉1 𝑉1 PROPN cana-3627 53 23 , , PUNCT cana-3627 53 24 we we PRON cana-3627 53 25 have have VERB cana-3627 53 26 οΏ½ οΏ½ NOUN cana-3627 53 27 Μ… Μ… NOUN cana-3627 53 28 οΏ½ οΏ½ NOUN cana-3627 53 29 (𝑑 (𝑑 NOUN cana-3627 53 30 ) ) PUNCT cana-3627 53 31 βˆͺ βˆͺ ADP cana-3627 53 32 { { PUNCT cana-3627 53 33 𝑦 𝑦 NOUN cana-3627 53 34 } } PUNCT cana-3627 53 35 is be AUX cana-3627 53 36 a a DET cana-3627 53 37 ( ( PUNCT cana-3627 53 38 𝑑 𝑑 NOUN cana-3627 53 39 : : PUNCT cana-3627 53 40 2 2 NUM cana-3627 53 41 ) ) PUNCT cana-3627 53 42 βˆ’ βˆ’ NOUN cana-3627 53 43 set set VERB cana-3627 53 44 in in ADP cana-3627 53 45 Ξ³ Ξ³ PROPN cana-3627 53 46 . . PROPN cana-3627 54 1 2 2 NUM cana-3627 54 2 . . X cana-3627 54 3 𝑦 𝑦 NUM cana-3627 54 4 βˆ‰ βˆ‰ PROPN cana-3627 54 5 οΏ½ οΏ½ PROPN cana-3627 54 6 Μ… Μ… NOUN cana-3627 54 7 οΏ½ οΏ½ NOUN cana-3627 54 8 (π‘₯ (π‘₯ NOUN cana-3627 54 9 ) ) PUNCT cana-3627 54 10 and and CCONJ cana-3627 54 11 c. c. PROPN cana-3627 54 12 𝑦 𝑦 PROPN cana-3627 54 13 ∈ ∈ PROPN cana-3627 54 14 𝐷γ(π‘Ÿ 𝐷γ(π‘Ÿ PROPN cana-3627 54 15 ) ) PUNCT cana-3627 54 16 for for ADP cana-3627 54 17 some some DET cana-3627 54 18 π‘Ÿ π‘Ÿ NOUN cana-3627 54 19 ∈ ∈ NOUN cana-3627 54 20 𝑉2 𝑉2 NOUN cana-3627 54 21 or or CCONJ cana-3627 54 22 if if SCONJ cana-3627 54 23 𝑁(𝑦 𝑁(𝑦 NOUN cana-3627 54 24 ) ) PUNCT cana-3627 54 25 ∩ ∩ NOUN cana-3627 54 26 𝑉2 𝑉2 NOUN cana-3627 54 27 = = SYM cana-3627 54 28 βˆ… βˆ… NOUN cana-3627 54 29 , , PUNCT cana-3627 54 30 𝑦 𝑦 NOUN cana-3627 54 31 ∈ ∈ NOUN cana-3627 54 32 𝐷γ(π‘Ÿ 𝐷γ(π‘Ÿ PROPN cana-3627 54 33 ) ) PUNCT cana-3627 54 34 , , PUNCT cana-3627 54 35 for for ADP cana-3627 54 36 some some DET cana-3627 54 37 π‘Ÿ π‘Ÿ NOUN cana-3627 54 38 ∈ ∈ NOUN cana-3627 54 39 𝑉1 𝑉1 NOUN cana-3627 54 40 where where SCONJ cana-3627 54 41 |𝑁(𝑦 |𝑁(𝑦 NOUN cana-3627 54 42 ) ) PUNCT cana-3627 54 43 ∩ ∩ NOUN cana-3627 54 44 𝑉1| 𝑉1| X cana-3627 54 45 > > X cana-3627 54 46 2 2 NUM cana-3627 54 47 if if SCONJ cana-3627 54 48 | | NOUN cana-3627 54 49 οΏ½ οΏ½ NOUN cana-3627 54 50 Μ… Μ… NOUN cana-3627 54 51 οΏ½ οΏ½ NOUN cana-3627 54 52 (π‘Ÿ (π‘Ÿ NOUN cana-3627 54 53 ) ) PUNCT cana-3627 54 54 βˆͺ βˆͺ NOUN cana-3627 54 55 { { PUNCT cana-3627 54 56 𝑦}| 𝑦}| NOUN cana-3627 54 57 > > SYM cana-3627 54 58 1 1 NUM cana-3627 54 59 and and CCONJ cana-3627 54 60 οΏ½ οΏ½ NOUN cana-3627 54 61 Μ… Μ… NOUN cana-3627 54 62 οΏ½ οΏ½ NOUN cana-3627 54 63 (π‘Ÿ (π‘Ÿ NOUN cana-3627 54 64 ) ) PUNCT cana-3627 54 65 βˆͺ βˆͺ ADP cana-3627 54 66 { { PUNCT cana-3627 54 67 𝑦 𝑦 NOUN cana-3627 54 68 } } PUNCT cana-3627 54 69 does do AUX cana-3627 54 70 not not PART cana-3627 54 71 induce induce VERB cana-3627 54 72 a a DET cana-3627 54 73 clique clique NOUN cana-3627 54 74 and and CCONJ cana-3627 54 75 οΏ½ οΏ½ NOUN cana-3627 54 76 Μ… Μ… NOUN cana-3627 54 77 οΏ½ οΏ½ NOUN cana-3627 54 78 (π‘₯ (π‘₯ NOUN cana-3627 54 79 ) ) PUNCT cana-3627 54 80 βˆ’ βˆ’ PROPN cana-3627 54 81 { { PUNCT cana-3627 54 82 𝑦 𝑦 NOUN cana-3627 54 83 } } PUNCT cana-3627 54 84 is be AUX cana-3627 54 85 a a DET cana-3627 54 86 ( ( PUNCT cana-3627 54 87 π‘₯ π‘₯ NOUN cana-3627 54 88 : : PUNCT cana-3627 54 89 𝑖 𝑖 X cana-3627 54 90 ) ) PUNCT cana-3627 54 91 βˆ’ βˆ’ NOUN cana-3627 54 92 set set VERB cana-3627 54 93 in in ADP cana-3627 54 94 Ξ³ Ξ³ PRON cana-3627 54 95 where where SCONJ cana-3627 54 96 𝑔(π‘₯ 𝑔(π‘₯ NOUN cana-3627 54 97 ) ) PUNCT cana-3627 54 98 = = SYM cana-3627 54 99 𝑖 𝑖 PROPN cana-3627 54 100 or or CCONJ cana-3627 54 101 d. d. PROPN cana-3627 54 102 π‘Ÿ π‘Ÿ NOUN cana-3627 54 103 ∈ ∈ PROPN cana-3627 54 104 𝑉1 𝑉1 PROPN cana-3627 54 105 , , PUNCT cana-3627 54 106 οΏ½ οΏ½ NOUN cana-3627 54 107 Μ… Μ… NOUN cana-3627 54 108 οΏ½ οΏ½ NOUN cana-3627 54 109 (π‘Ÿ (π‘Ÿ NOUN cana-3627 54 110 ) ) PUNCT cana-3627 54 111 βˆͺ βˆͺ ADP cana-3627 54 112 { { PUNCT cana-3627 54 113 𝑦 𝑦 NOUN cana-3627 54 114 } } PUNCT cana-3627 54 115 is be AUX cana-3627 54 116 a a DET cana-3627 54 117 ( ( PUNCT cana-3627 54 118 π‘Ÿ π‘Ÿ NOUN cana-3627 54 119 : : PUNCT cana-3627 54 120 2 2 NUM cana-3627 54 121 ) ) PUNCT cana-3627 54 122 βˆ’ βˆ’ NOUN cana-3627 54 123 set set NOUN cana-3627 54 124 and and CCONJ cana-3627 54 125 we we PRON cana-3627 54 126 can can AUX cana-3627 54 127 find find VERB cana-3627 54 128 π‘š π‘š X cana-3627 54 129 ∈ ∈ NOUN cana-3627 54 130 𝑉1⋃𝑉2 𝑉1⋃𝑉2 NOUN cana-3627 54 131 such such ADJ cana-3627 54 132 that that SCONJ cana-3627 54 133 π‘₯ π‘₯ PROPN cana-3627 54 134 ∈ ∈ PROPN cana-3627 54 135 𝐷γ(π‘š 𝐷γ(π‘š NOUN cana-3627 54 136 ) ) PUNCT cana-3627 54 137 . . PUNCT cana-3627 55 1 proof proof NOUN cana-3627 55 2 . . PUNCT cana-3627 56 1 assume assume VERB cana-3627 56 2 that that SCONJ cana-3627 56 3 𝑦 𝑦 PROPN cana-3627 56 4 ∈ ∈ PROPN cana-3627 56 5 οΏ½ οΏ½ PROPN cana-3627 56 6 Μ… Μ… NOUN cana-3627 56 7 οΏ½ οΏ½ NOUN cana-3627 56 8 (π‘₯ (π‘₯ NOUN cana-3627 56 9 ) ) PUNCT cana-3627 56 10 , , PUNCT cana-3627 56 11 π‘₯ π‘₯ PROPN cana-3627 56 12 ∈ ∈ PROPN cana-3627 56 13 𝑉2 𝑉2 NOUN cana-3627 56 14 . . PUNCT cana-3627 57 1 let let AUX cana-3627 57 2 οΏ½ οΏ½ PRON cana-3627 57 3 Μ… Μ… VERB cana-3627 57 4 οΏ½ οΏ½ NOUN cana-3627 57 5 (π‘₯ (π‘₯ NOUN cana-3627 57 6 ) ) PUNCT cana-3627 57 7 βˆ’ βˆ’ PROPN cana-3627 57 8 { { PUNCT cana-3627 57 9 𝑦 𝑦 NOUN cana-3627 57 10 } } PUNCT cana-3627 57 11 be be AUX cana-3627 57 12 a a DET cana-3627 57 13 ( ( PUNCT cana-3627 57 14 π‘₯ π‘₯ NOUN cana-3627 57 15 : : SYM cana-3627 57 16 1 1 NUM cana-3627 57 17 ) ) PUNCT cana-3627 57 18 βˆ’ βˆ’ NOUN cana-3627 57 19 set set VERB cana-3627 57 20 in in ADP cana-3627 57 21 Ξ³ Ξ³ PROPN cana-3627 57 22 . . PROPN cana-3627 58 1 consider consider VERB cana-3627 58 2 the the DET cana-3627 58 3 function function NOUN cana-3627 58 4 𝑔′ 𝑔′ ADV cana-3627 58 5 defined define VERB cana-3627 58 6 on on ADP cana-3627 58 7 𝑉 𝑉 PROPN cana-3627 58 8 and and CCONJ cana-3627 58 9 having have VERB cana-3627 58 10 values value NOUN cana-3627 58 11 in in ADP cana-3627 58 12 { { PUNCT cana-3627 58 13 0,1,2 0,1,2 NOUN cana-3627 58 14 } } PUNCT cana-3627 58 15 by by ADP cana-3627 58 16 𝑔′(𝑦 𝑔′(𝑦 NOUN cana-3627 58 17 ) ) PUNCT cana-3627 58 18 = = SYM cana-3627 58 19 1 1 NUM cana-3627 58 20 , , PUNCT cana-3627 58 21 𝑔′(π‘₯ 𝑔′(π‘₯ PUNCT cana-3627 58 22 ) ) PUNCT cana-3627 58 23 = = SYM cana-3627 58 24 1 1 NUM cana-3627 58 25 and and CCONJ cana-3627 58 26 𝑔′(𝑧 𝑔′(𝑧 PROPN cana-3627 58 27 ) ) PUNCT cana-3627 58 28 = = SYM cana-3627 58 29 𝑔(𝑧 𝑔(𝑧 PROPN cana-3627 58 30 ) ) PUNCT cana-3627 58 31 for for ADP cana-3627 58 32 all all DET cana-3627 58 33 𝑧 𝑧 DET cana-3627 58 34 ∈ ∈ PROPN cana-3627 58 35 𝑉 𝑉 PROPN cana-3627 58 36 βˆ’ βˆ’ PROPN cana-3627 58 37 { { PUNCT cana-3627 58 38 π‘₯ π‘₯ NOUN cana-3627 58 39 , , PUNCT cana-3627 58 40 𝑦 𝑦 NOUN cana-3627 58 41 } } PUNCT cana-3627 58 42 . . PUNCT cana-3627 59 1 the the DET cana-3627 59 2 function function NOUN cana-3627 59 3 𝑔′ 𝑔′ ADV cana-3627 59 4 is be AUX cana-3627 59 5 a a DET cana-3627 59 6 π‘Šπ‘…π·πΉ π‘Šπ‘…π·πΉ NOUN cana-3627 59 7 on on ADP cana-3627 59 8 Ξ³ Ξ³ NOUN cana-3627 59 9 βˆ’ βˆ’ PROPN cana-3627 59 10 𝑒 𝑒 ADP cana-3627 59 11 having have VERB cana-3627 59 12 weight weight NOUN cana-3627 59 13 π›Ύπ‘Ÿ(Ξ³ π›Ύπ‘Ÿ(Ξ³ NOUN cana-3627 59 14 ) ) PUNCT cana-3627 59 15 . . PUNCT cana-3627 60 1 since since SCONJ cana-3627 60 2 𝑒 𝑒 PROPN cana-3627 60 3 βˆ‰ βˆ‰ PROPN cana-3627 60 4 πΈβˆ’ πΈβˆ’ ADJ cana-3627 60 5 , , PUNCT cana-3627 60 6 𝑔′ 𝑔′ ADV cana-3627 60 7 is be AUX cana-3627 60 8 a a DET cana-3627 60 9 π›Ύπ‘Ÿ π›Ύπ‘Ÿ ADJ cana-3627 60 10 βˆ’ βˆ’ NOUN cana-3627 60 11 function function NOUN cana-3627 60 12 defined define VERB cana-3627 60 13 on on ADP cana-3627 60 14 Ξ³ Ξ³ PROPN cana-3627 60 15 βˆ’ βˆ’ PROPN cana-3627 60 16 𝑒. 𝑒. PROPN cana-3627 60 17 therefore therefore ADV cana-3627 60 18 , , PUNCT cana-3627 60 19 π›Ύπ‘Ÿ(Ξ³ π›Ύπ‘Ÿ(Ξ³ NUM cana-3627 60 20 βˆ’ βˆ’ PROPN cana-3627 60 21 𝑒 𝑒 NOUN cana-3627 60 22 ) ) PUNCT cana-3627 60 23 = = SYM cana-3627 60 24 π›Ύπ‘Ÿ(Ξ³ π›Ύπ‘Ÿ(Ξ³ NOUN cana-3627 60 25 ) ) PUNCT cana-3627 60 26 . . PUNCT cana-3627 61 1 so so ADV cana-3627 61 2 , , PUNCT cana-3627 61 3 𝑒 𝑒 PROPN cana-3627 61 4 ∈ ∈ PROPN cana-3627 61 5 𝐸0 𝐸0 NOUN cana-3627 61 6 . . PUNCT cana-3627 62 1 assume assume VERB cana-3627 62 2 now now ADV cana-3627 62 3 that that SCONJ cana-3627 62 4 , , PUNCT cana-3627 62 5 π‘₯ π‘₯ DET cana-3627 62 6 ∈ ∈ NOUN cana-3627 62 7 𝑉1 𝑉1 NOUN cana-3627 62 8 . . PUNCT cana-3627 63 1 if if SCONJ cana-3627 63 2 π‘₯ π‘₯ PROPN cana-3627 63 3 ∈ ∈ PROPN cana-3627 63 4 𝐷𝐺(𝑀 𝐷𝐺(𝑀 NOUN cana-3627 63 5 ) ) PUNCT cana-3627 63 6 for for ADP cana-3627 63 7 some some DET cana-3627 63 8 𝑀 𝑀 PRON cana-3627 63 9 ∈ ∈ PROPN cana-3627 63 10 𝑉1⋃𝑉2 𝑉1⋃𝑉2 NOUN cana-3627 63 11 then then ADV cana-3627 63 12 there there PRON cana-3627 63 13 exists exist VERB cana-3627 63 14 no no DET cana-3627 63 15 𝑧 𝑧 PRON cana-3627 63 16 ∈ ∈ PROPN cana-3627 63 17 𝑉0⋂𝑁(π‘₯ 𝑉0⋂𝑁(π‘₯ NUM cana-3627 63 18 ) ) PUNCT cana-3627 63 19 with with ADP cana-3627 63 20 𝑁(𝑧 𝑁(𝑧 ADJ cana-3627 63 21 ) ) PUNCT cana-3627 63 22 ∩ ∩ NOUN cana-3627 63 23 𝑉2 𝑉2 NOUN cana-3627 63 24 = = SYM cana-3627 63 25 βˆ… βˆ… NOUN cana-3627 63 26 and and CCONJ cana-3627 63 27 for for ADP cana-3627 63 28 every every DET cana-3627 63 29 𝑑 𝑑 PROPN cana-3627 63 30 ∈ ∈ PROPN cana-3627 63 31 𝑁(𝑧)⋂𝑉1 𝑁(𝑧)⋂𝑉1 PROPN cana-3627 63 32 , , PUNCT cana-3627 63 33 if if SCONJ cana-3627 63 34 we we PRON cana-3627 63 35 have have VERB cana-3627 63 36 οΏ½ οΏ½ NOUN cana-3627 63 37 Μ… Μ… NOUN cana-3627 63 38 οΏ½ οΏ½ NOUN cana-3627 63 39 (𝑑 (𝑑 NOUN cana-3627 63 40 ) ) PUNCT cana-3627 63 41 βˆͺ βˆͺ ADP cana-3627 63 42 { { PUNCT cana-3627 63 43 𝑦 𝑦 NOUN cana-3627 63 44 } } PUNCT cana-3627 63 45 is be AUX cana-3627 63 46 a a DET cana-3627 63 47 ( ( PUNCT cana-3627 63 48 𝑑 𝑑 NOUN cana-3627 64 1 : : PUNCT cana-3627 64 2 2 2 NUM cana-3627 64 3 ) ) PUNCT cana-3627 64 4 βˆ’ βˆ’ NOUN cana-3627 64 5 set set VERB cana-3627 64 6 in in ADP cana-3627 64 7 Ξ³ Ξ³ PROPN cana-3627 65 1 , , PUNCT cana-3627 65 2 then then ADV cana-3627 65 3 the the DET cana-3627 65 4 function function NOUN cana-3627 65 5 𝑔′ 𝑔′ ADV cana-3627 65 6 defined define VERB cana-3627 65 7 on on ADP cana-3627 65 8 𝑉 𝑉 PROPN cana-3627 65 9 having have VERB cana-3627 65 10 values value NOUN cana-3627 65 11 in in ADP cana-3627 65 12 { { PUNCT cana-3627 65 13 0,1,2 0,1,2 NOUN cana-3627 65 14 } } PUNCT cana-3627 65 15 defined define VERB cana-3627 65 16 by by ADP cana-3627 65 17 𝑔′(𝑦 𝑔′(𝑦 NOUN cana-3627 65 18 ) ) PUNCT cana-3627 65 19 = = SYM cana-3627 65 20 1 1 NUM cana-3627 65 21 , , PUNCT cana-3627 65 22 𝑔′(π‘₯ 𝑔′(π‘₯ PUNCT cana-3627 65 23 ) ) PUNCT cana-3627 65 24 = = SYM cana-3627 65 25 0 0 NUM cana-3627 65 26 , , PUNCT cana-3627 65 27 𝑔′(𝑠 𝑔′(𝑠 X cana-3627 65 28 ) ) PUNCT cana-3627 65 29 = = SYM cana-3627 65 30 𝑔(𝑠 𝑔(𝑠 NOUN cana-3627 65 31 ) ) PUNCT cana-3627 65 32 for for ADP cana-3627 65 33 all all DET cana-3627 65 34 𝑠 𝑠 PROPN cana-3627 65 35 ∈ ∈ PROPN cana-3627 65 36 𝑉 𝑉 PROPN cana-3627 65 37 βˆ’ βˆ’ PROPN cana-3627 65 38 { { PUNCT cana-3627 65 39 π‘₯ π‘₯ NOUN cana-3627 65 40 , , PUNCT cana-3627 65 41 𝑦 𝑦 NOUN cana-3627 65 42 } } PUNCT cana-3627 65 43 is be AUX cana-3627 65 44 a a DET cana-3627 65 45 π‘Šπ‘…π·πΉ π‘Šπ‘…π·πΉ NOUN cana-3627 65 46 defined define VERB cana-3627 65 47 on on ADP cana-3627 65 48 Ξ³ Ξ³ NOUN cana-3627 65 49 βˆ’ βˆ’ PROPN cana-3627 65 50 𝑒 𝑒 ADP cana-3627 65 51 having have VERB cana-3627 65 52 weight weight NOUN cana-3627 65 53 π›Ύπ‘Ÿ(Ξ³ π›Ύπ‘Ÿ(Ξ³ NOUN cana-3627 65 54 ) ) PUNCT cana-3627 65 55 . . PUNCT cana-3627 66 1 since since SCONJ cana-3627 66 2 𝑒 𝑒 PROPN cana-3627 66 3 βˆ‰ βˆ‰ PROPN cana-3627 66 4 πΈβˆ’ πΈβˆ’ ADJ cana-3627 66 5 we we PRON cana-3627 66 6 have have VERB cana-3627 66 7 𝑔′ 𝑔′ ADV cana-3627 66 8 is be AUX cana-3627 66 9 a a DET cana-3627 66 10 π›Ύπ‘Ÿ π›Ύπ‘Ÿ ADJ cana-3627 66 11 βˆ’ βˆ’ NOUN cana-3627 66 12 function function NOUN cana-3627 66 13 defined define VERB cana-3627 66 14 on on ADP cana-3627 66 15 Ξ³ Ξ³ PROPN cana-3627 66 16 βˆ’ βˆ’ PROPN cana-3627 66 17 𝑒. 𝑒. PROPN cana-3627 66 18 therefore therefore ADV cana-3627 66 19 , , PUNCT cana-3627 66 20 π›Ύπ‘Ÿ(Ξ³ π›Ύπ‘Ÿ(Ξ³ NUM cana-3627 66 21 βˆ’ βˆ’ PROPN cana-3627 66 22 𝑒 𝑒 NOUN cana-3627 66 23 ) ) PUNCT cana-3627 66 24 = = SYM cana-3627 66 25 π›Ύπ‘Ÿ(Ξ³ π›Ύπ‘Ÿ(Ξ³ NOUN cana-3627 66 26 ) ) PUNCT cana-3627 66 27 . . PUNCT cana-3627 67 1 so so ADV cana-3627 67 2 , , PUNCT cana-3627 67 3 𝑒 𝑒 PROPN cana-3627 67 4 ∈ ∈ PROPN cana-3627 67 5 𝐸0 𝐸0 NOUN cana-3627 67 6 . . PUNCT cana-3627 68 1 assume assume VERB cana-3627 68 2 now now ADV cana-3627 68 3 that that SCONJ cana-3627 68 4 𝑦 𝑦 PROPN cana-3627 68 5 βˆ‰ βˆ‰ PROPN cana-3627 68 6 οΏ½ οΏ½ PROPN cana-3627 68 7 Μ… Μ… NOUN cana-3627 68 8 οΏ½ οΏ½ NOUN cana-3627 68 9 (π‘₯ (π‘₯ NOUN cana-3627 68 10 ) ) PUNCT cana-3627 68 11 . . PUNCT cana-3627 69 1 assume assume VERB cana-3627 69 2 that that SCONJ cana-3627 69 3 there there PRON cana-3627 69 4 is be VERB cana-3627 69 5 some some DET cana-3627 69 6 π‘Ÿ π‘Ÿ NUM cana-3627 69 7 ∈ ∈ PROPN cana-3627 69 8 𝑉2 𝑉2 NOUN cana-3627 69 9 . . PUNCT cana-3627 70 1 then then ADV cana-3627 70 2 , , PUNCT cana-3627 70 3 𝑒 𝑒 PROPN cana-3627 70 4 ∈ ∈ PROPN cana-3627 70 5 𝐸0 𝐸0 NOUN cana-3627 70 6 . . PUNCT cana-3627 71 1 let let VERB cana-3627 71 2 𝑁(𝑦)⋂𝑉2 𝑁(𝑦)⋂𝑉2 NOUN cana-3627 71 3 = = SYM cana-3627 71 4 βˆ… βˆ… NOUN cana-3627 71 5 , , PUNCT cana-3627 71 6 𝑦 𝑦 PROPN cana-3627 71 7 ∈ ∈ PROPN cana-3627 71 8 οΏ½ οΏ½ PROPN cana-3627 71 9 Μ… Μ… NOUN cana-3627 71 10 οΏ½ οΏ½ NOUN cana-3627 71 11 (π‘Ÿ (π‘Ÿ NOUN cana-3627 71 12 ) ) PUNCT cana-3627 71 13 for for ADP cana-3627 71 14 some some DET cana-3627 71 15 π‘Ÿ π‘Ÿ PRON cana-3627 71 16 ∈ ∈ NOUN cana-3627 71 17 𝑉1 𝑉1 NOUN cana-3627 71 18 where where SCONJ cana-3627 71 19 |𝑁(𝑦 |𝑁(𝑦 NOUN cana-3627 71 20 ) ) PUNCT cana-3627 71 21 ∩ ∩ NOUN cana-3627 71 22 𝑉1| 𝑉1| X cana-3627 71 23 > > X cana-3627 71 24 2 2 NUM cana-3627 71 25 if if SCONJ cana-3627 71 26 | | NOUN cana-3627 71 27 οΏ½ οΏ½ NOUN cana-3627 71 28 Μ… Μ… NOUN cana-3627 71 29 οΏ½ οΏ½ NOUN cana-3627 71 30 (π‘Ÿ (π‘Ÿ NOUN cana-3627 71 31 ) ) PUNCT cana-3627 71 32 βˆͺ βˆͺ NOUN cana-3627 71 33 { { PUNCT cana-3627 71 34 𝑦}| 𝑦}| NOUN cana-3627 71 35 > > X cana-3627 71 36 1 1 NUM cana-3627 71 37 , , PUNCT cana-3627 71 38 οΏ½ οΏ½ NOUN cana-3627 71 39 Μ… Μ… NOUN cana-3627 71 40 οΏ½ οΏ½ NOUN cana-3627 71 41 (π‘Ÿ (π‘Ÿ NOUN cana-3627 71 42 ) ) PUNCT cana-3627 71 43 βˆͺ βˆͺ ADP cana-3627 71 44 { { PUNCT cana-3627 71 45 𝑦 𝑦 NOUN cana-3627 71 46 } } PUNCT cana-3627 71 47 does do AUX cana-3627 71 48 not not PART cana-3627 71 49 induce induce VERB cana-3627 71 50 a a DET cana-3627 71 51 clique clique NOUN cana-3627 71 52 in in ADP cana-3627 71 53 such such DET cana-3627 71 54 a a DET cana-3627 71 55 manner manner NOUN cana-3627 71 56 that that PRON cana-3627 71 57 οΏ½ οΏ½ NOUN cana-3627 71 58 Μ… Μ… NOUN cana-3627 71 59 οΏ½ οΏ½ NOUN cana-3627 71 60 (π‘₯ (π‘₯ NOUN cana-3627 71 61 ) ) PUNCT cana-3627 71 62 βˆ’ βˆ’ PROPN cana-3627 71 63 { { PUNCT cana-3627 71 64 𝑦 𝑦 NOUN cana-3627 71 65 } } PUNCT cana-3627 71 66 happens happen VERB cana-3627 71 67 to to PART cana-3627 71 68 be be AUX cana-3627 71 69 a a DET cana-3627 71 70 ( ( PUNCT cana-3627 71 71 𝑒 𝑒 NOUN cana-3627 71 72 : : PUNCT cana-3627 71 73 𝑖 𝑖 NUM cana-3627 71 74 ) ) PUNCT cana-3627 71 75 βˆ’ βˆ’ NOUN cana-3627 71 76 set set NOUN cana-3627 71 77 , , PUNCT cana-3627 71 78 where where SCONJ cana-3627 71 79 𝑔(π‘₯ 𝑔(π‘₯ NOUN cana-3627 71 80 ) ) PUNCT cana-3627 71 81 = = SYM cana-3627 72 1 𝑖 𝑖 PROPN cana-3627 72 2 , , PUNCT cana-3627 72 3 𝑖 𝑖 NOUN cana-3627 72 4 = = SYM cana-3627 72 5 1 1 NUM cana-3627 72 6 , , PUNCT cana-3627 72 7 2 2 NUM cana-3627 72 8 . . PUNCT cana-3627 73 1 in in ADP cana-3627 73 2 this this DET cana-3627 73 3 case case NOUN cana-3627 73 4 , , PUNCT cana-3627 73 5 𝑔 𝑔 PROPN cana-3627 73 6 is be AUX cana-3627 73 7 a a DET cana-3627 73 8 π‘Šπ‘…π·πΉ π‘Šπ‘…π·πΉ NOUN cana-3627 73 9 defined define VERB cana-3627 73 10 on on ADP cana-3627 73 11 Ξ³ Ξ³ PROPN cana-3627 73 12 βˆ’ βˆ’ PROPN cana-3627 73 13 𝑒. 𝑒. VERB cana-3627 73 14 again again ADV cana-3627 73 15 since since SCONJ cana-3627 73 16 𝑒 𝑒 PROPN cana-3627 73 17 βˆ‰ βˆ‰ PROPN cana-3627 73 18 πΈβˆ’ πΈβˆ’ ADJ cana-3627 73 19 , , PUNCT cana-3627 73 20 we we PRON cana-3627 73 21 have have VERB cana-3627 73 22 that that PRON cana-3627 73 23 𝑔′ 𝑔′ NOUN cana-3627 73 24 is be AUX cana-3627 73 25 a a DET cana-3627 73 26 π›Ύπ‘Ÿ π›Ύπ‘Ÿ ADJ cana-3627 73 27 βˆ’ βˆ’ NOUN cana-3627 73 28 function function NOUN cana-3627 73 29 defined define VERB cana-3627 73 30 on on ADP cana-3627 73 31 Ξ³ Ξ³ PROPN cana-3627 73 32 βˆ’ βˆ’ PROPN cana-3627 73 33 𝑒. 𝑒. PROPN cana-3627 73 34 therefore therefore ADV cana-3627 73 35 , , PUNCT cana-3627 73 36 π›Ύπ‘Ÿ(Ξ³ π›Ύπ‘Ÿ(Ξ³ NUM cana-3627 73 37 βˆ’ βˆ’ PROPN cana-3627 73 38 𝑒 𝑒 NOUN cana-3627 73 39 ) ) PUNCT cana-3627 73 40 = = SYM cana-3627 73 41 π›Ύπ‘Ÿ(Ξ³ π›Ύπ‘Ÿ(Ξ³ NOUN cana-3627 73 42 ) ) PUNCT cana-3627 73 43 . . PUNCT cana-3627 74 1 consequently consequently ADV cana-3627 74 2 , , PUNCT cana-3627 74 3 𝑒 𝑒 PROPN cana-3627 74 4 ∈ ∈ PROPN cana-3627 74 5 𝐸0 𝐸0 NOUN cana-3627 74 6 . . PUNCT cana-3627 75 1 if if SCONJ cana-3627 75 2 π‘Ÿ π‘Ÿ PRON cana-3627 75 3 ∈ ∈ PROPN cana-3627 75 4 𝑉1 𝑉1 NOUN cana-3627 75 5 , , PUNCT cana-3627 75 6 and and CCONJ cana-3627 75 7 οΏ½ οΏ½ NOUN cana-3627 75 8 Μ… Μ… NOUN cana-3627 75 9 οΏ½ οΏ½ NOUN cana-3627 75 10 (π‘Ÿ (π‘Ÿ NOUN cana-3627 75 11 ) ) PUNCT cana-3627 75 12 βˆͺ βˆͺ ADP cana-3627 75 13 { { PUNCT cana-3627 75 14 𝑦 𝑦 NOUN cana-3627 75 15 } } PUNCT cana-3627 75 16 is be AUX cana-3627 75 17 a a DET cana-3627 75 18 ( ( PUNCT cana-3627 75 19 π‘Ÿ π‘Ÿ NOUN cana-3627 75 20 : : PUNCT cana-3627 75 21 2 2 NUM cana-3627 75 22 ) ) PUNCT cana-3627 75 23 βˆ’ βˆ’ NOUN cana-3627 75 24 set set VERB cana-3627 75 25 in in ADP cana-3627 75 26 Ξ³ Ξ³ NOUN cana-3627 75 27 and and CCONJ cana-3627 75 28 if if SCONJ cana-3627 75 29 we we PRON cana-3627 75 30 can can AUX cana-3627 75 31 find find VERB cana-3627 75 32 π‘š π‘š ADP cana-3627 75 33 ∈ ∈ PROPN cana-3627 75 34 𝑉1 𝑉1 NOUN cana-3627 75 35 βˆͺ βˆͺ NOUN cana-3627 75 36 𝑉2 𝑉2 NOUN cana-3627 75 37 in in ADP cana-3627 75 38 such such DET cana-3627 75 39 a a DET cana-3627 75 40 manner manner NOUN cana-3627 75 41 that that PRON cana-3627 75 42 π‘₯ π‘₯ DET cana-3627 75 43 ∈ ∈ PROPN cana-3627 75 44 𝐷𝐺(π‘š 𝐷𝐺(π‘š NOUN cana-3627 75 45 ) ) PUNCT cana-3627 75 46 , , PUNCT cana-3627 75 47 then then ADV cana-3627 75 48 the the DET cana-3627 75 49 function function NOUN cana-3627 75 50 𝑔′ 𝑔′ ADV cana-3627 75 51 defined define VERB cana-3627 75 52 on on ADP cana-3627 75 53 𝑉 𝑉 PROPN cana-3627 75 54 having have VERB cana-3627 75 55 values value NOUN cana-3627 75 56 in in ADP cana-3627 75 57 { { PUNCT cana-3627 75 58 0,1,2 0,1,2 NUM cana-3627 75 59 } } PUNCT cana-3627 75 60 so so ADV cana-3627 75 61 defined define VERB cana-3627 75 62 that that SCONJ cana-3627 75 63 𝑔′(π‘Ÿ 𝑔′(π‘Ÿ NUM cana-3627 75 64 ) ) PUNCT cana-3627 75 65 = = SYM cana-3627 75 66 2 2 NUM cana-3627 75 67 , , PUNCT cana-3627 75 68 𝑔′(π‘₯ 𝑔′(π‘₯ PUNCT cana-3627 75 69 ) ) PUNCT cana-3627 75 70 = = SYM cana-3627 75 71 0 0 NUM cana-3627 75 72 , , PUNCT cana-3627 75 73 𝑔′(𝑧 𝑔′(𝑧 X cana-3627 75 74 ) ) PUNCT cana-3627 75 75 = = SYM cana-3627 75 76 𝑔(𝑧 𝑔(𝑧 PROPN cana-3627 75 77 ) ) PUNCT cana-3627 75 78 when when SCONJ cana-3627 75 79 𝑧 𝑧 PRON cana-3627 75 80 ∈ ∈ PROPN cana-3627 75 81 𝑉 𝑉 PROPN cana-3627 75 82 βˆ’ βˆ’ PROPN cana-3627 75 83 { { PUNCT cana-3627 75 84 π‘Ÿ π‘Ÿ NOUN cana-3627 75 85 , , PUNCT cana-3627 75 86 π‘₯ π‘₯ X cana-3627 75 87 } } PUNCT cana-3627 75 88 will will AUX cana-3627 75 89 be be AUX cana-3627 75 90 a a DET cana-3627 75 91 π‘Šπ‘…π·πΉ π‘Šπ‘…π·πΉ NOUN cana-3627 75 92 defined define VERB cana-3627 75 93 on on ADP cana-3627 75 94 Ξ³ Ξ³ NOUN cana-3627 75 95 βˆ’ βˆ’ PROPN cana-3627 75 96 𝑒 𝑒 AUX cana-3627 75 97 having have VERB cana-3627 75 98 π›Ύπ‘Ÿ(Ξ³ π›Ύπ‘Ÿ(Ξ³ NOUN cana-3627 75 99 ) ) PUNCT cana-3627 75 100 as as ADP cana-3627 75 101 its its PRON cana-3627 75 102 weight weight NOUN cana-3627 75 103 . . PUNCT cana-3627 76 1 arguing argue VERB cana-3627 76 2 as as ADP cana-3627 76 3 before before ADV cana-3627 76 4 , , PUNCT cana-3627 76 5 we we PRON cana-3627 76 6 have have VERB cana-3627 76 7 𝑒 𝑒 PROPN cana-3627 76 8 ∈ ∈ PROPN cana-3627 76 9 𝐸0 𝐸0 NOUN cana-3627 76 10 . . PUNCT cana-3627 77 1 conversely conversely ADV cana-3627 77 2 , , PUNCT cana-3627 77 3 let let VERB cana-3627 77 4 𝑒 𝑒 PROPN cana-3627 77 5 ∈ ∈ PROPN cana-3627 77 6 𝐸0 𝐸0 NOUN cana-3627 77 7 . . PUNCT cana-3627 78 1 if if SCONJ cana-3627 78 2 𝑦 𝑦 PROPN cana-3627 78 3 ∈ ∈ PROPN cana-3627 78 4 οΏ½ οΏ½ PROPN cana-3627 78 5 Μ… Μ… NOUN cana-3627 78 6 οΏ½ οΏ½ NOUN cana-3627 78 7 (π‘₯ (π‘₯ NOUN cana-3627 78 8 ) ) PUNCT cana-3627 78 9 , , PUNCT cana-3627 78 10 then then ADV cana-3627 78 11 𝑒 𝑒 PROPN cana-3627 78 12 ∈ ∈ PROPN cana-3627 78 13 𝐸+ 𝐸+ PUNCT cana-3627 78 14 if if SCONJ cana-3627 78 15 none none NOUN cana-3627 78 16 of of ADP cana-3627 78 17 the the DET cana-3627 78 18 conditions condition NOUN cana-3627 78 19 a a PRON cana-3627 78 20 or or CCONJ cana-3627 78 21 b b NOUN cana-3627 78 22 is be AUX cana-3627 78 23 satisfied satisfied ADJ cana-3627 78 24 . . PUNCT cana-3627 79 1 similarly similarly ADV cana-3627 79 2 , , PUNCT cana-3627 79 3 if if SCONJ cana-3627 79 4 𝑦 𝑦 PROPN cana-3627 79 5 ∈ ∈ PROPN cana-3627 79 6 οΏ½ οΏ½ PROPN cana-3627 79 7 Μ… Μ… NOUN cana-3627 79 8 οΏ½ οΏ½ NOUN cana-3627 79 9 (π‘₯ (π‘₯ NOUN cana-3627 79 10 ) ) PUNCT cana-3627 79 11 , , PUNCT cana-3627 79 12 then then ADV cana-3627 79 13 𝑒 𝑒 PROPN cana-3627 79 14 ∈ ∈ PROPN cana-3627 79 15 𝐸+ 𝐸+ PUNCT cana-3627 79 16 if if SCONJ cana-3627 79 17 none none NOUN cana-3627 79 18 of of ADP cana-3627 79 19 the the DET cana-3627 79 20 conditions condition NOUN cana-3627 79 21 c c NOUN cana-3627 79 22 or or CCONJ cana-3627 79 23 d d PROPN cana-3627 79 24 is be AUX cana-3627 79 25 satisfied satisfied ADJ cana-3627 79 26 . . PUNCT cana-3627 80 1 this this PRON cana-3627 80 2 proves prove VERB cana-3627 80 3 the the DET cana-3627 80 4 theorem theorem NOUN cana-3627 80 5 completely completely ADV cana-3627 80 6 . . PUNCT cana-3627 81 1 ο‚ž ο‚ž PROPN cana-3627 81 2 theorem theorem VERB cana-3627 81 3 2.2 2.2 NUM cana-3627 81 4 . . PUNCT cana-3627 82 1 for for ADP cana-3627 82 2 any any DET cana-3627 82 3 graph graph NOUN cana-3627 83 1 Ξ³ Ξ³ ADP cana-3627 83 2 having have VERB cana-3627 83 3 an an DET cana-3627 83 4 edge edge NOUN cana-3627 83 5 π‘₯𝑦 π‘₯𝑦 NOUN cana-3627 83 6 (= (= NOUN cana-3627 83 7 𝑒 𝑒 PROPN cana-3627 83 8 ) ) PUNCT cana-3627 83 9 , , PUNCT cana-3627 83 10 𝑒 𝑒 PROPN cana-3627 83 11 ∈ ∈ PROPN cana-3627 83 12 𝐸+ 𝐸+ PUNCT cana-3627 83 13 if if SCONJ cana-3627 83 14 and and CCONJ cana-3627 83 15 only only ADV cana-3627 83 16 if if SCONJ cana-3627 83 17 𝑒 𝑒 PROPN cana-3627 83 18 βˆ‰ βˆ‰ PROPN cana-3627 83 19 𝐸0 𝐸0 PROPN cana-3627 83 20 . . PUNCT cana-3627 84 1 proof proof NOUN cana-3627 84 2 . . PUNCT cana-3627 85 1 the the DET cana-3627 85 2 result result NOUN cana-3627 85 3 is be AUX cana-3627 85 4 a a DET cana-3627 85 5 consequence consequence NOUN cana-3627 85 6 of of ADP cana-3627 85 7 the the DET cana-3627 85 8 facts fact NOUN cana-3627 85 9 , , PUNCT cana-3627 85 10 𝐸 𝐸 PROPN cana-3627 85 11 = = PUNCT cana-3627 85 12 πΈβˆ’ πΈβˆ’ ADJ cana-3627 85 13 βˆͺ βˆͺ ADJ cana-3627 85 14 𝐸0 𝐸0 NOUN cana-3627 85 15 βˆͺ βˆͺ ADP cana-3627 85 16 𝐸+ 𝐸+ PROPN cana-3627 85 17 , , PUNCT cana-3627 85 18 πΈβˆ’ πΈβˆ’ ADJ cana-3627 85 19 = = NOUN cana-3627 85 20 βˆ… βˆ… NOUN cana-3627 85 21 and and CCONJ cana-3627 85 22 𝐸0 𝐸0 ADJ cana-3627 85 23 ∩ ∩ NOUN cana-3627 85 24 𝐸+ 𝐸+ PUNCT cana-3627 85 25 = = VERB cana-3627 85 26 βˆ…. βˆ…. PRON cana-3627 85 27 ο‚ž ο‚ž ADJ cana-3627 85 28 references reference NOUN cana-3627 85 29 [ [ X cana-3627 85 30 1 1 NUM cana-3627 85 31 ] ] X cana-3627 85 32 e.w e.w PROPN cana-3627 85 33 . . PROPN cana-3627 85 34 chambers chambers PROPN cana-3627 85 35 et et PROPN cana-3627 85 36 al al PROPN cana-3627 85 37 . . PROPN cana-3627 85 38 , , PUNCT cana-3627 85 39 extremal extremal ADJ cana-3627 85 40 problems problem NOUN cana-3627 85 41 for for ADP cana-3627 85 42 roman roman ADJ cana-3627 85 43 domination domination NOUN cana-3627 85 44 , , PUNCT cana-3627 85 45 siam siam ADJ cana-3627 85 46 journal journal NOUN cana-3627 85 47 on on ADP cana-3627 85 48 discrete discrete ADJ cana-3627 85 49 mathematics mathematic NOUN cana-3627 85 50 , , PUNCT cana-3627 85 51 volume volume NOUN cana-3627 85 52 23(3 23(3 NUM cana-3627 85 53 ) ) PUNCT cana-3627 85 54 ( ( PUNCT cana-3627 85 55 2009 2009 NUM cana-3627 85 56 ) ) PUNCT cana-3627 85 57 , , PUNCT cana-3627 85 58 pp pp ADP cana-3627 85 59 . . PUNCT cana-3627 86 1 1575 1575 NUM cana-3627 86 2 - - SYM cana-3627 86 3 1586 1586 NUM cana-3627 86 4 . . PUNCT cana-3627 87 1 [ [ X cana-3627 87 2 2 2 NUM cana-3627 87 3 ] ] SYM cana-3627 87 4 e.j e.j PROPN cana-3627 87 5 . . PROPN cana-3627 87 6 cockayne cockayne PROPN cana-3627 87 7 , , PUNCT cana-3627 87 8 p.a p.a PROPN cana-3627 87 9 . . PUNCT cana-3627 87 10 dreyer dreyer PROPN cana-3627 87 11 , , PUNCT cana-3627 87 12 s.m s.m PROPN cana-3627 87 13 . . PROPN cana-3627 87 14 hedetniemi hedetniemi PROPN cana-3627 87 15 and and CCONJ cana-3627 87 16 s.t s.t PROPN cana-3627 87 17 . . PROPN cana-3627 87 18 hedetniemi hedetniemi PROPN cana-3627 87 19 , , PUNCT cana-3627 87 20 roman roman ADJ cana-3627 87 21 domination domination NOUN cana-3627 87 22 in in ADP cana-3627 87 23 graphs graph NOUN cana-3627 87 24 , , PUNCT cana-3627 87 25 discrete discrete ADJ cana-3627 87 26 mathematics mathematic NOUN cana-3627 87 27 , , PUNCT cana-3627 87 28 278 278 NUM cana-3627 87 29 ( ( PUNCT cana-3627 87 30 2004 2004 NUM cana-3627 87 31 ) ) PUNCT cana-3627 87 32 , , PUNCT cana-3627 87 33 11 11 NUM cana-3627 87 34 - - SYM cana-3627 87 35 22 22 NUM cana-3627 87 36 . . PUNCT cana-3627 88 1 [ [ X cana-3627 88 2 3 3 X cana-3627 88 3 ] ] X cana-3627 88 4 m.a m.a PROPN cana-3627 88 5 . . PROPN cana-3627 88 6 henning henning PROPN cana-3627 88 7 and and CCONJ cana-3627 88 8 s.t s.t PROPN cana-3627 88 9 . . PROPN cana-3627 88 10 hedetniemi hedetniemi PROPN cana-3627 88 11 , , PUNCT cana-3627 88 12 defending defend VERB cana-3627 88 13 the the DET cana-3627 88 14 roman roman ADJ cana-3627 88 15 empire empire NOUN cana-3627 88 16 a a DET cana-3627 88 17 new new ADJ cana-3627 88 18 strategy strategy NOUN cana-3627 88 19 , , PUNCT cana-3627 88 20 discrete discrete ADJ cana-3627 88 21 mathematics mathematic NOUN cana-3627 88 22 , , PUNCT cana-3627 88 23 266(1 266(1 NUM cana-3627 88 24 - - SYM cana-3627 88 25 3 3 NUM cana-3627 88 26 ) ) PUNCT cana-3627 88 27 ( ( PUNCT cana-3627 88 28 2003 2003 NUM cana-3627 88 29 ) ) PUNCT cana-3627 88 30 , , PUNCT cana-3627 88 31 239 239 NUM cana-3627 88 32 - - SYM cana-3627 88 33 251 251 NUM cana-3627 88 34 . . PUNCT cana-3627 89 1 communications communication NOUN cana-3627 89 2 on on ADP cana-3627 89 3 applied apply VERB cana-3627 89 4 nonlinear nonlinear ADJ cana-3627 89 5 analysis analysis NOUN cana-3627 89 6 issn issn NOUN cana-3627 89 7 : : PUNCT cana-3627 89 8 1074 1074 NUM cana-3627 89 9 - - PUNCT cana-3627 89 10 133x 133x NUM cana-3627 89 11 vol vol NOUN cana-3627 89 12 32 32 NUM cana-3627 89 13 no no NOUN cana-3627 89 14 . . PUNCT cana-3627 90 1 8s 8s PROPN cana-3627 90 2 ( ( PUNCT cana-3627 90 3 2025 2025 NUM cana-3627 90 4 ) ) PUNCT cana-3627 90 5 185 185 NUM cana-3627 90 6 https://internationalpubls.com https://internationalpubls.com X cana-3627 91 1 [ [ X cana-3627 91 2 4 4 NUM cana-3627 91 3 ] ] PUNCT cana-3627 91 4 a. a. NOUN cana-3627 91 5 klobucar klobucar NOUN cana-3627 91 6 and and CCONJ cana-3627 91 7 i. i. PROPN cana-3627 91 8 puljic puljic PROPN cana-3627 91 9 , , PUNCT cana-3627 91 10 some some PRON cana-3627 91 11 results result NOUN cana-3627 91 12 for for ADP cana-3627 91 13 roman roman ADJ cana-3627 91 14 domination domination NOUN cana-3627 91 15 number number NOUN cana-3627 91 16 on on ADP cana-3627 91 17 cardinal cardinal ADJ cana-3627 91 18 product product NOUN cana-3627 91 19 of of ADP cana-3627 91 20 paths path NOUN cana-3627 91 21 and and CCONJ cana-3627 91 22 cycles cycle NOUN cana-3627 91 23 , , PUNCT cana-3627 91 24 kragujevac kragujevac PROPN cana-3627 91 25 journal journal NOUN cana-3627 91 26 of of ADP cana-3627 91 27 mathematics mathematics PROPN cana-3627 91 28 volume volume NOUN cana-3627 91 29 38(1 38(1 NOUN cana-3627 91 30 ) ) PUNCT cana-3627 91 31 ( ( PUNCT cana-3627 91 32 2014 2014 NUM cana-3627 91 33 ) ) PUNCT cana-3627 91 34 , , PUNCT cana-3627 91 35 pp pp ADP cana-3627 91 36 . . PUNCT cana-3627 92 1 83–94 83–94 NUM cana-3627 92 2 . . PUNCT cana-3627 93 1 [ [ X cana-3627 93 2 5 5 X cana-3627 93 3 ] ] X cana-3627 93 4 majid majid PROPN cana-3627 93 5 hajian hajian PROPN cana-3627 93 6 and and CCONJ cana-3627 93 7 nader nader VERB cana-3627 93 8 jafari jafari PROPN cana-3627 93 9 rad rad PROPN cana-3627 93 10 , , PUNCT cana-3627 93 11 on on ADP cana-3627 93 12 the the DET cana-3627 93 13 roman roman ADJ cana-3627 93 14 domination domination NOUN cana-3627 93 15 stable stable ADJ cana-3627 93 16 graphs graph NOUN cana-3627 93 17 , , PUNCT cana-3627 93 18 discussiones discussione NOUN cana-3627 93 19 mathematicae mathematicae PROPN cana-3627 93 20 graph graph NOUN cana-3627 93 21 theory theory NOUN cana-3627 93 22 , , PUNCT cana-3627 93 23 volume volume NOUN cana-3627 93 24 37 37 NUM cana-3627 93 25 ( ( PUNCT cana-3627 93 26 2017 2017 NUM cana-3627 93 27 ) ) PUNCT cana-3627 93 28 , , PUNCT cana-3627 93 29 859 859 NUM cana-3627 93 30 - - SYM cana-3627 93 31 871 871 NUM cana-3627 93 32 . . PUNCT cana-3627 94 1 [ [ X cana-3627 94 2 6 6 NUM cana-3627 94 3 ] ] PUNCT cana-3627 94 4 b. b. PROPN cana-3627 94 5 mahavir mahavir PROPN cana-3627 94 6 et et PROPN cana-3627 94 7 al al PROPN cana-3627 94 8 . . PROPN cana-3627 94 9 , , PUNCT cana-3627 94 10 an an DET cana-3627 94 11 algorithm algorithm NOUN cana-3627 94 12 to to PART cana-3627 94 13 recognize recognize VERB cana-3627 94 14 weak weak ADJ cana-3627 94 15 roman roman ADJ cana-3627 94 16 domination domination NOUN cana-3627 94 17 stable stable ADJ cana-3627 94 18 trees tree NOUN cana-3627 94 19 under under ADP cana-3627 94 20 vertex vertex NOUN cana-3627 94 21 deletion deletion NOUN cana-3627 94 22 , , PUNCT cana-3627 94 23 discrete discrete ADJ cana-3627 94 24 mathematics mathematic NOUN cana-3627 94 25 , , PUNCT cana-3627 94 26 algorithms algorithm NOUN cana-3627 94 27 and and CCONJ cana-3627 94 28 applications application NOUN cana-3627 94 29 , , PUNCT cana-3627 94 30 vol.12 vol.12 NOUN cana-3627 94 31 , , PUNCT cana-3627 94 32 no no NOUN cana-3627 94 33 . . NOUN cana-3627 94 34 04 04 NUM cana-3627 94 35 , , PUNCT cana-3627 94 36 2050049 2050049 NUM cana-3627 94 37 ( ( PUNCT cana-3627 94 38 2020 2020 NUM cana-3627 94 39 ) ) PUNCT cana-3627 94 40 . . PUNCT cana-3627 95 1 [ [ X cana-3627 95 2 7 7 X cana-3627 95 3 ] ] X cana-3627 95 4 b.p b.p PROPN cana-3627 95 5 . . PROPN cana-3627 95 6 mobaraky mobaraky PROPN cana-3627 95 7 and and CCONJ cana-3627 95 8 s.m s.m PROPN cana-3627 95 9 . . PROPN cana-3627 95 10 sheikholeslami sheikholeslami PROPN cana-3627 95 11 , , PUNCT cana-3627 95 12 bounds bound VERB cana-3627 95 13 on on ADP cana-3627 95 14 roman roman ADJ cana-3627 95 15 domination domination NOUN cana-3627 95 16 numbers number NOUN cana-3627 95 17 of of ADP cana-3627 95 18 graphs graph NOUN cana-3627 95 19 , , PUNCT cana-3627 95 20 matematicki matematicki NOUN cana-3627 95 21 vesnik vesnik X cana-3627 95 22 60(4 60(4 NUM cana-3627 95 23 ) ) PUNCT cana-3627 95 24 ( ( PUNCT cana-3627 95 25 2008 2008 NUM cana-3627 95 26 ) ) PUNCT cana-3627 95 27 , , PUNCT cana-3627 95 28 247 247 NUM cana-3627 95 29 - - SYM cana-3627 95 30 253 253 NUM cana-3627 95 31 . . PUNCT cana-3627 96 1 [ [ X cana-3627 96 2 8 8 NUM cana-3627 96 3 ] ] X cana-3627 96 4 polona polona NOUN cana-3627 96 5 pavlic pavlic NOUN cana-3627 96 6 , , PUNCT cana-3627 96 7 roman roman ADJ cana-3627 96 8 domination domination NOUN cana-3627 96 9 number number NOUN cana-3627 96 10 of of ADP cana-3627 96 11 the the DET cana-3627 96 12 cartesian cartesian ADJ cana-3627 96 13 products product NOUN cana-3627 96 14 of of ADP cana-3627 96 15 paths path NOUN cana-3627 96 16 and and CCONJ cana-3627 96 17 cycles cycle NOUN cana-3627 96 18 , , PUNCT cana-3627 96 19 the the DET cana-3627 96 20 electronic electronic ADJ cana-3627 96 21 journal journal NOUN cana-3627 96 22 of of ADP cana-3627 96 23 combinatorics combinatoric NOUN cana-3627 96 24 , , PUNCT cana-3627 96 25 19(3 19(3 NUM cana-3627 96 26 ) ) PUNCT cana-3627 96 27 ( ( PUNCT cana-3627 96 28 2012 2012 NUM cana-3627 96 29 ) ) PUNCT cana-3627 96 30 , , PUNCT cana-3627 96 31 pp pp ADP cana-3627 96 32 . . PUNCT cana-3627 97 1 1 1 NUM cana-3627 97 2 - - SYM cana-3627 97 3 37 37 NUM cana-3627 97 4 . . PUNCT cana-3627 98 1 [ [ X cana-3627 98 2 9 9 NUM cana-3627 98 3 ] ] PUNCT cana-3627 98 4 f. f. PROPN cana-3627 98 5 ramezani ramezani PROPN cana-3627 98 6 et et PROPN cana-3627 98 7 al al PROPN cana-3627 98 8 . . PROPN cana-3627 98 9 , , PUNCT cana-3627 98 10 on on ADP cana-3627 98 11 the the DET cana-3627 98 12 roman roman ADJ cana-3627 98 13 domination domination NOUN cana-3627 98 14 number number NOUN cana-3627 98 15 of of ADP cana-3627 98 16 generalized generalized ADJ cana-3627 98 17 sierpinski sierpinski ADJ cana-3627 98 18 graphs graph NOUN cana-3627 98 19 , , PUNCT cana-3627 98 20 filomat filomat NOUN cana-3627 98 21 31(20 31(20 NUM cana-3627 98 22 ) ) PUNCT cana-3627 98 23 ( ( PUNCT cana-3627 98 24 2017 2017 NUM cana-3627 98 25 ) ) PUNCT cana-3627 98 26 , , PUNCT cana-3627 98 27 pp pp ADJ cana-3627 98 28 . . PUNCT cana-3627 99 1 6515–6528 6515–6528 NUM cana-3627 99 2 . . PUNCT cana-3627 100 1 [ [ X cana-3627 100 2 10 10 NUM cana-3627 100 3 ] ] X cana-3627 100 4 p. p. NOUN cana-3627 100 5 roushini roushini PROPN cana-3627 100 6 leely leely ADV cana-3627 100 7 pushpam pushpam VERB cana-3627 100 8 and and CCONJ cana-3627 100 9 t.n.m t.n.m NOUN cana-3627 100 10 . . PUNCT cana-3627 101 1 malini malini PROPN cana-3627 101 2 mai mai PROPN cana-3627 101 3 , , PUNCT cana-3627 101 4 weak weak ADJ cana-3627 101 5 roman roman ADJ cana-3627 101 6 domination domination NOUN cana-3627 101 7 in in ADP cana-3627 101 8 graphs graph NOUN cana-3627 101 9 , , PUNCT cana-3627 101 10 discussiones discussione NOUN cana-3627 101 11 mathematicae mathematicae VERB cana-3627 101 12 , , PUNCT cana-3627 101 13 graph graph NOUN cana-3627 101 14 theory theory NOUN cana-3627 101 15 31 31 NUM cana-3627 101 16 , , PUNCT cana-3627 101 17 ( ( PUNCT cana-3627 101 18 2011 2011 NUM cana-3627 101 19 ) ) PUNCT cana-3627 101 20 , , PUNCT cana-3627 101 21 115 115 NUM cana-3627 101 22 - - SYM cana-3627 101 23 128 128 NUM cana-3627 101 24 . . PUNCT cana-3627 102 1 [ [ X cana-3627 102 2 11 11 NUM cana-3627 102 3 ] ] PUNCT cana-3627 102 4 p. p. NOUN cana-3627 102 5 roushini roushini PROPN cana-3627 102 6 leely leely ADV cana-3627 102 7 pushpam pushpam VERB cana-3627 102 8 and and CCONJ cana-3627 102 9 m. m. NOUN cana-3627 102 10 kamalam kamalam PROPN cana-3627 102 11 , , PUNCT cana-3627 102 12 stability stability NOUN cana-3627 102 13 of of ADP cana-3627 102 14 weak weak ADJ cana-3627 102 15 roman roman ADJ cana-3627 102 16 domination domination NOUN cana-3627 102 17 upon upon SCONJ cana-3627 102 18 vertex vertex NOUN cana-3627 102 19 deletion deletion NOUN cana-3627 102 20 , , PUNCT cana-3627 102 21 asian asian ADJ cana-3627 102 22 journal journal NOUN cana-3627 102 23 of of ADP cana-3627 102 24 mathematics mathematics PROPN cana-3627 102 25 and and CCONJ cana-3627 102 26 computer computer NOUN cana-3627 102 27 research research NOUN cana-3627 102 28 , , PUNCT cana-3627 102 29 25(2 25(2 NUM cana-3627 102 30 ) ) PUNCT cana-3627 102 31 ( ( PUNCT cana-3627 102 32 2018 2018 NUM cana-3627 102 33 ) ) PUNCT cana-3627 102 34 , , PUNCT cana-3627 103 1 97–105 97–105 PROPN cana-3627 103 2 . . PUNCT cana-3627 104 1 [ [ X cana-3627 104 2 12 12 NUM cana-3627 104 3 ] ] X cana-3627 104 4 p. p. NOUN cana-3627 104 5 roushini roushini PROPN cana-3627 104 6 leely leely ADV cana-3627 104 7 pushpam pushpam VERB cana-3627 104 8 and and CCONJ cana-3627 104 9 m. m. NOUN cana-3627 104 10 kamalam kamalam PROPN cana-3627 104 11 , , PUNCT cana-3627 104 12 effect effect NOUN cana-3627 104 13 of of ADP cana-3627 104 14 vertex vertex NOUN cana-3627 104 15 deletion deletion NOUN cana-3627 104 16 on on ADP cana-3627 104 17 the the DET cana-3627 104 18 weak weak ADJ cana-3627 104 19 roman roman ADJ cana-3627 104 20 domination domination NOUN cana-3627 104 21 number number NOUN cana-3627 104 22 of of ADP cana-3627 104 23 a a DET cana-3627 104 24 graph graph NOUN cana-3627 104 25 , , PUNCT cana-3627 104 26 akce akce ADJ cana-3627 104 27 international international ADJ cana-3627 104 28 journal journal NOUN cana-3627 104 29 of of ADP cana-3627 104 30 graphs graph NOUN cana-3627 104 31 and and CCONJ cana-3627 104 32 combinatorics combinatoric NOUN cana-3627 104 33 , , PUNCT cana-3627 104 34 volume volume NOUN cana-3627 104 35 16 16 NUM cana-3627 104 36 , , PUNCT cana-3627 104 37 issue issue NOUN cana-3627 104 38 2 2 NUM cana-3627 104 39 , , PUNCT cana-3627 104 40 august august PROPN cana-3627 104 41 2019 2019 NUM cana-3627 104 42 , , PUNCT cana-3627 104 43 pp pp ADP cana-3627 104 44 204212 204212 NUM cana-3627 104 45 . . PUNCT