id sid tid token lemma pos cana-5232 1 1 communications communication NOUN cana-5232 1 2 on on ADP cana-5232 1 3 applied apply VERB cana-5232 1 4 nonlinear nonlinear ADJ cana-5232 1 5 analysis analysis NOUN cana-5232 1 6 issn issn NOUN cana-5232 1 7 : : PUNCT cana-5232 1 8 1074 1074 NUM cana-5232 1 9 - - PUNCT cana-5232 1 10 133x 133x NUM cana-5232 1 11 vol vol VERB cana-5232 1 12 32 32 NUM cana-5232 1 13 no no NOUN cana-5232 1 14 . . PUNCT cana-5232 2 1 10s 10 NOUN cana-5232 2 2 ( ( PUNCT cana-5232 2 3 2025 2025 NUM cana-5232 2 4 ) ) PUNCT cana-5232 2 5 1296 1296 NUM cana-5232 2 6 https://internationalpubls.com https://internationalpubls.com X cana-5232 2 7 closely closely ADV cana-5232 2 8 connected connect VERB cana-5232 2 9 domination domination NOUN cana-5232 2 10 number number NOUN cana-5232 2 11 in in ADP cana-5232 2 12 corona corona NOUN cana-5232 2 13 product product NOUN cana-5232 2 14 of of ADP cana-5232 2 15 graphs graph NOUN cana-5232 2 16 gurusamy gurusamy NOUN cana-5232 2 17 p1 p1 NOUN cana-5232 2 18 , , PUNCT cana-5232 2 19 * * PUNCT cana-5232 2 20 angel angel NOUN cana-5232 2 21 joy joy NOUN cana-5232 2 22 r𝟐 r𝟐 PROPN cana-5232 2 23 , , PUNCT cana-5232 2 24 1 1 NUM cana-5232 2 25 research research NOUN cana-5232 2 26 scholar scholar NOUN cana-5232 2 27 , , PUNCT cana-5232 2 28 department department NOUN cana-5232 2 29 of of ADP cana-5232 2 30 mathematics mathematic NOUN cana-5232 2 31 , , PUNCT cana-5232 2 32 sri sri PROPN cana-5232 2 33 g.v.g g.v.g ADP cana-5232 2 34 visalakshi visalakshi PROPN cana-5232 2 35 college college PROPN cana-5232 2 36 for for ADP cana-5232 2 37 women woman NOUN cana-5232 2 38 , , PUNCT cana-5232 2 39 udumalpet udumalpet ADJ cana-5232 2 40 . . PUNCT cana-5232 3 1 * * PUNCT cana-5232 3 2 assistant assistant NOUN cana-5232 3 3 professor professor NOUN cana-5232 3 4 , , PUNCT cana-5232 3 5 department department NOUN cana-5232 3 6 of of ADP cana-5232 3 7 mathematics mathematics PROPN cana-5232 3 8 . . PUNCT cana-5232 4 1 government government NOUN cana-5232 4 2 arts art NOUN cana-5232 4 3 and and CCONJ cana-5232 4 4 science science NOUN cana-5232 4 5 college college PROPN cana-5232 4 6 , , PUNCT cana-5232 4 7 kangeyam kangeyam NOUN cana-5232 4 8 . . PUNCT cana-5232 5 1 2 2 NUM cana-5232 5 2 assistant assistant NOUN cana-5232 5 3 professor professor NOUN cana-5232 5 4 , , PUNCT cana-5232 5 5 department department NOUN cana-5232 5 6 of of ADP cana-5232 5 7 mathematics mathematic NOUN cana-5232 5 8 , , PUNCT cana-5232 5 9 sri sri PROPN cana-5232 5 10 g.v.g g.v.g ADP cana-5232 5 11 visalakshi visalakshi PROPN cana-5232 5 12 college college PROPN cana-5232 5 13 for for ADP cana-5232 5 14 women woman NOUN cana-5232 5 15 , , PUNCT cana-5232 5 16 udumalpet udumalpet ADJ cana-5232 5 17 e.mail e.mail ADJ cana-5232 5 18 : : PUNCT cana-5232 5 19 1gurusamymathsgasckgm@gmail.com 1gurusamymathsgasckgm@gmail.com NUM cana-5232 5 20 , , PUNCT cana-5232 5 21 e. e. PROPN cana-5232 5 22 mail mail PROPN cana-5232 5 23 : : PUNCT cana-5232 5 24 2angeljoy@gvgvc.ac.in 2angeljoy@gvgvc.ac.in NUM cana-5232 5 25 article article NOUN cana-5232 5 26 history history NOUN cana-5232 5 27 : : PUNCT cana-5232 5 28 received receive VERB cana-5232 5 29 : : PUNCT cana-5232 5 30 12 12 NUM cana-5232 5 31 - - SYM cana-5232 5 32 01 01 NUM cana-5232 5 33 - - PUNCT cana-5232 5 34 2025 2025 NUM cana-5232 5 35 revised revise VERB cana-5232 5 36 : : PUNCT cana-5232 5 37 15 15 NUM cana-5232 5 38 - - NUM cana-5232 5 39 02 02 NUM cana-5232 5 40 - - PUNCT cana-5232 5 41 2025 2025 NUM cana-5232 5 42 accepted accept VERB cana-5232 5 43 : : PUNCT cana-5232 5 44 01 01 NUM cana-5232 5 45 - - SYM cana-5232 5 46 03 03 NUM cana-5232 5 47 - - PUNCT cana-5232 5 48 2025 2025 NUM cana-5232 5 49 abstract abstract NOUN cana-5232 5 50 : : PUNCT cana-5232 5 51 let let VERB cana-5232 5 52 𝐺 𝐺 PROPN cana-5232 5 53 = = SYM cana-5232 5 54 αˆΊπ‘‰ αˆΊπ‘‰ NOUN cana-5232 5 55 , , PUNCT cana-5232 5 56 𝐸ሻ 𝐸ሻ AUX cana-5232 5 57 be be AUX cana-5232 5 58 a a DET cana-5232 5 59 simple simple ADJ cana-5232 5 60 connected connected ADJ cana-5232 5 61 graph graph NOUN cana-5232 5 62 . . PUNCT cana-5232 6 1 the the DET cana-5232 6 2 vertices vertex NOUN cana-5232 6 3 𝑒 𝑒 PART cana-5232 6 4 , , PUNCT cana-5232 6 5 𝑣 𝑣 PRON cana-5232 6 6 ∈ ∈ PROPN cana-5232 6 7 𝑉 𝑉 PROPN cana-5232 6 8 are be AUX cana-5232 6 9 closely closely ADV cana-5232 6 10 connected connect VERB cana-5232 6 11 if if SCONJ cana-5232 6 12 atleast atleast ADJ cana-5232 6 13 one one NUM cana-5232 6 14 of of ADP cana-5232 6 15 the the DET cana-5232 6 16 shortest short ADJ cana-5232 6 17 paths path NOUN cana-5232 6 18 connecting connect VERB cana-5232 6 19 them they PRON cana-5232 6 20 is be AUX cana-5232 6 21 not not PART cana-5232 6 22 a a DET cana-5232 6 23 cut cut ADJ cana-5232 6 24 path path NOUN cana-5232 6 25 . . PUNCT cana-5232 7 1 a a DET cana-5232 7 2 set set ADJ cana-5232 7 3 𝑆 𝑆 PROPN cana-5232 7 4 of of ADP cana-5232 7 5 vertices vertex NOUN cana-5232 7 6 of of ADP cana-5232 7 7 a a DET cana-5232 7 8 simple simple ADJ cana-5232 7 9 graph graph NOUN cana-5232 7 10 𝐺 𝐺 NOUN cana-5232 7 11 = = SYM cana-5232 7 12 αˆΊπ‘‰ αˆΊπ‘‰ NOUN cana-5232 7 13 , , PUNCT cana-5232 7 14 𝐸ሻ 𝐸ሻ PROPN cana-5232 7 15 is be AUX cana-5232 7 16 a a DET cana-5232 7 17 cc cc NOUN cana-5232 7 18 - - ADJ cana-5232 7 19 dominating dominating ADJ cana-5232 7 20 set set NOUN cana-5232 7 21 ( ( PUNCT cana-5232 7 22 closely closely ADV cana-5232 7 23 connected connect VERB cana-5232 7 24 dominating dominating NOUN cana-5232 7 25 set set NOUN cana-5232 7 26 ) ) PUNCT cana-5232 7 27 if if SCONJ cana-5232 7 28 for for ADP cana-5232 7 29 every every DET cana-5232 7 30 vertex vertex NOUN cana-5232 7 31 𝑣 𝑣 ADP cana-5232 7 32 ∈ ∈ NOUN cana-5232 7 33 𝑉\𝑆 𝑉\𝑆 NOUN cana-5232 7 34 there there ADV cana-5232 7 35 exist exist VERB cana-5232 7 36 a a DET cana-5232 7 37 vertex vertex NOUN cana-5232 7 38 𝑒 𝑒 ADP cana-5232 7 39 ∈ ∈ PROPN cana-5232 7 40 𝑆 𝑆 PROPN cana-5232 7 41 such such ADJ cana-5232 7 42 that that SCONJ cana-5232 7 43 π›€π‘π‘αˆΊπ‘’ π›€π‘π‘αˆΊπ‘’ PROPN cana-5232 7 44 , , PUNCT cana-5232 7 45 π‘£αˆ» π‘£αˆ» X cana-5232 7 46 β‰₯ β‰₯ NOUN cana-5232 7 47 1 1 NUM cana-5232 7 48 , , PUNCT cana-5232 7 49 where where SCONJ cana-5232 7 50 π›€π‘π‘αˆΊπ‘’ π›€π‘π‘αˆΊπ‘’ PROPN cana-5232 7 51 , , PUNCT cana-5232 7 52 π‘£αˆ» π‘£αˆ» ADJ cana-5232 7 53 is be AUX cana-5232 7 54 the the DET cana-5232 7 55 number number NOUN cana-5232 7 56 of of ADP cana-5232 7 57 shortest short ADJ cana-5232 7 58 paths path NOUN cana-5232 7 59 connecting connect VERB cana-5232 7 60 𝑒 𝑒 NOUN cana-5232 7 61 and and CCONJ cana-5232 7 62 𝑣 𝑣 PROPN cana-5232 7 63 except except SCONJ cana-5232 7 64 the the DET cana-5232 7 65 cut cut NOUN cana-5232 7 66 paths path NOUN cana-5232 7 67 . . PUNCT cana-5232 8 1 the the DET cana-5232 8 2 minimum minimum ADJ cana-5232 8 3 cordiality cordiality NOUN cana-5232 8 4 of of ADP cana-5232 8 5 a a DET cana-5232 8 6 cc cc NOUN cana-5232 8 7 - - ADJ cana-5232 8 8 dominating dominating ADJ cana-5232 8 9 set set NOUN cana-5232 8 10 is be AUX cana-5232 8 11 called call VERB cana-5232 8 12 the the DET cana-5232 8 13 cc cc NOUN cana-5232 8 14 - - PUNCT cana-5232 8 15 domination domination NOUN cana-5232 8 16 number number NOUN cana-5232 8 17 , , PUNCT cana-5232 8 18 denoted denote VERB cana-5232 8 19 by by ADP cana-5232 8 20 π›Ύπ‘π‘αˆΊπΊαˆ» π›Ύπ‘π‘αˆΊπΊαˆ» PROPN cana-5232 8 21 . . PUNCT cana-5232 9 1 this this DET cana-5232 9 2 paper paper NOUN cana-5232 9 3 evaluates evaluate VERB cana-5232 9 4 ccdomination ccdomination NOUN cana-5232 9 5 number number NOUN cana-5232 9 6 in in ADP cana-5232 9 7 corona corona NOUN cana-5232 9 8 product product NOUN cana-5232 9 9 of of ADP cana-5232 9 10 some some DET cana-5232 9 11 standard standard ADJ cana-5232 9 12 graphs graph NOUN cana-5232 9 13 . . PUNCT cana-5232 10 1 objectives objective NOUN cana-5232 10 2 : : PUNCT cana-5232 10 3 closely closely ADV cana-5232 10 4 connected connected ADJ cana-5232 10 5 domination domination NOUN cana-5232 10 6 is be AUX cana-5232 10 7 new new ADJ cana-5232 10 8 direction direction NOUN cana-5232 10 9 of of ADP cana-5232 10 10 domination domination NOUN cana-5232 10 11 in in ADP cana-5232 10 12 graphs graph NOUN cana-5232 10 13 , , PUNCT cana-5232 10 14 here here ADV cana-5232 10 15 find find VERB cana-5232 10 16 closely closely ADV cana-5232 10 17 connected connected ADJ cana-5232 10 18 domination domination NOUN cana-5232 10 19 number number NOUN cana-5232 10 20 for for ADP cana-5232 10 21 corona corona NOUN cana-5232 10 22 product product NOUN cana-5232 10 23 of of ADP cana-5232 10 24 path path NOUN cana-5232 10 25 , , PUNCT cana-5232 10 26 cycle cycle NOUN cana-5232 10 27 with with ADP cana-5232 10 28 some some DET cana-5232 10 29 graphs graph NOUN cana-5232 10 30 methods method NOUN cana-5232 10 31 : : PUNCT cana-5232 10 32 consider consider VERB cana-5232 10 33 the the DET cana-5232 10 34 graph graph NOUN cana-5232 10 35 g g NOUN cana-5232 10 36 is be AUX cana-5232 10 37 undirected undirected ADJ cana-5232 10 38 connected connected ADJ cana-5232 10 39 simple simple ADJ cana-5232 10 40 graph graph NOUN cana-5232 10 41 . . PUNCT cana-5232 11 1 the the DET cana-5232 11 2 closely closely ADV cana-5232 11 3 connected connected ADJ cana-5232 11 4 domination domination NOUN cana-5232 11 5 number number NOUN cana-5232 11 6 ( ( PUNCT cana-5232 11 7 ccdomination ccdomination NOUN cana-5232 11 8 number number NOUN cana-5232 11 9 ) ) PUNCT cana-5232 11 10 for for ADP cana-5232 11 11 product product NOUN cana-5232 11 12 of of ADP cana-5232 11 13 graphs graph NOUN cana-5232 11 14 , , PUNCT cana-5232 11 15 which which PRON cana-5232 11 16 is be AUX cana-5232 11 17 represented represent VERB cana-5232 11 18 as as ADP cana-5232 11 19 π›Ύπ‘π‘αˆΊπΊαˆ» π›Ύπ‘π‘αˆΊπΊαˆ» PROPN cana-5232 11 20 is be AUX cana-5232 11 21 the the DET cana-5232 11 22 minimum minimum ADJ cana-5232 11 23 cardinality cardinality NOUN cana-5232 11 24 of of ADP cana-5232 11 25 closely closely ADV cana-5232 11 26 connected connect VERB cana-5232 11 27 dominating dominating NOUN cana-5232 11 28 set set NOUN cana-5232 11 29 . . PUNCT cana-5232 12 1 keywords keyword NOUN cana-5232 12 2 : : PUNCT cana-5232 12 3 closely closely ADV cana-5232 12 4 - - PUNCT cana-5232 12 5 connected connect VERB cana-5232 12 6 vertices vertex NOUN cana-5232 12 7 , , PUNCT cana-5232 12 8 cc cc NOUN cana-5232 12 9 - - NOUN cana-5232 12 10 degree degree NOUN cana-5232 12 11 of of ADP cana-5232 12 12 a a DET cana-5232 12 13 vertex vertex NOUN cana-5232 12 14 , , PUNCT cana-5232 12 15 cc cc NOUN cana-5232 12 16 - - NOUN cana-5232 12 17 domination domination NOUN cana-5232 12 18 number number NOUN cana-5232 12 19 , , PUNCT cana-5232 12 20 corona corona NOUN cana-5232 12 21 product product NOUN cana-5232 12 22 of of ADP cana-5232 12 23 graphs graph NOUN cana-5232 12 24 2020 2020 NUM cana-5232 12 25 mathematics mathematic NOUN cana-5232 12 26 subject subject ADJ cana-5232 12 27 classification classification NOUN cana-5232 12 28 : : PUNCT cana-5232 12 29 05c40 05c40 NUM cana-5232 12 30 , , PUNCT cana-5232 12 31 05c07 05c07 NOUN cana-5232 12 32 , , PUNCT cana-5232 12 33 05c69 05c69 NUM cana-5232 12 34 , , PUNCT cana-5232 12 35 05c76 05c76 PRON cana-5232 12 36 . . PUNCT cana-5232 13 1 1 1 X cana-5232 13 2 . . X cana-5232 13 3 introduction introduction NOUN cana-5232 13 4 in in ADP cana-5232 13 5 our our PRON cana-5232 13 6 day day NOUN cana-5232 13 7 - - PUNCT cana-5232 13 8 to to ADP cana-5232 13 9 - - PUNCT cana-5232 13 10 day day NOUN cana-5232 13 11 life life NOUN cana-5232 13 12 , , PUNCT cana-5232 13 13 shortest short ADJ cana-5232 13 14 route route NOUN cana-5232 13 15 is be AUX cana-5232 13 16 a a DET cana-5232 13 17 path path NOUN cana-5232 13 18 between between ADP cana-5232 13 19 two two NUM cana-5232 13 20 destinations destination NOUN cana-5232 13 21 which which PRON cana-5232 13 22 traverses traverse VERB cana-5232 13 23 the the DET cana-5232 13 24 minimal minimal ADJ cana-5232 13 25 distance distance NOUN cana-5232 13 26 over over ADP cana-5232 13 27 the the DET cana-5232 13 28 network network NOUN cana-5232 13 29 . . PUNCT cana-5232 14 1 shortest short ADJ cana-5232 14 2 route route NOUN cana-5232 14 3 between between ADP cana-5232 14 4 different different ADJ cana-5232 14 5 location location NOUN cana-5232 14 6 can can AUX cana-5232 14 7 not not PART cana-5232 14 8 always always ADV cana-5232 14 9 be be AUX cana-5232 14 10 preferable preferable ADJ cana-5232 14 11 and and CCONJ cana-5232 14 12 we we PRON cana-5232 14 13 prefer prefer VERB cana-5232 14 14 routes route NOUN cana-5232 14 15 that that PRON cana-5232 14 16 optimize optimize VERB cana-5232 14 17 the the DET cana-5232 14 18 cost cost NOUN cana-5232 14 19 and and CCONJ cana-5232 14 20 time time NOUN cana-5232 14 21 . . PUNCT cana-5232 15 1 this this PRON cana-5232 15 2 is be AUX cana-5232 15 3 possible possible ADJ cana-5232 15 4 only only ADV cana-5232 15 5 if if SCONJ cana-5232 15 6 another another DET cana-5232 15 7 path path NOUN cana-5232 15 8 exists exist VERB cana-5232 15 9 between between ADP cana-5232 15 10 the the DET cana-5232 15 11 locations location NOUN cana-5232 15 12 , , PUNCT cana-5232 15 13 even even ADV cana-5232 15 14 if if SCONJ cana-5232 15 15 the the DET cana-5232 15 16 shortest short ADJ cana-5232 15 17 route route NOUN cana-5232 15 18 is be AUX cana-5232 15 19 not not PART cana-5232 15 20 approachable approachable ADJ cana-5232 15 21 . . PUNCT cana-5232 16 1 in in ADP cana-5232 16 2 graph graph NOUN cana-5232 16 3 theoretically theoretically ADV cana-5232 16 4 , , PUNCT cana-5232 16 5 places place NOUN cana-5232 16 6 are be AUX cana-5232 16 7 considered consider VERB cana-5232 16 8 as as ADP cana-5232 16 9 vertices vertex NOUN cana-5232 16 10 and and CCONJ cana-5232 16 11 paths path NOUN cana-5232 16 12 are be AUX cana-5232 16 13 edges edge NOUN cana-5232 16 14 in in ADP cana-5232 16 15 a a DET cana-5232 16 16 graph graph NOUN cana-5232 16 17 . . PUNCT cana-5232 17 1 then then ADV cana-5232 17 2 our our PRON cana-5232 17 3 aim aim NOUN cana-5232 17 4 is be AUX cana-5232 17 5 to to PART cana-5232 17 6 find find VERB cana-5232 17 7 for for ADP cana-5232 17 8 those those DET cana-5232 17 9 vertex vertex NOUN cana-5232 17 10 pairs pair NOUN cana-5232 17 11 which which PRON cana-5232 17 12 does do AUX cana-5232 17 13 not not PART cana-5232 17 14 alter alter VERB cana-5232 17 15 the the DET cana-5232 17 16 connectivity connectivity NOUN cana-5232 17 17 of of ADP cana-5232 17 18 the the DET cana-5232 17 19 graph graph NOUN cana-5232 17 20 . . PUNCT cana-5232 18 1 more more ADV cana-5232 18 2 precisely precisely ADV cana-5232 18 3 the the DET cana-5232 18 4 vertex vertex NOUN cana-5232 18 5 pairs pair NOUN cana-5232 18 6 do do AUX cana-5232 18 7 not not PART cana-5232 18 8 disconnect disconnect VERB cana-5232 18 9 the the DET cana-5232 18 10 graph graph NOUN cana-5232 18 11 even even ADV cana-5232 18 12 if if SCONJ cana-5232 18 13 the the DET cana-5232 18 14 shortest short ADJ cana-5232 18 15 path path NOUN cana-5232 18 16 between between ADP cana-5232 18 17 them they PRON cana-5232 18 18 is be AUX cana-5232 18 19 deleted delete VERB cana-5232 18 20 . . PUNCT cana-5232 19 1 for for ADP cana-5232 19 2 this this DET cana-5232 19 3 idea idea NOUN cana-5232 19 4 k. k. PROPN cana-5232 19 5 priya priya PROPN cana-5232 19 6 and and CCONJ cana-5232 19 7 v. v. PROPN cana-5232 19 8 anilkumar anilkumar PROPN cana-5232 19 9 introduced introduce VERB cana-5232 19 10 closely closely ADV cana-5232 19 11 connected connect VERB cana-5232 19 12 vertices vertex NOUN cana-5232 19 13 in in ADP cana-5232 19 14 [ [ X cana-5232 19 15 4 4 NUM cana-5232 19 16 ] ] PUNCT cana-5232 19 17 . . PUNCT cana-5232 20 1 let let VERB cana-5232 20 2 𝐺 𝐺 NOUN cana-5232 20 3 = = SYM cana-5232 20 4 αˆΊπ‘‰ αˆΊπ‘‰ NOUN cana-5232 20 5 , , PUNCT cana-5232 20 6 𝐸ሻ 𝐸ሻ PROPN cana-5232 20 7 , , PUNCT cana-5232 20 8 the the DET cana-5232 20 9 vertices vertex NOUN cana-5232 20 10 𝑒 𝑒 PART cana-5232 20 11 , , PUNCT cana-5232 20 12 𝑣 𝑣 PRON cana-5232 20 13 ∈ ∈ PROPN cana-5232 20 14 𝑉 𝑉 PROPN cana-5232 20 15 are be AUX cana-5232 20 16 closely closely ADV cana-5232 20 17 connected connect VERB cana-5232 20 18 if if SCONJ cana-5232 20 19 atleast atleast ADJ cana-5232 20 20 one one NUM cana-5232 20 21 of of ADP cana-5232 20 22 the the DET cana-5232 20 23 shortest short ADJ cana-5232 20 24 paths path NOUN cana-5232 20 25 connecting connect VERB cana-5232 20 26 them they PRON cana-5232 20 27 is be AUX cana-5232 20 28 not not PART cana-5232 20 29 a a DET cana-5232 20 30 cut cut ADJ cana-5232 20 31 path path NOUN cana-5232 20 32 . . PUNCT cana-5232 21 1 the the DET cana-5232 21 2 concept concept NOUN cana-5232 21 3 of of ADP cana-5232 21 4 domination domination NOUN cana-5232 21 5 discussed discuss VERB cana-5232 21 6 in in ADP cana-5232 21 7 [ [ X cana-5232 21 8 2,3 2,3 NUM cana-5232 21 9 ] ] PUNCT cana-5232 21 10 entirely entirely ADV cana-5232 21 11 depending depend VERB cana-5232 21 12 upon upon SCONJ cana-5232 21 13 the the DET cana-5232 21 14 adjacency adjacency NOUN cana-5232 21 15 property property NOUN cana-5232 21 16 of of ADP cana-5232 21 17 vertices vertex NOUN cana-5232 21 18 in in ADP cana-5232 21 19 a a DET cana-5232 21 20 graph graph NOUN cana-5232 21 21 . . PUNCT cana-5232 22 1 but but CCONJ cana-5232 22 2 adjacency adjacency NOUN cana-5232 22 3 is be AUX cana-5232 22 4 not not PART cana-5232 22 5 at at ADV cana-5232 22 6 all all ADV cana-5232 22 7 sufficient sufficient ADJ cana-5232 22 8 to to PART cana-5232 22 9 characterize characterize VERB cana-5232 22 10 the the DET cana-5232 22 11 vertex vertex NOUN cana-5232 22 12 pairs pair NOUN cana-5232 22 13 in in ADP cana-5232 22 14 a a DET cana-5232 22 15 graph graph NOUN cana-5232 22 16 as as ADP cana-5232 22 17 the the DET cana-5232 22 18 deletion deletion NOUN cana-5232 22 19 of of ADP cana-5232 22 20 the the DET cana-5232 22 21 edges edge NOUN cana-5232 22 22 linked link VERB cana-5232 22 23 by by ADP cana-5232 22 24 adjacent adjacent ADJ cana-5232 22 25 vertices vertex NOUN cana-5232 22 26 may may AUX cana-5232 22 27 or or CCONJ cana-5232 22 28 may may AUX cana-5232 22 29 not not PART cana-5232 22 30 disconnect disconnect VERB cana-5232 22 31 the the DET cana-5232 22 32 graph graph NOUN cana-5232 22 33 . . PUNCT cana-5232 23 1 a a DET cana-5232 23 2 set set NOUN cana-5232 23 3 𝑆 𝑆 PROPN cana-5232 23 4 βŠ† βŠ† NUM cana-5232 23 5 π‘‰αˆΊπΊαˆ» π‘‰αˆΊπΊαˆ» PROPN cana-5232 23 6 is be AUX cana-5232 23 7 called call VERB cana-5232 23 8 a a DET cana-5232 23 9 dominating dominating NOUN cana-5232 23 10 set set NOUN cana-5232 23 11 of of ADP cana-5232 23 12 𝐺 𝐺 PROPN cana-5232 23 13 if if SCONJ cana-5232 23 14 every every DET cana-5232 23 15 vertex vertex NOUN cana-5232 23 16 in in ADP cana-5232 23 17 π‘‰αˆΊπΊαˆ» π‘‰αˆΊπΊαˆ» PROPN cana-5232 23 18 βˆ’ βˆ’ PROPN cana-5232 23 19 𝑆 𝑆 PROPN cana-5232 23 20 is be AUX cana-5232 23 21 adjacent adjacent ADJ cana-5232 23 22 to to ADP cana-5232 23 23 some some DET cana-5232 23 24 vertex vertex NOUN cana-5232 23 25 in in ADP cana-5232 23 26 𝑆. 𝑆. PROPN cana-5232 23 27 the the DET cana-5232 23 28 domination domination NOUN cana-5232 23 29 number number NOUN cana-5232 23 30 π›ΎαˆΊπΊαˆ» π›ΎαˆΊπΊαˆ» NOUN cana-5232 23 31 of of ADP cana-5232 23 32 𝐺 𝐺 PROPN cana-5232 23 33 is be AUX cana-5232 23 34 the the DET cana-5232 23 35 minimum minimum ADJ cana-5232 23 36 cardinality cardinality NOUN cana-5232 23 37 of of ADP cana-5232 23 38 its its PRON cana-5232 23 39 dominating dominating NOUN cana-5232 23 40 sets set NOUN cana-5232 23 41 . . PUNCT cana-5232 24 1 the the DET cana-5232 24 2 concept concept NOUN cana-5232 24 3 of of ADP cana-5232 24 4 closely closely ADV cana-5232 24 5 connected connected ADJ cana-5232 24 6 domination domination NOUN cana-5232 24 7 introduced introduce VERB cana-5232 24 8 by by ADP cana-5232 24 9 k. k. PROPN cana-5232 24 10 priya priya PROPN cana-5232 24 11 and and CCONJ cana-5232 24 12 v. v. ADP cana-5232 24 13 anilkumar anilkumar PROPN cana-5232 24 14 in in ADP cana-5232 24 15 [ [ X cana-5232 24 16 4 4 NUM cana-5232 24 17 ] ] PUNCT cana-5232 24 18 . . PUNCT cana-5232 25 1 mailto:gurusamymathsgasckgm@gmail.com mailto:gurusamymathsgasckgm@gmail.com X cana-5232 25 2 mailto:angeljoy@gvgvc.ac.in mailto:angeljoy@gvgvc.ac.in NOUN cana-5232 25 3 https://d.docs.live.net/dfc4d13c8eba2327/desktop/cc-%20domination%20zip%20file.zip https://d.docs.live.net/dfc4d13c8eba2327/desktop/cc-%20domination%20zip%20file.zip VERB cana-5232 25 4 https://d.docs.live.net/dfc4d13c8eba2327/desktop/cc-%20domination%20zip%20file.zip https://d.docs.live.net/dfc4d13c8eba2327/desktop/cc-%20domination%20zip%20file.zip NOUN cana-5232 25 5 https://d.docs.live.net/dfc4d13c8eba2327/desktop/cc-%20domination%20zip%20file.zip https://d.docs.live.net/dfc4d13c8eba2327/desktop/cc-%20domination%20zip%20file.zip NOUN cana-5232 25 6 https://d.docs.live.net/dfc4d13c8eba2327/desktop/cc-%20domination%20zip%20file.zip https://d.docs.live.net/dfc4d13c8eba2327/desktop/cc-%20domination%20zip%20file.zip NOUN cana-5232 25 7 https://d.docs.live.net/dfc4d13c8eba2327/desktop/cc-%20domination%20zip%20file.zip https://d.docs.live.net/dfc4d13c8eba2327/desktop/cc-%20domination%20zip%20file.zip NOUN cana-5232 25 8 https://d.docs.live.net/dfc4d13c8eba2327/desktop/cc-%20domination%20zip%20file.zip https://d.docs.live.net/dfc4d13c8eba2327/desktop/cc-%20domination%20zip%20file.zip NOUN cana-5232 25 9 https://d.docs.live.net/dfc4d13c8eba2327/desktop/cc-%20domination%20zip%20file.zip https://d.docs.live.net/dfc4d13c8eba2327/desktop/cc-%20domination%20zip%20file.zip NOUN cana-5232 25 10 communications communication NOUN cana-5232 25 11 on on ADP cana-5232 25 12 applied apply VERB cana-5232 25 13 nonlinear nonlinear ADJ cana-5232 25 14 analysis analysis NOUN cana-5232 25 15 issn issn NOUN cana-5232 25 16 : : PUNCT cana-5232 25 17 1074 1074 NUM cana-5232 25 18 - - PUNCT cana-5232 25 19 133x 133x NUM cana-5232 25 20 vol vol VERB cana-5232 25 21 32 32 NUM cana-5232 25 22 no no NOUN cana-5232 25 23 . . PUNCT cana-5232 26 1 10s 10 NOUN cana-5232 26 2 ( ( PUNCT cana-5232 26 3 2025 2025 NUM cana-5232 26 4 ) ) PUNCT cana-5232 26 5 1297 1297 NUM cana-5232 26 6 https://internationalpubls.com https://internationalpubls.com X cana-5232 26 7 a a DET cana-5232 26 8 subset subset ADJ cana-5232 26 9 𝑆 𝑆 PROPN cana-5232 26 10 of of ADP cana-5232 26 11 𝑉 𝑉 PROPN cana-5232 26 12 is be AUX cana-5232 26 13 closely closely ADV cana-5232 26 14 connected connect VERB cana-5232 26 15 dominating dominating NOUN cana-5232 26 16 set(abbreviated set(abbreviate VERB cana-5232 26 17 as as ADP cana-5232 26 18 cc cc NOUN cana-5232 26 19 - - ADJ cana-5232 26 20 dominating dominating ADJ cana-5232 26 21 set set NOUN cana-5232 26 22 ) ) PUNCT cana-5232 26 23 if if SCONJ cana-5232 26 24 for for ADP cana-5232 26 25 every every DET cana-5232 26 26 vertex vertex NOUN cana-5232 26 27 𝑣 𝑣 ADP cana-5232 26 28 ∈ ∈ NOUN cana-5232 26 29 𝑉\𝑆 𝑉\𝑆 NOUN cana-5232 26 30 there there ADV cana-5232 26 31 exist exist VERB cana-5232 26 32 a a DET cana-5232 26 33 vertex vertex NOUN cana-5232 26 34 𝑒 𝑒 ADP cana-5232 26 35 ∈ ∈ PROPN cana-5232 26 36 𝑆 𝑆 PROPN cana-5232 26 37 such such ADJ cana-5232 26 38 that that DET cana-5232 26 39 π›€πΆπΆαˆΊπ‘’ π›€πΆπΆαˆΊπ‘’ PROPN cana-5232 26 40 , , PUNCT cana-5232 26 41 π‘£αˆ» π‘£αˆ» X cana-5232 26 42 β‰₯ β‰₯ NOUN cana-5232 26 43 1 1 NUM cana-5232 26 44 where where SCONJ cana-5232 26 45 π›€πΆπΆαˆΊπ‘’ π›€πΆπΆαˆΊπ‘’ NOUN cana-5232 26 46 , , PUNCT cana-5232 26 47 π‘£αˆ» π‘£αˆ» PROPN cana-5232 26 48 is be AUX cana-5232 26 49 cardinality cardinality NOUN cana-5232 26 50 of of ADP cana-5232 26 51 { { PUNCT cana-5232 26 52 𝑝|π‘ƒαˆΊπ‘’ 𝑝|π‘ƒαˆΊπ‘’ INTJ cana-5232 26 53 , , PUNCT cana-5232 26 54 π‘£αˆ»|𝑝 π‘£αˆ»|𝑝 NOUN cana-5232 26 55 is be AUX cana-5232 26 56 a a DET cana-5232 26 57 not not PART cana-5232 26 58 a a DET cana-5232 26 59 cut cut ADJ cana-5232 26 60 path path NOUN cana-5232 26 61 in in ADP cana-5232 26 62 𝐺 𝐺 PROPN cana-5232 26 63 } } PUNCT cana-5232 26 64 , , PUNCT cana-5232 26 65 where where SCONJ cana-5232 26 66 π‘ƒαˆΊπ‘’ π‘ƒαˆΊπ‘’ PROPN cana-5232 26 67 , , PUNCT cana-5232 26 68 π‘£αˆ» π‘£αˆ» ADJ cana-5232 26 69 is be AUX cana-5232 26 70 the the DET cana-5232 26 71 set set NOUN cana-5232 26 72 of of ADP cana-5232 26 73 all all DET cana-5232 26 74 shortest short ADJ cana-5232 26 75 paths path NOUN cana-5232 26 76 linking link VERB cana-5232 26 77 𝑒 𝑒 NOUN cana-5232 26 78 and and CCONJ cana-5232 26 79 𝑣 𝑣 X cana-5232 26 80 in in ADP cana-5232 26 81 𝐺. 𝐺. NOUN cana-5232 26 82 the the DET cana-5232 26 83 minimum minimum ADJ cana-5232 26 84 cardinality cardinality NOUN cana-5232 26 85 of of ADP cana-5232 26 86 cc cc NOUN cana-5232 26 87 - - ADJ cana-5232 26 88 dominating dominating ADJ cana-5232 26 89 set set NOUN cana-5232 26 90 is be AUX cana-5232 26 91 called call VERB cana-5232 26 92 cc cc NOUN cana-5232 26 93 - - PUNCT cana-5232 26 94 domination domination NOUN cana-5232 26 95 number number NOUN cana-5232 26 96 , , PUNCT cana-5232 26 97 denoted denote VERB cana-5232 26 98 by by ADP cana-5232 26 99 π›Ύπ‘π‘αˆΊπΊαˆ» π›Ύπ‘π‘αˆΊπΊαˆ» PROPN cana-5232 26 100 . . PUNCT cana-5232 27 1 the the DET cana-5232 27 2 open open ADJ cana-5232 27 3 cc cc NOUN cana-5232 27 4 - - NOUN cana-5232 27 5 neighbourhood neighbourhood NOUN cana-5232 27 6 of of ADP cana-5232 27 7 a a DET cana-5232 27 8 vertex vertex NOUN cana-5232 27 9 𝑣 𝑣 ADP cana-5232 27 10 ∈ ∈ PROPN cana-5232 27 11 𝑉 𝑉 PROPN cana-5232 27 12 is be AUX cana-5232 27 13 the the DET cana-5232 27 14 set set NOUN cana-5232 27 15 π‘π‘π‘αˆΊπ‘£αˆ» π‘π‘π‘αˆΊπ‘£αˆ» PROPN cana-5232 27 16 = = PUNCT cana-5232 27 17 { { PUNCT cana-5232 27 18 𝑒 𝑒 PRON cana-5232 27 19 ∈ ∈ PROPN cana-5232 27 20 𝑉 𝑉 PROPN cana-5232 27 21 : : PUNCT cana-5232 27 22 π›€π‘π‘αˆΊπ‘’ π›€π‘π‘αˆΊπ‘’ PROPN cana-5232 27 23 , , PUNCT cana-5232 27 24 π‘£αˆ» π‘£αˆ» X cana-5232 27 25 β‰₯ β‰₯ NOUN cana-5232 27 26 1 1 NUM cana-5232 27 27 } } PUNCT cana-5232 27 28 , , PUNCT cana-5232 27 29 whereas whereas SCONJ cana-5232 27 30 the the DET cana-5232 27 31 closed closed ADJ cana-5232 27 32 cc cc NOUN cana-5232 27 33 - - NOUN cana-5232 27 34 neighbourhood neighbourhood NOUN cana-5232 27 35 of of ADP cana-5232 27 36 𝑉 𝑉 PROPN cana-5232 27 37 is be AUX cana-5232 27 38 defined define VERB cana-5232 27 39 as as ADP cana-5232 27 40 𝑁𝑐𝑐[𝑣 𝑁𝑐𝑐[𝑣 PROPN cana-5232 27 41 ] ] X cana-5232 28 1 = = PUNCT cana-5232 28 2 π‘πΆπΆαˆΊπ‘£αˆ» π‘πΆπΆαˆΊπ‘£αˆ» PROPN cana-5232 28 3 βˆͺ βˆͺ ADV cana-5232 28 4 { { PUNCT cana-5232 28 5 𝑣 𝑣 NOUN cana-5232 28 6 } } PUNCT cana-5232 28 7 . . PUNCT cana-5232 29 1 the the DET cana-5232 29 2 cardinality cardinality NOUN cana-5232 29 3 of of ADP cana-5232 29 4 π‘πΆπΆαˆΊπ‘£αˆ» π‘πΆπΆαˆΊπ‘£αˆ» PROPN cana-5232 29 5 is be AUX cana-5232 29 6 the the DET cana-5232 29 7 cc cc NOUN cana-5232 29 8 - - NOUN cana-5232 29 9 degree degree NOUN cana-5232 29 10 of of ADP cana-5232 29 11 𝑣 𝑣 NOUN cana-5232 29 12 , , PUNCT cana-5232 29 13 denoted denote VERB cana-5232 29 14 by by ADP cana-5232 29 15 degπΆπΆαˆΊπ‘£αˆ» degπΆπΆαˆΊπ‘£αˆ» PROPN cana-5232 29 16 . . PUNCT cana-5232 30 1 the the DET cana-5232 30 2 vertex vertex NOUN cana-5232 30 3 𝑣 𝑣 PROPN cana-5232 30 4 is be AUX cana-5232 30 5 said say VERB cana-5232 30 6 to to PART cana-5232 30 7 be be AUX cana-5232 30 8 cc cc VERB cana-5232 30 9 - - ADJ cana-5232 30 10 isolated isolated ADJ cana-5232 30 11 if if SCONJ cana-5232 30 12 π‘π‘π‘αˆΊπ‘£αˆ» π‘π‘π‘αˆΊπ‘£αˆ» PROPN cana-5232 30 13 = = PUNCT cana-5232 30 14 πœ™. πœ™. NOUN cana-5232 30 15 2 2 NUM cana-5232 30 16 . . PUNCT cana-5232 30 17 definitions definition NOUN cana-5232 30 18 and and CCONJ cana-5232 30 19 previous previous ADJ cana-5232 30 20 results result NOUN cana-5232 30 21 definition definition NOUN cana-5232 30 22 2.1 2.1 NUM cana-5232 30 23 [ [ X cana-5232 30 24 4 4 NUM cana-5232 30 25 ] ] PUNCT cana-5232 30 26 a a DET cana-5232 30 27 subset subset NOUN cana-5232 30 28 𝑆 𝑆 PROPN cana-5232 30 29 of of ADP cana-5232 30 30 𝑉 𝑉 PROPN cana-5232 30 31 is be AUX cana-5232 30 32 closely closely ADV cana-5232 30 33 connected connect VERB cana-5232 30 34 dominating dominating NOUN cana-5232 30 35 set(abbreviated set(abbreviate VERB cana-5232 30 36 as as ADP cana-5232 30 37 cc cc NOUN cana-5232 30 38 - - ADJ cana-5232 30 39 dominating dominating ADJ cana-5232 30 40 set set NOUN cana-5232 30 41 ) ) PUNCT cana-5232 30 42 if if SCONJ cana-5232 30 43 for for ADP cana-5232 30 44 every every DET cana-5232 30 45 vertex vertex NOUN cana-5232 30 46 𝑣 𝑣 ADP cana-5232 30 47 ∈ ∈ NOUN cana-5232 30 48 𝑉\𝑆 𝑉\𝑆 NOUN cana-5232 30 49 there there ADV cana-5232 30 50 exist exist VERB cana-5232 30 51 a a DET cana-5232 30 52 vertex vertex NOUN cana-5232 30 53 𝑒 𝑒 ADP cana-5232 30 54 ∈ ∈ PROPN cana-5232 30 55 𝑆 𝑆 PROPN cana-5232 30 56 such such ADJ cana-5232 30 57 that that DET cana-5232 30 58 π›€πΆπΆαˆΊπ‘’ π›€πΆπΆαˆΊπ‘’ PROPN cana-5232 30 59 , , PUNCT cana-5232 30 60 π‘£αˆ» π‘£αˆ» X cana-5232 30 61 β‰₯ β‰₯ NOUN cana-5232 30 62 1 1 NUM cana-5232 30 63 where where SCONJ cana-5232 30 64 π›€πΆπΆαˆΊπ‘’ π›€πΆπΆαˆΊπ‘’ NOUN cana-5232 30 65 , , PUNCT cana-5232 30 66 π‘£αˆ» π‘£αˆ» PROPN cana-5232 30 67 is be AUX cana-5232 30 68 cardinality cardinality NOUN cana-5232 30 69 of of ADP cana-5232 30 70 { { PUNCT cana-5232 30 71 𝑝|π‘ƒαˆΊπ‘’ 𝑝|π‘ƒαˆΊπ‘’ INTJ cana-5232 30 72 , , PUNCT cana-5232 30 73 π‘£αˆ»|𝑝 π‘£αˆ»|𝑝 NOUN cana-5232 30 74 is be AUX cana-5232 30 75 a a DET cana-5232 30 76 not not PART cana-5232 30 77 a a DET cana-5232 30 78 cut cut ADJ cana-5232 30 79 path path NOUN cana-5232 30 80 in in ADP cana-5232 30 81 𝐺 𝐺 PROPN cana-5232 30 82 } } PUNCT cana-5232 30 83 , , PUNCT cana-5232 30 84 where where SCONJ cana-5232 30 85 π‘ƒαˆΊπ‘’ π‘ƒαˆΊπ‘’ PROPN cana-5232 30 86 , , PUNCT cana-5232 30 87 π‘£αˆ» π‘£αˆ» ADJ cana-5232 30 88 is be AUX cana-5232 30 89 the the DET cana-5232 30 90 set set NOUN cana-5232 30 91 of of ADP cana-5232 30 92 all all DET cana-5232 30 93 shortest short ADJ cana-5232 30 94 paths path NOUN cana-5232 30 95 linking link VERB cana-5232 30 96 𝑒 𝑒 NOUN cana-5232 30 97 and and CCONJ cana-5232 30 98 𝑣 𝑣 X cana-5232 30 99 in in ADP cana-5232 30 100 𝐺. 𝐺. NOUN cana-5232 30 101 definition definition NOUN cana-5232 30 102 2.2[4 2.2[4 NOUN cana-5232 30 103 ] ] X cana-5232 30 104 the the DET cana-5232 30 105 minimum minimum ADJ cana-5232 30 106 cardinality cardinality NOUN cana-5232 30 107 of of ADP cana-5232 30 108 cc cc NOUN cana-5232 30 109 - - ADJ cana-5232 30 110 dominating dominating ADJ cana-5232 30 111 set set NOUN cana-5232 30 112 is be AUX cana-5232 30 113 called call VERB cana-5232 30 114 cc cc NOUN cana-5232 30 115 - - PUNCT cana-5232 30 116 domination domination NOUN cana-5232 30 117 number number NOUN cana-5232 30 118 , , PUNCT cana-5232 30 119 denoted denote VERB cana-5232 30 120 by by ADP cana-5232 30 121 π›Ύπ‘π‘αˆΊπΊαˆ» π›Ύπ‘π‘αˆΊπΊαˆ» PROPN cana-5232 30 122 . . PUNCT cana-5232 31 1 in in ADP cana-5232 31 2 this this DET cana-5232 31 3 paper paper NOUN cana-5232 31 4 , , PUNCT cana-5232 31 5 we we PRON cana-5232 31 6 investigate investigate VERB cana-5232 31 7 the the DET cana-5232 31 8 cc cc NOUN cana-5232 31 9 - - NOUN cana-5232 31 10 domination domination NOUN cana-5232 31 11 number number NOUN cana-5232 31 12 of of ADP cana-5232 31 13 corona corona NOUN cana-5232 31 14 product product NOUN cana-5232 31 15 of of ADP cana-5232 31 16 paths path NOUN cana-5232 31 17 , , PUNCT cana-5232 31 18 cycles cycle NOUN cana-5232 31 19 and and CCONJ cana-5232 31 20 some some DET cana-5232 31 21 standard standard ADJ cana-5232 31 22 graphs graph NOUN cana-5232 31 23 . . PUNCT cana-5232 32 1 for for ADP cana-5232 32 2 a a DET cana-5232 32 3 graph graph NOUN cana-5232 32 4 𝐺 𝐺 NOUN cana-5232 32 5 of of ADP cana-5232 32 6 order order NOUN cana-5232 32 7 𝑛 𝑛 ADP cana-5232 32 8 , , PUNCT cana-5232 32 9 the the DET cana-5232 32 10 following follow VERB cana-5232 32 11 are be AUX cana-5232 32 12 some some DET cana-5232 32 13 basic basic ADJ cana-5232 32 14 results result NOUN cana-5232 32 15 of of ADP cana-5232 32 16 π›Ύπ‘π‘αˆΊπΊαˆ» π›Ύπ‘π‘αˆΊπΊαˆ» PROPN cana-5232 32 17 in in ADP cana-5232 32 18 [ [ X cana-5232 32 19 4 4 NUM cana-5232 32 20 ] ] PUNCT cana-5232 32 21 . . PUNCT cana-5232 33 1 1 1 NUM cana-5232 33 2 . . SYM cana-5232 33 3 1 1 NUM cana-5232 33 4 ≀ ≀ NUM cana-5232 33 5 π›Ύπ‘π‘αˆΊπΊαˆ» π›Ύπ‘π‘αˆΊπΊαˆ» PROPN cana-5232 33 6 ≀ ≀ ADJ cana-5232 33 7 𝑛. 𝑛. NOUN cana-5232 33 8 2 2 NUM cana-5232 33 9 . . PUNCT cana-5232 34 1 π›Ύπ‘π‘αˆΊπΊαˆ» π›Ύπ‘π‘αˆΊπΊαˆ» PROPN cana-5232 34 2 = = PUNCT cana-5232 35 1 𝑛 𝑛 PROPN cana-5232 35 2 iff iff PROPN cana-5232 35 3 𝐺 𝐺 PROPN cana-5232 35 4 is be AUX cana-5232 35 5 acyclic acyclic ADJ cana-5232 35 6 . . PUNCT cana-5232 36 1 3 3 X cana-5232 36 2 . . X cana-5232 36 3 for for ADP cana-5232 36 4 path path NOUN cana-5232 36 5 𝑃𝑛 𝑃𝑛 PROPN cana-5232 36 6 , , PUNCT cana-5232 36 7 π›Ύπ‘π‘αˆΊπ‘ƒπ‘›αˆ» π›Ύπ‘π‘αˆΊπ‘ƒπ‘›αˆ» PROPN cana-5232 36 8 = = SYM cana-5232 36 9 𝑛. 𝑛. NOUN cana-5232 36 10 4 4 NUM cana-5232 36 11 . . X cana-5232 37 1 if if SCONJ cana-5232 37 2 𝐺 𝐺 PROPN cana-5232 37 3 has have VERB cana-5232 37 4 no no DET cana-5232 37 5 cut cut NOUN cana-5232 37 6 edge edge NOUN cana-5232 37 7 , , PUNCT cana-5232 37 8 then then ADV cana-5232 37 9 π›Ύπ‘π‘αˆΊπΊαˆ» π›Ύπ‘π‘αˆΊπΊαˆ» PROPN cana-5232 37 10 ≀ ≀ PROPN cana-5232 37 11 π›ΎαˆΊπΊαˆ» π›ΎαˆΊπΊαˆ» PROPN cana-5232 37 12 . . PUNCT cana-5232 38 1 theorem theorem VERB cana-5232 38 2 2.3.[4 2.3.[4 NUM cana-5232 38 3 ] ] PUNCT cana-5232 38 4 let let VERB cana-5232 38 5 𝐺 𝐺 PRON cana-5232 38 6 be be AUX cana-5232 38 7 a a DET cana-5232 38 8 graph graph NOUN cana-5232 38 9 and and CCONJ cana-5232 38 10 𝑒 𝑒 PROPN cana-5232 38 11 ∈ ∈ PROPN cana-5232 38 12 π‘‰αˆΊπΊαˆ» π‘‰αˆΊπΊαˆ» PROPN cana-5232 38 13 be be AUX cana-5232 38 14 cc cc VERB cana-5232 38 15 - - VERB cana-5232 38 16 isolated isolated ADJ cana-5232 38 17 . . PUNCT cana-5232 39 1 then then ADV cana-5232 39 2 𝑒 𝑒 NOUN cana-5232 39 3 belongs belong VERB cana-5232 39 4 to to ADP cana-5232 39 5 every every DET cana-5232 39 6 ccdominating ccdominating NOUN cana-5232 39 7 set set NOUN cana-5232 39 8 of of ADP cana-5232 39 9 𝐺. 𝐺. NOUN cana-5232 39 10 theorem theorem ADJ cana-5232 39 11 2.4.[4 2.4.[4 NUM cana-5232 39 12 ] ] PUNCT cana-5232 39 13 for for ADP cana-5232 39 14 the the DET cana-5232 39 15 complete complete ADJ cana-5232 39 16 graph graph NOUN cana-5232 39 17 𝐾𝑛. 𝐾𝑛. PROPN cana-5232 39 18 π›Ύπ‘π‘αˆΊπΎπ‘›αˆ» π›Ύπ‘π‘αˆΊπΎπ‘›αˆ» NOUN cana-5232 39 19 = = PUNCT cana-5232 39 20 { { PUNCT cana-5232 39 21 2 2 NUM cana-5232 39 22 if if SCONJ cana-5232 39 23 𝑛 𝑛 ADJ cana-5232 39 24 = = SYM cana-5232 39 25 2 2 NUM cana-5232 39 26 1 1 NUM cana-5232 39 27 if if SCONJ cana-5232 39 28 𝑛 𝑛 DET cana-5232 39 29 β‰  β‰  PROPN cana-5232 39 30 2 2 NUM cana-5232 39 31 theorem theorem VERB cana-5232 39 32 2.5 2.5 NUM cana-5232 39 33 . . PUNCT cana-5232 40 1 for for ADP cana-5232 40 2 the the DET cana-5232 40 3 cycle cycle NOUN cana-5232 40 4 𝐢𝑛 𝐢𝑛 PROPN cana-5232 40 5 , , PUNCT cana-5232 40 6 π›Ύπ‘π‘αˆΊπΆπ‘›αˆ» π›Ύπ‘π‘αˆΊπΆπ‘›αˆ» ADV cana-5232 40 7 = = PUNCT cana-5232 40 8 π›ΎαˆΊπΆπ‘›αˆ» π›ΎαˆΊπΆπ‘›αˆ» ADV cana-5232 40 9 = = PUNCT cana-5232 40 10 ⌈ ⌈ NOUN cana-5232 40 11 𝑛 𝑛 DET cana-5232 40 12 3 3 NUM cana-5232 40 13 βŒ‰ βŒ‰ NOUN cana-5232 40 14 , , PUNCT cana-5232 40 15 βˆ€π‘› βˆ€π‘› NOUN cana-5232 40 16 ∈ ∈ NOUN cana-5232 40 17 𝑁. 𝑁. PROPN cana-5232 40 18 proof proof NOUN cana-5232 40 19 . . PUNCT cana-5232 41 1 in in ADP cana-5232 41 2 𝐢𝑛 𝐢𝑛 PROPN cana-5232 41 3 every every DET cana-5232 41 4 vertex vertex NOUN cana-5232 41 5 is be AUX cana-5232 41 6 closely closely ADV cana-5232 41 7 connected connect VERB cana-5232 41 8 only only ADV cana-5232 41 9 with with ADP cana-5232 41 10 its its PRON cana-5232 41 11 adjacent adjacent ADJ cana-5232 41 12 vertices vertex NOUN cana-5232 41 13 , , PUNCT cana-5232 41 14 so so SCONJ cana-5232 41 15 we we PRON cana-5232 41 16 conclude conclude VERB cana-5232 41 17 the the DET cana-5232 41 18 above above ADJ cana-5232 41 19 result result NOUN cana-5232 41 20 . . PUNCT cana-5232 42 1 3 3 X cana-5232 42 2 . . X cana-5232 42 3 closely closely ADV cana-5232 42 4 connected connect VERB cana-5232 42 5 domination domination NOUN cana-5232 42 6 number number NOUN cana-5232 42 7 in in ADP cana-5232 42 8 corona corona NOUN cana-5232 42 9 product product NOUN cana-5232 42 10 of of ADP cana-5232 42 11 graphs graph NOUN cana-5232 42 12 the the DET cana-5232 42 13 corona corona NOUN cana-5232 42 14 of of ADP cana-5232 42 15 two two NUM cana-5232 42 16 graphs graph NOUN cana-5232 42 17 𝐺1 𝐺1 NOUN cana-5232 42 18 and and CCONJ cana-5232 42 19 𝐺2 𝐺2 NOUN cana-5232 42 20 has have AUX cana-5232 42 21 been be AUX cana-5232 42 22 defined define VERB cana-5232 42 23 by by ADP cana-5232 42 24 frucht frucht NOUN cana-5232 42 25 and and CCONJ cana-5232 42 26 harary harary NOUN cana-5232 42 27 in in ADP cana-5232 42 28 [ [ X cana-5232 42 29 1 1 NUM cana-5232 42 30 ] ] PUNCT cana-5232 42 31 to to PART cana-5232 42 32 be be AUX cana-5232 42 33 the the DET cana-5232 42 34 graph graph NOUN cana-5232 42 35 𝐺 𝐺 PROPN cana-5232 42 36 formed form VERB cana-5232 42 37 from from ADP cana-5232 42 38 one one NUM cana-5232 42 39 copy copy NOUN cana-5232 42 40 of of ADP cana-5232 42 41 𝐺1 𝐺1 PROPN cana-5232 42 42 and and CCONJ cana-5232 42 43 |π‘‰αˆΊπΊ1ሻ| |π‘‰αˆΊπΊ1ሻ| PROPN cana-5232 42 44 copies copy NOUN cana-5232 42 45 of of ADP cana-5232 42 46 𝐺2 𝐺2 ADJ cana-5232 42 47 , , PUNCT cana-5232 42 48 where where SCONJ cana-5232 42 49 the the DET cana-5232 42 50 𝑖th 𝑖th PROPN cana-5232 42 51 vertex vertex NOUN cana-5232 42 52 of of ADP cana-5232 42 53 𝐺1 𝐺1 PROPN cana-5232 42 54 is be AUX cana-5232 42 55 adjacent adjacent ADJ cana-5232 42 56 to to ADP cana-5232 42 57 every every DET cana-5232 42 58 vertex vertex NOUN cana-5232 42 59 in in ADP cana-5232 42 60 the the DET cana-5232 42 61 𝑖th 𝑖th PROPN cana-5232 42 62 copy copy NOUN cana-5232 42 63 of of ADP cana-5232 42 64 𝐺2 𝐺2 NOUN cana-5232 42 65 and and CCONJ cana-5232 42 66 is be AUX cana-5232 42 67 denoted denote VERB cana-5232 42 68 by by ADP cana-5232 42 69 𝐺 𝐺 PROPN cana-5232 42 70 = = PROPN cana-5232 42 71 𝐺1 𝐺1 NOUN cana-5232 42 72 ∘ ∘ NOUN cana-5232 42 73 𝐺2 𝐺2 PROPN cana-5232 42 74 . . PUNCT cana-5232 43 1 theorem theorem VERB cana-5232 43 2 3.1[2 3.1[2 NOUN cana-5232 43 3 ] ] PUNCT cana-5232 43 4 for for ADP cana-5232 43 5 the the DET cana-5232 43 6 path path NOUN cana-5232 43 7 𝐺 𝐺 PROPN cana-5232 43 8 = = SYM cana-5232 43 9 𝑃𝑛 𝑃𝑛 PROPN cana-5232 43 10 , , PUNCT cana-5232 43 11 π›ΎαˆΊπΊαˆ» π›ΎαˆΊπΊαˆ» PROPN cana-5232 43 12 = = SYM cana-5232 43 13 ⌈ ⌈ ADP cana-5232 43 14 𝑛 𝑛 ADP cana-5232 43 15 2 2 NUM cana-5232 43 16 βŒ‰. βŒ‰. ADJ cana-5232 43 17 communications communication NOUN cana-5232 43 18 on on ADP cana-5232 43 19 applied apply VERB cana-5232 43 20 nonlinear nonlinear ADJ cana-5232 43 21 analysis analysis NOUN cana-5232 43 22 issn issn NOUN cana-5232 43 23 : : PUNCT cana-5232 43 24 1074 1074 NUM cana-5232 43 25 - - PUNCT cana-5232 43 26 133x 133x NUM cana-5232 43 27 vol vol VERB cana-5232 43 28 32 32 NUM cana-5232 43 29 no no NOUN cana-5232 43 30 . . PUNCT cana-5232 44 1 10s 10 NOUN cana-5232 44 2 ( ( PUNCT cana-5232 44 3 2025 2025 NUM cana-5232 44 4 ) ) PUNCT cana-5232 44 5 1298 1298 NUM cana-5232 44 6 https://internationalpubls.com https://internationalpubls.com X cana-5232 44 7 theorem theorem VERB cana-5232 44 8 3.2.[4 3.2.[4 NUM cana-5232 44 9 ] ] PUNCT cana-5232 44 10 for for ADP cana-5232 44 11 the the DET cana-5232 44 12 path path NOUN cana-5232 44 13 𝐺 𝐺 PROPN cana-5232 44 14 = = SYM cana-5232 44 15 𝑃𝑛 𝑃𝑛 PROPN cana-5232 44 16 , , PUNCT cana-5232 44 17 π›Ύπ‘π‘αˆΊπΊαˆ» π›Ύπ‘π‘αˆΊπΊαˆ» NOUN cana-5232 44 18 = = SYM cana-5232 44 19 𝑛. 𝑛. NOUN cana-5232 44 20 theorem theorem VERB cana-5232 44 21 3.3 3.3 NUM cana-5232 44 22 . . PUNCT cana-5232 45 1 for for ADP cana-5232 45 2 𝑛 𝑛 PRON cana-5232 45 3 β‰₯ β‰₯ NUM cana-5232 45 4 1 1 NUM cana-5232 45 5 , , PUNCT cana-5232 45 6 π›Ύπ‘π‘αˆΊπ‘ƒπ‘› π›Ύπ‘π‘αˆΊπ‘ƒπ‘› PROPN cana-5232 45 7 ∘ ∘ NOUN cana-5232 45 8 𝐾1ሻ 𝐾1ሻ NOUN cana-5232 45 9 = = SYM cana-5232 45 10 2𝑛. 2𝑛. NUM cana-5232 45 11 proof proof NOUN cana-5232 45 12 . . PUNCT cana-5232 46 1 let let VERB cana-5232 46 2 𝐺 𝐺 NOUN cana-5232 46 3 = = PUNCT cana-5232 46 4 𝑃𝑛 𝑃𝑛 PROPN cana-5232 46 5 ∘ ∘ ADJ cana-5232 46 6 𝐾1 𝐾1 PROPN cana-5232 46 7 , , PUNCT cana-5232 46 8 let let VERB cana-5232 46 9 π‘‰αˆΊπ‘ƒπ‘›αˆ» π‘‰αˆΊπ‘ƒπ‘›αˆ» PROPN cana-5232 46 10 = = SYM cana-5232 46 11 { { PUNCT cana-5232 46 12 𝑠1 𝑠1 PROPN cana-5232 46 13 , , PUNCT cana-5232 46 14 𝑠2 𝑠2 NOUN cana-5232 46 15 , , PUNCT cana-5232 46 16 … … PUNCT cana-5232 46 17 , , PUNCT cana-5232 46 18 𝑠𝑛 𝑠𝑛 NOUN cana-5232 46 19 } } PUNCT cana-5232 46 20 and and CCONJ cana-5232 46 21 𝑑𝑖 𝑑𝑖 PRON cana-5232 46 22 be be AUX cana-5232 46 23 the the DET cana-5232 46 24 vertices vertex NOUN cana-5232 46 25 of of ADP cana-5232 46 26 the the DET cana-5232 46 27 𝑖th 𝑖th PROPN cana-5232 46 28 copy copy NOUN cana-5232 46 29 of of ADP cana-5232 46 30 𝐾1 𝐾1 PROPN cana-5232 46 31 joined join VERB cana-5232 46 32 to to ADP cana-5232 46 33 𝑠𝑖. 𝑠𝑖. NOUN cana-5232 46 34 then then ADV cana-5232 46 35 π‘‰αˆΊπΊαˆ» π‘‰αˆΊπΊαˆ» PROPN cana-5232 46 36 = = SYM cana-5232 46 37 { { PUNCT cana-5232 46 38 𝑠1 𝑠1 PROPN cana-5232 46 39 , , PUNCT cana-5232 46 40 𝑠2 𝑠2 NOUN cana-5232 46 41 , , PUNCT cana-5232 46 42 … … PUNCT cana-5232 46 43 , , PUNCT cana-5232 46 44 𝑠𝑛 𝑠𝑛 NOUN cana-5232 46 45 } } PUNCT cana-5232 46 46 βˆͺ βˆͺ ADJ cana-5232 46 47 { { PUNCT cana-5232 46 48 𝑑1 𝑑1 NOUN cana-5232 46 49 , , PUNCT cana-5232 46 50 𝑑2 𝑑2 PROPN cana-5232 46 51 , , PUNCT cana-5232 46 52 β‹― β‹― PROPN cana-5232 46 53 , , PUNCT cana-5232 46 54 𝑑𝑛 𝑑𝑛 ADJ cana-5232 46 55 } } PUNCT cana-5232 46 56 . . PUNCT cana-5232 47 1 every every DET cana-5232 47 2 vertex vertex NOUN cana-5232 47 3 in in ADP cana-5232 47 4 π‘‰αˆΊπΊαˆ» π‘‰αˆΊπΊαˆ» PROPN cana-5232 47 5 is be AUX cana-5232 47 6 cc cc NOUN cana-5232 47 7 - - ADJ cana-5232 47 8 isolated isolate VERB cana-5232 47 9 vertex vertex NOUN cana-5232 47 10 , , PUNCT cana-5232 47 11 by by ADP cana-5232 47 12 the the DET cana-5232 47 13 theorem theorem NOUN cana-5232 47 14 1 1 NUM cana-5232 47 15 we we PRON cana-5232 47 16 have have VERB cana-5232 47 17 all all DET cana-5232 47 18 the the DET cana-5232 47 19 cc cc NOUN cana-5232 47 20 - - ADJ cana-5232 47 21 isolated isolate VERB cana-5232 47 22 vertices vertex NOUN cana-5232 47 23 must must AUX cana-5232 47 24 belong belong VERB cana-5232 47 25 to to ADP cana-5232 47 26 every every DET cana-5232 47 27 cc cc NOUN cana-5232 47 28 - - ADJ cana-5232 47 29 dominating dominating ADJ cana-5232 47 30 set set NOUN cana-5232 47 31 . . PUNCT cana-5232 48 1 the the DET cana-5232 48 2 only only ADJ cana-5232 48 3 one one NUM cana-5232 48 4 cc cc NOUN cana-5232 48 5 - - ADJ cana-5232 48 6 dominating dominating ADJ cana-5232 48 7 set set NOUN cana-5232 48 8 is be AUX cana-5232 48 9 π‘‰αˆΊπΊαˆ» π‘‰αˆΊπΊαˆ» PROPN cana-5232 48 10 . . PUNCT cana-5232 49 1 therefore therefore ADV cana-5232 49 2 π›Ύπ‘π‘αˆΊπΊαˆ» π›Ύπ‘π‘αˆΊπΊαˆ» PROPN cana-5232 49 3 = = SYM cana-5232 49 4 |π‘‰αˆΊπΊαˆ»| |π‘‰αˆΊπΊαˆ»| PROPN cana-5232 49 5 = = SYM cana-5232 49 6 2𝑛. 2𝑛. NOUN cana-5232 49 7 theorem theorem VERB cana-5232 49 8 3.4 3.4 NUM cana-5232 49 9 . . PUNCT cana-5232 50 1 for for ADP cana-5232 50 2 𝑛 𝑛 PRON cana-5232 50 3 β‰₯ β‰₯ NUM cana-5232 50 4 1 1 NUM cana-5232 50 5 , , PUNCT cana-5232 50 6 π›Ύπ‘π‘αˆΊπ‘ƒπ‘› π›Ύπ‘π‘αˆΊπ‘ƒπ‘› VERB cana-5232 50 7 ∘ ∘ NOUN cana-5232 50 8 𝐾2ሻ 𝐾2ሻ PROPN cana-5232 50 9 = = PUNCT cana-5232 50 10 𝑛. 𝑛. NOUN cana-5232 50 11 proof proof NOUN cana-5232 50 12 . . PUNCT cana-5232 51 1 let let VERB cana-5232 51 2 𝐺 𝐺 NOUN cana-5232 51 3 = = PUNCT cana-5232 51 4 𝑃𝑛 𝑃𝑛 PROPN cana-5232 51 5 ∘ ∘ ADV cana-5232 51 6 𝐾2 𝐾2 NOUN cana-5232 51 7 . . PUNCT cana-5232 52 1 let let VERB cana-5232 52 2 π‘‰αˆΊπ‘ƒπ‘›αˆ» π‘‰αˆΊπ‘ƒπ‘›αˆ» PROPN cana-5232 52 3 = = SYM cana-5232 52 4 { { PUNCT cana-5232 52 5 𝑠1 𝑠1 PROPN cana-5232 52 6 , , PUNCT cana-5232 52 7 𝑠2 𝑠2 NOUN cana-5232 52 8 , , PUNCT cana-5232 52 9 … … PUNCT cana-5232 52 10 , , PUNCT cana-5232 52 11 𝑠𝑛 𝑠𝑛 NOUN cana-5232 52 12 } } PUNCT cana-5232 52 13 and and CCONJ cana-5232 52 14 { { PUNCT cana-5232 52 15 𝑑𝑖1 𝑑𝑖1 PROPN cana-5232 52 16 , , PUNCT cana-5232 52 17 𝑑𝑖2 𝑑𝑖2 PROPN cana-5232 52 18 } } PUNCT cana-5232 52 19 be be AUX cana-5232 52 20 the the DET cana-5232 52 21 vertex vertex NOUN cana-5232 52 22 set set NOUN cana-5232 52 23 of of ADP cana-5232 52 24 the the DET cana-5232 52 25 𝑖th 𝑖th PROPN cana-5232 52 26 copy copy NOUN cana-5232 52 27 of of ADP cana-5232 52 28 𝐾2 𝐾2 NOUN cana-5232 52 29 attached attach VERB cana-5232 52 30 with with ADP cana-5232 52 31 𝑠𝑖. 𝑠𝑖. NOUN cana-5232 52 32 then then ADV cana-5232 52 33 π‘‰αˆΊπΊαˆ» π‘‰αˆΊπΊαˆ» PROPN cana-5232 52 34 = = SYM cana-5232 52 35 { { PUNCT cana-5232 52 36 𝑠1 𝑠1 PROPN cana-5232 52 37 , , PUNCT cana-5232 52 38 𝑠2 𝑠2 NOUN cana-5232 52 39 , , PUNCT cana-5232 52 40 𝑠3 𝑠3 NOUN cana-5232 52 41 , , PUNCT cana-5232 52 42 … … PUNCT cana-5232 52 43 , , PUNCT cana-5232 52 44 𝑠𝑛 𝑠𝑛 NOUN cana-5232 52 45 } } PUNCT cana-5232 52 46 βˆͺ βˆͺ NOUN cana-5232 52 47 { { PUNCT cana-5232 52 48 𝑑11 𝑑11 PROPN cana-5232 52 49 , , PUNCT cana-5232 52 50 𝑑12 𝑑12 NOUN cana-5232 52 51 } } PUNCT cana-5232 52 52 βˆͺ βˆͺ NOUN cana-5232 52 53 { { PUNCT cana-5232 52 54 𝑑21 𝑑21 NOUN cana-5232 52 55 , , PUNCT cana-5232 52 56 𝑑22 𝑑22 ADV cana-5232 52 57 } } PUNCT cana-5232 52 58 βˆͺ βˆͺ ADV cana-5232 52 59 β‹― β‹― PRON cana-5232 52 60 { { PUNCT cana-5232 52 61 𝑑𝑛1 𝑑𝑛1 NOUN cana-5232 52 62 , , PUNCT cana-5232 52 63 𝑑𝑛2 𝑑𝑛2 NOUN cana-5232 52 64 } } PUNCT cana-5232 52 65     SPACE cana-5232 52 66 π‘Žπ‘›π‘‘|π‘‰αˆΊπΊαˆ»| π‘Žπ‘›π‘‘|π‘‰αˆΊπΊαˆ»| PROPN cana-5232 52 67 = = SYM cana-5232 52 68 3𝑛. 3𝑛. NUM cana-5232 52 69 in in ADP cana-5232 52 70 figure figure NOUN cana-5232 52 71 1,every 1,every NUM cana-5232 52 72 vertex vertex NOUN cana-5232 52 73 𝑠𝑖 𝑠𝑖 NOUN cana-5232 52 74 of of ADP cana-5232 52 75 𝑃𝑛 𝑃𝑛 PROPN cana-5232 52 76 is be AUX cana-5232 52 77 closely closely ADV cana-5232 52 78 connected connect VERB cana-5232 52 79 with with ADP cana-5232 52 80 𝑑𝑖1 𝑑𝑖1 NOUN cana-5232 52 81 and and CCONJ cana-5232 52 82 𝑑𝑖2 𝑑𝑖2 NOUN cana-5232 52 83 but but CCONJ cana-5232 52 84 not not PART cana-5232 52 85 closely closely ADV cana-5232 52 86 connected connect VERB cana-5232 52 87 with with ADP cana-5232 52 88 π‘ π‘–βˆ’1 π‘ π‘–βˆ’1 PROPN cana-5232 52 89 and and CCONJ cana-5232 52 90 𝑠𝑖+1 𝑠𝑖+1 NUM cana-5232 52 91 . . NOUN cana-5232 53 1 i.e. i.e. X cana-5232 53 2 , , PUNCT cana-5232 53 3 π›€π‘π‘αˆΊπ‘ π‘– π›€π‘π‘αˆΊπ‘ π‘– PROPN cana-5232 53 4 , , PUNCT cana-5232 53 5 𝑑𝑖1ሻ 𝑑𝑖1ሻ PROPN cana-5232 53 6 β‰₯ β‰₯ NUM cana-5232 53 7 1 1 NUM cana-5232 53 8 , , PUNCT cana-5232 53 9 π›€π‘π‘αˆΊπ‘ π‘– π›€π‘π‘αˆΊπ‘ π‘– PROPN cana-5232 53 10 , , PUNCT cana-5232 53 11 𝑑𝑖2ሻ 𝑑𝑖2ሻ PROPN cana-5232 53 12 β‰₯ β‰₯ NOUN cana-5232 53 13 1 1 NUM cana-5232 53 14 and and CCONJ cana-5232 53 15 π›€π‘π‘αˆΊπ‘ π‘– π›€π‘π‘αˆΊπ‘ π‘– PROPN cana-5232 53 16 , , PUNCT cana-5232 53 17 π‘ π‘–βˆ’1ሻ π‘ π‘–βˆ’1ሻ PROPN cana-5232 53 18 = = SYM cana-5232 53 19 0 0 NUM cana-5232 53 20 and and CCONJ cana-5232 53 21 π›€π‘π‘αˆΊπ‘ π‘– π›€π‘π‘αˆΊπ‘ π‘– PROPN cana-5232 53 22 , , PUNCT cana-5232 53 23 𝑠𝑖+1ሻ 𝑠𝑖+1ሻ NOUN cana-5232 53 24 = = SYM cana-5232 53 25 0 0 X cana-5232 53 26 . . PUNCT cana-5232 54 1 so so ADV cana-5232 54 2 , , PUNCT cana-5232 54 3 one one NUM cana-5232 54 4 of of ADP cana-5232 54 5 the the DET cana-5232 54 6 minimums minimum NOUN cana-5232 54 7 𝐢𝐢-dominating 𝐢𝐢-dominate VERB cana-5232 54 8 set set NOUN cana-5232 54 9 𝐷 𝐷 NOUN cana-5232 54 10 = = SYM cana-5232 54 11 { { PUNCT cana-5232 54 12 𝑠1 𝑠1 PROPN cana-5232 54 13 , , PUNCT cana-5232 54 14 𝑠2 𝑠2 NOUN cana-5232 54 15 , , PUNCT cana-5232 54 16 … … PUNCT cana-5232 54 17 , , PUNCT cana-5232 54 18 𝑠𝑛 𝑠𝑛 NOUN cana-5232 54 19 } } PUNCT cana-5232 54 20 . . PUNCT cana-5232 55 1 since since SCONJ cana-5232 55 2 all all DET cana-5232 55 3 vertices vertex NOUN cana-5232 55 4 in in ADP cana-5232 55 5 𝐺 𝐺 PROPN cana-5232 55 6 is be AUX cana-5232 55 7 closely closely ADV cana-5232 55 8 connected connect VERB cana-5232 55 9 to to ADP cana-5232 55 10 at at ADV cana-5232 55 11 least least ADV cana-5232 55 12 one one NUM cana-5232 55 13 vertex vertex NOUN cana-5232 55 14 in in ADP cana-5232 55 15 𝐷. 𝐷. PROPN cana-5232 55 16 on on ADP cana-5232 55 17 removing remove VERB cana-5232 55 18 one one NUM cana-5232 55 19 vertex vertex NOUN cana-5232 55 20 in in ADP cana-5232 55 21 𝐷 𝐷 PROPN cana-5232 55 22 , , PUNCT cana-5232 55 23 it it PRON cana-5232 55 24 is be AUX cana-5232 55 25 not not PART cana-5232 55 26 a a DET cana-5232 55 27 cc cc NOUN cana-5232 55 28 - - ADJ cana-5232 55 29 dominating dominating ADJ cana-5232 55 30 set set NOUN cana-5232 55 31 . . PUNCT cana-5232 56 1 hence hence ADV cana-5232 56 2 π›Ύπ‘π‘αˆΊπΊαˆ» π›Ύπ‘π‘αˆΊπΊαˆ» PROPN cana-5232 56 3 = = SYM cana-5232 56 4 |𝐷| |𝐷| NOUN cana-5232 56 5 = = PUNCT cana-5232 56 6 𝑛. 𝑛. NOUN cana-5232 56 7 . . PUNCT cana-5232 57 1 figure figure VERB cana-5232 57 2 1 1 NUM cana-5232 57 3 the the DET cana-5232 57 4 corona corona NOUN cana-5232 57 5 product product NOUN cana-5232 58 1 𝑷𝒏 𝑷𝒏 PROPN cana-5232 58 2 ∘ ∘ X cana-5232 58 3 π‘²πŸ π‘²πŸ NOUN cana-5232 58 4 theorem theorem VERB cana-5232 58 5 3.6 3.6 NUM cana-5232 58 6 . . PUNCT cana-5232 59 1 the the DET cana-5232 59 2 cc cc NOUN cana-5232 59 3 - - PUNCT cana-5232 59 4 domination domination NOUN cana-5232 59 5 number number NOUN cana-5232 59 6 of of ADP cana-5232 59 7 𝑃𝑛 𝑃𝑛 PROPN cana-5232 59 8 ∘ ∘ ADV cana-5232 60 1 πΎπ‘š πΎπ‘š PROPN cana-5232 60 2 is be AUX cana-5232 60 3 π›Ύπ‘π‘αˆΊπ‘ƒπ‘› π›Ύπ‘π‘αˆΊπ‘ƒπ‘› NOUN cana-5232 60 4 ∘ ∘ NUM cana-5232 60 5 πΎπ‘šαˆ» πΎπ‘šαˆ» PROPN cana-5232 60 6 = = PUNCT cana-5232 60 7 𝑛 𝑛 PROPN cana-5232 60 8 for for ADP cana-5232 60 9     SPACE cana-5232 60 10 𝑛 𝑛 PRON cana-5232 60 11 β‰₯ β‰₯ NUM cana-5232 60 12 1 1 NUM cana-5232 60 13 and and CCONJ cana-5232 60 14 π‘š π‘š X cana-5232 60 15 > > X cana-5232 60 16 1 1 X cana-5232 60 17 . . PUNCT cana-5232 61 1 proof proof NOUN cana-5232 61 2 . . PUNCT cana-5232 62 1 let let VERB cana-5232 62 2 𝐺 𝐺 NOUN cana-5232 62 3 = = PUNCT cana-5232 62 4 𝑃𝑛 𝑃𝑛 PROPN cana-5232 62 5 ∘ ∘ NOUN cana-5232 62 6 πΎπ‘š πΎπ‘š PROPN cana-5232 62 7 , , PUNCT cana-5232 62 8 π‘‰αˆΊπ‘ƒπ‘›αˆ» π‘‰αˆΊπ‘ƒπ‘›αˆ» PROPN cana-5232 62 9 = = SYM cana-5232 62 10 { { PUNCT cana-5232 62 11 𝑠1 𝑠1 PROPN cana-5232 62 12 , , PUNCT cana-5232 62 13 𝑠2 𝑠2 NOUN cana-5232 62 14 , , PUNCT cana-5232 62 15 … … PUNCT cana-5232 62 16 , , PUNCT cana-5232 62 17 𝑠𝑛 𝑠𝑛 NOUN cana-5232 62 18 } } PUNCT cana-5232 62 19 . . PUNCT cana-5232 63 1 let let VERB cana-5232 63 2 { { PUNCT cana-5232 63 3 𝑑𝑖1 𝑑𝑖1 PROPN cana-5232 63 4 , , PUNCT cana-5232 63 5 𝑑𝑖2 𝑑𝑖2 PROPN cana-5232 63 6 , , PUNCT cana-5232 63 7 𝑑𝑖3 𝑑𝑖3 PROPN cana-5232 63 8 , , PUNCT cana-5232 63 9 … … PUNCT cana-5232 63 10 , , PUNCT cana-5232 63 11 π‘‘π‘–π‘š π‘‘π‘–π‘š PROPN cana-5232 63 12 } } PUNCT cana-5232 63 13 be be VERB cana-5232 63 14 the the DET cana-5232 63 15 vertex vertex NOUN cana-5232 63 16 set set NOUN cana-5232 63 17 of of ADP cana-5232 63 18 the the DET cana-5232 63 19 𝑖th 𝑖th PROPN cana-5232 63 20 copy copy NOUN cana-5232 63 21 of of ADP cana-5232 63 22 πΎπ‘š πΎπ‘š PROPN cana-5232 63 23 joined join VERB cana-5232 63 24 with with ADP cana-5232 63 25 the the DET cana-5232 63 26 vertex vertex NOUN cana-5232 63 27 𝑠𝑖. 𝑠𝑖. NOUN cana-5232 63 28 ∴ ∴ PROPN cana-5232 63 29     SPACE cana-5232 63 30 π‘‰αˆΊπ‘ƒπ‘› π‘‰αˆΊπ‘ƒπ‘› PROPN cana-5232 63 31 ∘ ∘ NOUN cana-5232 63 32 πΎπ‘šαˆ» πΎπ‘šαˆ» PROPN cana-5232 63 33 = = SYM cana-5232 63 34 { { PUNCT cana-5232 63 35 𝑠1 𝑠1 PROPN cana-5232 63 36 , , PUNCT cana-5232 63 37 𝑠2 𝑠2 NOUN cana-5232 63 38 , , PUNCT cana-5232 63 39 … … PUNCT cana-5232 63 40 𝑠𝑛 𝑠𝑛 NOUN cana-5232 63 41 , , PUNCT cana-5232 63 42 𝑑11 𝑑11 PROPN cana-5232 63 43 , , PUNCT cana-5232 63 44 𝑑12 𝑑12 NOUN cana-5232 63 45 , , PUNCT cana-5232 63 46 … … PUNCT cana-5232 63 47 , , PUNCT cana-5232 63 48 𝑑1π‘š 𝑑1π‘š NOUN cana-5232 63 49 , , PUNCT cana-5232 63 50 𝑑21 𝑑21 NOUN cana-5232 63 51 , , PUNCT cana-5232 63 52 𝑑22 𝑑22 ADV cana-5232 63 53 , , PUNCT cana-5232 63 54 … … PUNCT cana-5232 63 55 , , PUNCT cana-5232 63 56 𝑑2π‘š 𝑑2π‘š NOUN cana-5232 63 57 , , PUNCT cana-5232 63 58 β‹― β‹― PROPN cana-5232 63 59 , , PUNCT cana-5232 63 60 𝑑𝑛1 𝑑𝑛1 ADV cana-5232 63 61 , , PUNCT cana-5232 63 62 𝑑𝑛2 𝑑𝑛2 PROPN cana-5232 63 63 , , PUNCT cana-5232 63 64 𝑑𝑛3 𝑑𝑛3 NOUN cana-5232 63 65 , , PUNCT cana-5232 63 66 … … PUNCT cana-5232 63 67 π‘‘π‘›π‘š π‘‘π‘›π‘š ADJ cana-5232 63 68 } } PUNCT cana-5232 63 69 for for ADP cana-5232 63 70 𝑗 𝑗 NOUN cana-5232 63 71 = = SYM cana-5232 63 72 1 1 NUM cana-5232 63 73 to to ADP cana-5232 63 74 π‘š π‘š ADP cana-5232 63 75 , , PUNCT cana-5232 63 76 𝑆𝑗 𝑆𝑗 PROPN cana-5232 63 77 = = PUNCT cana-5232 63 78 { { PUNCT cana-5232 63 79 𝑑1𝑗 𝑑1𝑗 PROPN cana-5232 63 80 , , PUNCT cana-5232 63 81 𝑑2𝑗 𝑑2𝑗 X cana-5232 63 82 , , PUNCT cana-5232 63 83 … … PUNCT cana-5232 63 84 , , PUNCT cana-5232 63 85 𝑑𝑛𝑗 𝑑𝑛𝑗 NOUN cana-5232 63 86 } } PUNCT cana-5232 63 87 and and CCONJ cana-5232 63 88 𝑆 𝑆 PROPN cana-5232 63 89 = = SYM cana-5232 63 90 { { PUNCT cana-5232 63 91 𝑠1 𝑠1 PROPN cana-5232 63 92 , , PUNCT cana-5232 63 93 𝑠2 𝑠2 NOUN cana-5232 63 94 , , PUNCT cana-5232 63 95 … … PUNCT cana-5232 63 96 , , PUNCT cana-5232 63 97 𝑠𝑛 𝑠𝑛 NOUN cana-5232 63 98 } } PUNCT cana-5232 63 99 are be AUX cana-5232 63 100 some some DET cana-5232 63 101 cc cc NOUN cana-5232 63 102 - - ADJ cana-5232 63 103 dominating dominating ADJ cana-5232 63 104 sets set NOUN cana-5232 63 105 of of ADP cana-5232 63 106 𝐺 𝐺 NOUN cana-5232 63 107 and and CCONJ cana-5232 63 108 |𝑆| |𝑆| VERB cana-5232 63 109 = = PUNCT cana-5232 63 110 |𝑆𝑗| |𝑆𝑗| PROPN cana-5232 63 111 = = SYM cana-5232 63 112 𝑛 𝑛 NOUN cana-5232 63 113 , , PUNCT cana-5232 63 114 since since SCONJ cana-5232 63 115 all all DET cana-5232 63 116 vertices vertex NOUN cana-5232 63 117 in in ADP cana-5232 63 118 𝐺 𝐺 PROPN cana-5232 63 119 is be AUX cana-5232 63 120 closely closely ADV cana-5232 63 121 connected connect VERB cana-5232 63 122 to to ADP cana-5232 63 123 atleast atleast VERB cana-5232 63 124 one one NUM cana-5232 63 125 vertex vertex NOUN cana-5232 63 126 in in ADP cana-5232 63 127 𝑆 𝑆 PROPN cana-5232 63 128 ( ( PUNCT cana-5232 63 129 or or CCONJ cana-5232 64 1 ) ) PUNCT cana-5232 64 2 𝑆𝑗. 𝑆𝑗. PROPN cana-5232 64 3 on on ADP cana-5232 64 4 removing remove VERB cana-5232 64 5 one one NUM cana-5232 64 6 vertex vertex NOUN cana-5232 64 7 in in ADP cana-5232 64 8 𝑆 𝑆 PROPN cana-5232 64 9 ( ( PUNCT cana-5232 64 10 or or CCONJ cana-5232 64 11 ) ) PUNCT cana-5232 64 12 𝑆𝑗 𝑆𝑗 PROPN cana-5232 64 13 , , PUNCT cana-5232 64 14 it it PRON cana-5232 64 15 is be AUX cana-5232 64 16 not not PART cana-5232 64 17 cc cc VERB cana-5232 64 18 - - ADJ cana-5232 64 19 dominating dominating ADJ cana-5232 64 20 set set NOUN cana-5232 64 21 . . PUNCT cana-5232 65 1 hence hence ADV cana-5232 65 2 , , PUNCT cana-5232 65 3 we we PRON cana-5232 65 4 conclude conclude VERB cana-5232 65 5 π›Ύπ‘π‘αˆΊπΊαˆ» π›Ύπ‘π‘αˆΊπΊαˆ» PROPN cana-5232 65 6 = = SYM cana-5232 65 7 𝑛. 𝑛. NOUN cana-5232 65 8 theorem theorem VERB cana-5232 65 9 3.7 3.7 NUM cana-5232 65 10 the the DET cana-5232 65 11 cc cc NOUN cana-5232 65 12 - - NOUN cana-5232 65 13 domination domination NOUN cana-5232 65 14 number number NOUN cana-5232 65 15 of of ADP cana-5232 65 16 πΆπ‘š πΆπ‘š PROPN cana-5232 65 17 ∘ ∘ NOUN cana-5232 65 18 𝐾1 𝐾1 PROPN cana-5232 65 19 is be AUX cana-5232 65 20 π›Ύπ‘π‘αˆΊπΆπ‘š π›Ύπ‘π‘αˆΊπΆπ‘š VERB cana-5232 65 21 ∘ ∘ PROPN cana-5232 65 22 𝐾1ሻ 𝐾1ሻ PROPN cana-5232 65 23 = = PUNCT cana-5232 65 24 π‘š π‘š PROPN cana-5232 65 25 + + SYM cana-5232 65 26 ⌈ ⌈ NOUN cana-5232 65 27 π‘š π‘š ADP cana-5232 65 28 3 3 NUM cana-5232 65 29 βŒ‰ βŒ‰ NOUN cana-5232 65 30 proof proof NOUN cana-5232 65 31 . . PUNCT cana-5232 66 1 let let VERB cana-5232 67 1 𝐺 𝐺 NOUN cana-5232 67 2 = = PUNCT cana-5232 67 3 πΆπ‘š πΆπ‘š NOUN cana-5232 67 4 ∘ ∘ NOUN cana-5232 67 5 𝐾1 𝐾1 NOUN cana-5232 67 6 and and CCONJ cana-5232 67 7 let let VERB cana-5232 67 8 π‘‰αˆΊπΊαˆ» π‘‰αˆΊπΊαˆ» VERB cana-5232 67 9 = = PUNCT cana-5232 67 10 { { PUNCT cana-5232 67 11 𝑒1 𝑒1 PROPN cana-5232 67 12 , , PUNCT cana-5232 67 13 𝑒2 𝑒2 PROPN cana-5232 67 14 , , PUNCT cana-5232 67 15 … … PUNCT cana-5232 67 16 , , PUNCT cana-5232 67 17 π‘’π‘š π‘’π‘š ADP cana-5232 67 18 } } PUNCT cana-5232 67 19 βˆͺ βˆͺ ADJ cana-5232 67 20 { { PUNCT cana-5232 67 21 𝑣1 𝑣1 PROPN cana-5232 67 22 , , PUNCT cana-5232 67 23 𝑣2 𝑣2 PROPN cana-5232 67 24 , , PUNCT cana-5232 67 25 … … PUNCT cana-5232 67 26 , , PUNCT cana-5232 67 27 π‘£π‘š π‘£π‘š VERB cana-5232 67 28 } } PUNCT cana-5232 67 29 . . PUNCT cana-5232 68 1 let let AUX cana-5232 68 2 𝑣𝑖 𝑣𝑖 ADV cana-5232 68 3 be be AUX cana-5232 68 4 the the DET cana-5232 68 5 vertices vertex NOUN cana-5232 68 6 of of ADP cana-5232 68 7 the the DET cana-5232 68 8 𝑖th 𝑖th PROPN cana-5232 68 9 copy copy NOUN cana-5232 68 10 of of ADP cana-5232 68 11 𝐾1 𝐾1 NOUN cana-5232 68 12 is be AUX cana-5232 68 13 joined join VERB cana-5232 68 14 to to ADP cana-5232 68 15 𝑒𝑖 𝑒𝑖 ADP cana-5232 68 16 , , PUNCT cana-5232 68 17 then then ADV cana-5232 68 18 |π‘‰αˆΊπΊαˆ»| |π‘‰αˆΊπΊαˆ»| PROPN cana-5232 69 1 = = SYM cana-5232 69 2 2π‘š. 2π‘š. NUM cana-5232 69 3 let let VERB cana-5232 69 4 𝐷 𝐷 PROPN cana-5232 69 5 = = SYM cana-5232 69 6 { { PUNCT cana-5232 69 7 𝑣1 𝑣1 PROPN cana-5232 69 8 , , PUNCT cana-5232 69 9 𝑣2 𝑣2 PROPN cana-5232 69 10 , , PUNCT cana-5232 69 11 … … PUNCT cana-5232 69 12 , , PUNCT cana-5232 69 13 π‘£π‘š π‘£π‘š AUX cana-5232 69 14 } } PUNCT cana-5232 69 15 be be AUX cana-5232 69 16 the the DET cana-5232 69 17 set set NOUN cana-5232 69 18 of of ADP cana-5232 69 19 all all DET cana-5232 69 20 pendent pendent ADJ cana-5232 69 21 vertices vertex NOUN cana-5232 69 22 of of ADP cana-5232 69 23 𝐺 𝐺 PROPN cana-5232 69 24 and and CCONJ cana-5232 69 25 is be AUX cana-5232 69 26 also also ADV cana-5232 69 27 cc cc VERB cana-5232 69 28 - - ADJ cana-5232 69 29 isolated isolate VERB cana-5232 69 30 vertices vertex NOUN cana-5232 69 31 . . PUNCT cana-5232 70 1 so so ADV cana-5232 70 2 𝐷 𝐷 PROPN cana-5232 70 3 is be AUX cana-5232 70 4 subset subset VERB cana-5232 70 5 of of ADP cana-5232 70 6 every every DET cana-5232 70 7 cc cc NOUN cana-5232 70 8 - - ADJ cana-5232 70 9 dominating dominating ADJ cana-5232 70 10 set set NOUN cana-5232 70 11 of of ADP cana-5232 70 12 𝐺. 𝐺. NOUN cana-5232 70 13 further further ADJ cana-5232 70 14 communications communication NOUN cana-5232 70 15 on on ADP cana-5232 70 16 applied apply VERB cana-5232 70 17 nonlinear nonlinear ADJ cana-5232 70 18 analysis analysis NOUN cana-5232 70 19 issn issn NOUN cana-5232 70 20 : : PUNCT cana-5232 70 21 1074 1074 NUM cana-5232 70 22 - - PUNCT cana-5232 70 23 133x 133x NUM cana-5232 70 24 vol vol VERB cana-5232 70 25 32 32 NUM cana-5232 70 26 no no NOUN cana-5232 70 27 . . PUNCT cana-5232 71 1 10s 10 NOUN cana-5232 71 2 ( ( PUNCT cana-5232 71 3 2025 2025 NUM cana-5232 71 4 ) ) PUNCT cana-5232 71 5 1299 1299 NUM cana-5232 71 6 https://internationalpubls.com https://internationalpubls.com X cana-5232 72 1 each each DET cana-5232 72 2 vertex vertex NOUN cana-5232 72 3 𝑒𝑖 𝑒𝑖 NOUN cana-5232 72 4 in in ADP cana-5232 72 5 𝐺 𝐺 PROPN cana-5232 72 6 is be AUX cana-5232 72 7 closely closely ADV cana-5232 72 8 connected connect VERB cana-5232 72 9 with with ADP cana-5232 72 10 its its PRON cana-5232 72 11 adjacent adjacent ADJ cana-5232 72 12 vertices vertex NOUN cana-5232 72 13 . . PUNCT cana-5232 73 1 i.e. i.e. X cana-5232 73 2 , , PUNCT cana-5232 73 3 π‘π‘π‘αˆΊπ‘’π‘–αˆ» π‘π‘π‘αˆΊπ‘’π‘–αˆ» PROPN cana-5232 73 4 = = PUNCT cana-5232 73 5 { { PUNCT cana-5232 73 6 π‘’π‘–βˆ’1 π‘’π‘–βˆ’1 PROPN cana-5232 73 7 , , PUNCT cana-5232 73 8 𝑒𝑖+1 𝑒𝑖+1 PROPN cana-5232 73 9 } } PUNCT cana-5232 73 10 for for ADP cana-5232 73 11 2 2 NUM cana-5232 73 12 ≀ ≀ NOUN cana-5232 73 13 𝑖 𝑖 SYM cana-5232 73 14 ≀ ≀ NUM cana-5232 73 15 𝑛 𝑛 PRON cana-5232 73 16 βˆ’ βˆ’ PROPN cana-5232 73 17 1 1 NUM cana-5232 73 18 . . PUNCT cana-5232 74 1 so so ADV cana-5232 74 2 , , PUNCT cana-5232 74 3 from from ADP cana-5232 74 4 the the DET cana-5232 74 5 figure.2 figure.2 PROPN cana-5232 74 6 , , PUNCT cana-5232 74 7 the the DET cana-5232 74 8 set set NOUN cana-5232 74 9 𝑆 𝑆 PROPN cana-5232 74 10 = = SYM cana-5232 74 11 { { PUNCT cana-5232 74 12 𝑣1 𝑣1 PROPN cana-5232 74 13 , , PUNCT cana-5232 74 14 𝑣2 𝑣2 PROPN cana-5232 74 15 , , PUNCT cana-5232 74 16 … … PUNCT cana-5232 74 17 , , PUNCT cana-5232 74 18 π‘£π‘š π‘£π‘š NOUN cana-5232 74 19 , , PUNCT cana-5232 74 20 𝑒2 𝑒2 PROPN cana-5232 74 21 , , PUNCT cana-5232 74 22 𝑒5 𝑒5 PROPN cana-5232 74 23 , , PUNCT cana-5232 74 24 … … PUNCT cana-5232 74 25 , , PUNCT cana-5232 74 26 π‘’π‘šβˆ’1 π‘’π‘šβˆ’1 PROPN cana-5232 74 27 } } PUNCT cana-5232 74 28 be be VERB cana-5232 74 29 the the DET cana-5232 74 30 one one NUM cana-5232 74 31 of of ADP cana-5232 74 32 the the DET cana-5232 74 33 minimum minimum ADJ cana-5232 74 34 cc cc NOUN cana-5232 74 35 - - ADJ cana-5232 74 36 dominating dominating ADJ cana-5232 74 37 set set NOUN cana-5232 74 38 of of ADP cana-5232 74 39 𝐺 𝐺 PROPN cana-5232 74 40 . . PUNCT cana-5232 75 1 by by ADP cana-5232 75 2 using use VERB cana-5232 75 3 theorem theorem NOUN cana-5232 75 4 1.3 1.3 NUM cana-5232 75 5 , , PUNCT cana-5232 75 6 the the DET cana-5232 75 7 cardinality cardinality NOUN cana-5232 75 8 of of ADP cana-5232 75 9 s s PROPN cana-5232 75 10 is be AUX cana-5232 75 11 π‘š π‘š PRON cana-5232 75 12 + + NOUN cana-5232 75 13 ⌈ ⌈ NOUN cana-5232 75 14 π‘š π‘š ADP cana-5232 75 15 3 3 NUM cana-5232 75 16 βŒ‰. βŒ‰. ADV cana-5232 75 17 so so ADV cana-5232 75 18 , , PUNCT cana-5232 75 19 every every DET cana-5232 75 20 vertex vertex NOUN cana-5232 75 21 in in ADP cana-5232 75 22 𝐺 𝐺 PROPN cana-5232 75 23 is be AUX cana-5232 75 24 closely closely ADV cana-5232 75 25 connected connect VERB cana-5232 75 26 to to ADP cana-5232 75 27 atleast atleast VERB cana-5232 75 28 one one NUM cana-5232 75 29 of of ADP cana-5232 75 30 the the DET cana-5232 75 31 vertices vertex NOUN cana-5232 75 32 in in ADP cana-5232 75 33 𝑆. 𝑆. PROPN cana-5232 75 34 hence hence ADV cana-5232 75 35 , , PUNCT cana-5232 75 36 we we PRON cana-5232 75 37 conclude conclude VERB cana-5232 75 38 π›Ύπ‘π‘αˆΊπΊαˆ» π›Ύπ‘π‘αˆΊπΊαˆ» PROPN cana-5232 75 39 = = PUNCT cana-5232 76 1 π‘š π‘š PROPN cana-5232 76 2 + + SYM cana-5232 76 3 ⌈ ⌈ NOUN cana-5232 76 4 π‘š π‘š ADP cana-5232 76 5 3 3 NUM cana-5232 76 6 βŒ‰ βŒ‰ NOUN cana-5232 76 7 theorem theorem VERB cana-5232 76 8 3.8 3.8 NUM cana-5232 76 9 . . PUNCT cana-5232 77 1 the the DET cana-5232 77 2 cc cc NOUN cana-5232 77 3 - - PUNCT cana-5232 77 4 domination domination NOUN cana-5232 77 5 number number NOUN cana-5232 77 6 of of ADP cana-5232 77 7 πΆπ‘š πΆπ‘š NOUN cana-5232 77 8 ∘ ∘ NOUN cana-5232 77 9 𝐾2 𝐾2 NOUN cana-5232 77 10 is be AUX cana-5232 77 11 π›Ύπ‘π‘αˆΊπΆπ‘š π›Ύπ‘π‘αˆΊπΆπ‘š VERB cana-5232 77 12 ∘ ∘ ADJ cana-5232 77 13 𝐾2ሻ 𝐾2ሻ PROPN cana-5232 78 1 = = PUNCT cana-5232 79 1 ⌈ ⌈ NOUN cana-5232 79 2 π‘š π‘š ADP cana-5232 79 3 3 3 NUM cana-5232 79 4 βŒ‰ βŒ‰ NOUN cana-5232 79 5     SPACE cana-5232 79 6 for for ADP cana-5232 79 7     SPACE cana-5232 79 8 π‘š π‘š PROPN cana-5232 79 9 β‰₯ β‰₯ NUM cana-5232 79 10 2 2 NUM cana-5232 79 11 . . PUNCT cana-5232 80 1 proof proof NOUN cana-5232 80 2 . . PUNCT cana-5232 81 1 let let VERB cana-5232 81 2 𝐺 𝐺 NOUN cana-5232 81 3 = = PUNCT cana-5232 82 1 πΆπ‘š πΆπ‘š NOUN cana-5232 82 2 ∘ ∘ NOUN cana-5232 82 3 𝐾2 𝐾2 NOUN cana-5232 82 4 , , PUNCT cana-5232 82 5 π‘‰αˆΊπΆπ‘šαˆ» π‘‰αˆΊπΆπ‘šαˆ» PROPN cana-5232 82 6 = = SYM cana-5232 82 7 { { PUNCT cana-5232 82 8 𝑒1 𝑒1 PROPN cana-5232 82 9 , , PUNCT cana-5232 82 10 𝑒2 𝑒2 PROPN cana-5232 82 11 , , PUNCT cana-5232 82 12 … … PUNCT cana-5232 82 13 , , PUNCT cana-5232 82 14 π‘’π‘š π‘’π‘š ADP cana-5232 82 15 } } PUNCT cana-5232 82 16 . . PUNCT cana-5232 83 1 let let VERB cana-5232 83 2 { { PUNCT cana-5232 83 3 𝑣𝑖1 𝑣𝑖1 VERB cana-5232 83 4 , , PUNCT cana-5232 83 5 𝑣𝑖2 𝑣𝑖2 ADV cana-5232 83 6 } } PUNCT cana-5232 83 7 be be AUX cana-5232 83 8 the the DET cana-5232 83 9 vertex vertex NOUN cana-5232 83 10 set set NOUN cana-5232 83 11 of of ADP cana-5232 83 12 the the DET cana-5232 83 13 𝑖th 𝑖th PROPN cana-5232 83 14 copy copy NOUN cana-5232 83 15 of of ADP cana-5232 83 16 𝐾2 𝐾2 NOUN cana-5232 83 17 joined join VERB cana-5232 83 18 with with ADP cana-5232 83 19 the the DET cana-5232 83 20 vertex vertex NOUN cana-5232 83 21 𝑒𝑖. 𝑒𝑖. NOUN cana-5232 83 22 ∴ ∴ PROPN cana-5232 83 23 π‘‰αˆΊπΊαˆ» π‘‰αˆΊπΊαˆ» PROPN cana-5232 83 24 = = PUNCT cana-5232 83 25 { { PUNCT cana-5232 83 26 𝑒1 𝑒1 PROPN cana-5232 83 27 , , PUNCT cana-5232 83 28 𝑒2 𝑒2 PROPN cana-5232 83 29 , , PUNCT cana-5232 83 30 … … PUNCT cana-5232 83 31 , , PUNCT cana-5232 83 32 π‘’π‘š π‘’π‘š ADP cana-5232 83 33 } } PUNCT cana-5232 83 34 βˆͺ βˆͺ ADJ cana-5232 83 35 { { PUNCT cana-5232 83 36 𝑣11 𝑣11 NUM cana-5232 83 37 , , PUNCT cana-5232 83 38 𝑣12 𝑣12 PROPN cana-5232 83 39 } } PUNCT cana-5232 83 40 βˆͺ βˆͺ VERB cana-5232 83 41 { { PUNCT cana-5232 83 42 𝑣21 𝑣21 NOUN cana-5232 83 43 , , PUNCT cana-5232 83 44 𝑣22 𝑣22 NOUN cana-5232 83 45 } } PUNCT cana-5232 83 46 βˆͺ βˆͺ NOUN cana-5232 83 47 β‹―βˆͺ β‹―βˆͺ PROPN cana-5232 83 48 { { PUNCT cana-5232 83 49 π‘£π‘š1 π‘£π‘š1 PROPN cana-5232 83 50 , , PUNCT cana-5232 83 51 π‘£π‘š2 π‘£π‘š2 NOUN cana-5232 83 52 } } PUNCT cana-5232 83 53 and and CCONJ cana-5232 83 54 |π‘‰αˆΊπΊαˆ»| |π‘‰αˆΊπΊαˆ»| PROPN cana-5232 83 55 = = SYM cana-5232 83 56 3π‘š. 3π‘š. VERB cana-5232 83 57 each each DET cana-5232 83 58 vertex vertex NOUN cana-5232 83 59 𝑒𝑖 𝑒𝑖 NOUN cana-5232 83 60 of of ADP cana-5232 83 61 πΆπ‘š πΆπ‘š PROPN cana-5232 83 62 is be AUX cana-5232 83 63 closely closely ADV cana-5232 83 64 connected connect VERB cana-5232 83 65 with with ADP cana-5232 83 66 π‘’π‘–βˆ’1 π‘’π‘–βˆ’1 PROPN cana-5232 83 67 , , PUNCT cana-5232 83 68 𝑣𝑖+1 𝑣𝑖+1 NUM cana-5232 83 69 , , PUNCT cana-5232 83 70 𝑣𝑖1 𝑣𝑖1 VERB cana-5232 83 71 , , PUNCT cana-5232 83 72 𝑣𝑖2 𝑣𝑖2 NOUN cana-5232 83 73 , , PUNCT cana-5232 83 74 π‘£αˆΊπ‘–βˆ’1ሻ1 π‘£αˆΊπ‘–βˆ’1ሻ1 NUM cana-5232 83 75 , , PUNCT cana-5232 83 76 π‘£αˆΊπ‘–βˆ’1ሻ2 π‘£αˆΊπ‘–βˆ’1ሻ2 NUM cana-5232 83 77 , , PUNCT cana-5232 83 78 π‘£αˆΊπ‘–+1ሻ1 π‘£αˆΊπ‘–+1ሻ1 NOUN cana-5232 83 79 and and CCONJ cana-5232 83 80 π‘£αˆΊπ‘–+1ሻ2 π‘£αˆΊπ‘–+1ሻ2 NOUN cana-5232 83 81 . . PUNCT cana-5232 84 1 so so ADV cana-5232 84 2 |π‘πΆπΆαˆΊπ‘’π‘–αˆ»| |π‘πΆπΆαˆΊπ‘’π‘–αˆ»| ADV cana-5232 84 3 = = SYM cana-5232 84 4 8 8 NUM cana-5232 84 5 . . PUNCT cana-5232 84 6     SPACE cana-5232 85 1 βˆ€π‘– βˆ€π‘– PROPN cana-5232 85 2 = = SYM cana-5232 85 3 1,2 1,2 NUM cana-5232 85 4 , , PUNCT cana-5232 85 5 … … PUNCT cana-5232 85 6 π‘š. π‘š. NOUN cana-5232 85 7 then then ADV cana-5232 85 8 the the DET cana-5232 85 9 set set NOUN cana-5232 85 10 𝐷1 𝐷1 NOUN cana-5232 85 11 = = SYM cana-5232 85 12 { { PUNCT cana-5232 85 13 𝑒1 𝑒1 PROPN cana-5232 85 14 , , PUNCT cana-5232 85 15 𝑒4 𝑒4 NOUN cana-5232 85 16 , , PUNCT cana-5232 85 17 β‹― β‹― NOUN cana-5232 85 18 , , PUNCT cana-5232 85 19 π‘’π‘šβˆ’2 π‘’π‘šβˆ’2 PROPN cana-5232 85 20 } } PUNCT cana-5232 85 21 and and CCONJ cana-5232 85 22 𝐷2 𝐷2 NOUN cana-5232 85 23 = = SYM cana-5232 85 24 { { PUNCT cana-5232 85 25 𝑒2 𝑒2 PROPN cana-5232 85 26 , , PUNCT cana-5232 85 27 𝑒5 𝑒5 PROPN cana-5232 85 28 , , PUNCT cana-5232 85 29 β‹― β‹― PROPN cana-5232 85 30 , , PUNCT cana-5232 85 31 π‘’π‘š π‘’π‘š ADP cana-5232 85 32 βˆ’ βˆ’ NOUN cana-5232 85 33 1 1 NUM cana-5232 85 34 } } PUNCT cana-5232 85 35 are be AUX cana-5232 85 36 some some DET cana-5232 85 37 cc cc NOUN cana-5232 85 38 - - ADJ cana-5232 85 39 dominating dominating ADJ cana-5232 85 40 sets set NOUN cana-5232 85 41 of of ADP cana-5232 85 42 𝐺 𝐺 PROPN cana-5232 85 43 and and CCONJ cana-5232 85 44 it it PRON cana-5232 85 45 is be AUX cana-5232 85 46 also also ADV cana-5232 85 47 minimum minimum ADJ cana-5232 85 48 cc cc NOUN cana-5232 85 49 - - ADJ cana-5232 85 50 dominating dominating ADJ cana-5232 85 51 set set NOUN cana-5232 85 52 . . PUNCT cana-5232 86 1 hence hence ADV cana-5232 86 2 π›Ύπ‘π‘αˆΊπΊαˆ» π›Ύπ‘π‘αˆΊπΊαˆ» PROPN cana-5232 86 3 = = SYM cana-5232 86 4 ⌈ ⌈ NOUN cana-5232 86 5 π‘š π‘š ADP cana-5232 86 6 3 3 NUM cana-5232 86 7 βŒ‰. βŒ‰. ADV cana-5232 86 8 theorem theorem VERB cana-5232 86 9 3.9 3.9 NUM cana-5232 86 10 . . PUNCT cana-5232 87 1 for for ADP cana-5232 87 2 𝑛 𝑛 PROPN cana-5232 87 3 β‰₯ β‰₯ NUM cana-5232 87 4 1 1 NUM cana-5232 87 5 and and CCONJ cana-5232 87 6 π‘š π‘š X cana-5232 87 7 > > X cana-5232 87 8 1 1 NUM cana-5232 87 9 , , PUNCT cana-5232 87 10 π›Ύπ‘π‘αˆΊπΆπ‘š π›Ύπ‘π‘αˆΊπΆπ‘š PROPN cana-5232 87 11 ∘ ∘ NUM cana-5232 88 1 πΎπ‘›αˆ» πΎπ‘›αˆ» PROPN cana-5232 88 2 = = PUNCT cana-5232 88 3 ⌈ ⌈ PROPN cana-5232 88 4 π‘š π‘š ADP cana-5232 88 5 3 3 NUM cana-5232 88 6 βŒ‰. βŒ‰. ADJ cana-5232 88 7 proof proof NOUN cana-5232 88 8 . . PUNCT cana-5232 89 1 let let VERB cana-5232 90 1 𝐺 𝐺 NOUN cana-5232 90 2 = = PUNCT cana-5232 91 1 πΆπ‘š πΆπ‘š ADP cana-5232 91 2 ∘ ∘ NOUN cana-5232 91 3 𝐾𝑛 𝐾𝑛 PROPN cana-5232 91 4 , , PUNCT cana-5232 91 5 and and CCONJ cana-5232 91 6 π‘‰αˆΊπΆπ‘šαˆ» π‘‰αˆΊπΆπ‘šαˆ» PROPN cana-5232 91 7 = = SYM cana-5232 91 8 { { PUNCT cana-5232 91 9 𝑒1 𝑒1 PROPN cana-5232 91 10 , , PUNCT cana-5232 91 11 𝑒2 𝑒2 PROPN cana-5232 91 12 , , PUNCT cana-5232 91 13 … … PUNCT cana-5232 91 14 , , PUNCT cana-5232 91 15 π‘’π‘š π‘’π‘š ADP cana-5232 91 16 } } PUNCT cana-5232 91 17 . . PUNCT cana-5232 92 1 let let VERB cana-5232 92 2 { { PUNCT cana-5232 92 3 𝑣𝑖1 𝑣𝑖1 VERB cana-5232 92 4 , , PUNCT cana-5232 92 5 𝑣𝑖2 𝑣𝑖2 NOUN cana-5232 92 6 , , PUNCT cana-5232 92 7 𝑣𝑖3 𝑣𝑖3 NOUN cana-5232 92 8 , , PUNCT cana-5232 92 9 β‹― β‹― PROPN cana-5232 92 10 , , PUNCT cana-5232 92 11 𝑣𝑖𝑛 𝑣𝑖𝑛 NOUN cana-5232 92 12 } } PUNCT cana-5232 92 13 be be VERB cana-5232 92 14 the the DET cana-5232 92 15 vertex vertex NOUN cana-5232 92 16 set set NOUN cana-5232 92 17 of of ADP cana-5232 92 18 the the DET cana-5232 92 19 𝑖th 𝑖th PROPN cana-5232 92 20 copy copy NOUN cana-5232 92 21 of of ADP cana-5232 92 22 𝐾𝑛 𝐾𝑛 PROPN cana-5232 92 23 joined join VERB cana-5232 92 24 with with ADP cana-5232 92 25 the the DET cana-5232 92 26 vertex vertex NOUN cana-5232 92 27 𝑒𝑖. 𝑒𝑖. NOUN cana-5232 92 28 so so ADV cana-5232 92 29 π‘‰αˆΊπΊαˆ» π‘‰αˆΊπΊαˆ» PROPN cana-5232 92 30 = = SYM cana-5232 92 31 { { PUNCT cana-5232 92 32 𝑒1 𝑒1 PROPN cana-5232 92 33 , , PUNCT cana-5232 92 34 𝑒2 𝑒2 PROPN cana-5232 92 35 , , PUNCT cana-5232 92 36 𝑒3 𝑒3 PROPN cana-5232 92 37 , , PUNCT cana-5232 92 38 … … PUNCT cana-5232 92 39 , , PUNCT cana-5232 92 40 π‘’π‘š π‘’π‘š NOUN cana-5232 92 41 , , PUNCT cana-5232 92 42 𝑣11 𝑣11 NOUN cana-5232 92 43 , , PUNCT cana-5232 92 44 𝑣12 𝑣12 PROPN cana-5232 92 45 , , PUNCT cana-5232 92 46 … … PUNCT cana-5232 92 47 , , PUNCT cana-5232 92 48 𝑣1𝑛 𝑣1𝑛 NOUN cana-5232 92 49 , , PUNCT cana-5232 92 50 𝑣21 𝑣21 NUM cana-5232 92 51 , , PUNCT cana-5232 92 52 𝑣22 𝑣22 NOUN cana-5232 92 53 , , PUNCT cana-5232 92 54 … … PUNCT cana-5232 92 55 , , PUNCT cana-5232 92 56 𝑣2𝑛 𝑣2𝑛 PROPN cana-5232 92 57 , , PUNCT cana-5232 92 58 … … PUNCT cana-5232 92 59 , , PUNCT cana-5232 92 60 π‘£π‘š1 π‘£π‘š1 NUM cana-5232 92 61 , , PUNCT cana-5232 92 62 π‘‰π‘š2 π‘‰π‘š2 NOUN cana-5232 92 63 , , PUNCT cana-5232 92 64 … … PUNCT cana-5232 92 65 , , PUNCT cana-5232 92 66 π‘£π‘šπ‘› π‘£π‘šπ‘› VERB cana-5232 92 67 } } PUNCT cana-5232 92 68 and and CCONJ cana-5232 92 69 |π‘‰αˆΊπΊαˆ»| |π‘‰αˆΊπΊαˆ»| PROPN cana-5232 93 1 = = SYM cana-5232 93 2 π‘šπ‘› π‘šπ‘› X cana-5232 94 1 + + NOUN cana-5232 94 2 π‘š. π‘š. ADJ cana-5232 94 3 for for ADP cana-5232 94 4 each each DET cana-5232 94 5 𝑒𝑖 𝑒𝑖 NOUN cana-5232 94 6 in in ADP cana-5232 94 7 πΆπ‘š πΆπ‘š PROPN cana-5232 94 8 is be AUX cana-5232 94 9 closely closely ADV cana-5232 94 10 connected connect VERB cana-5232 94 11 with with ADP cana-5232 94 12 the the DET cana-5232 94 13 following follow VERB cana-5232 94 14 vertices vertex NOUN cana-5232 94 15 π‘’π‘–βˆ’1 π‘’π‘–βˆ’1 PROPN cana-5232 94 16 , , PUNCT cana-5232 94 17 𝑒𝑖+1 𝑒𝑖+1 PROPN cana-5232 94 18 , , PUNCT cana-5232 94 19 αˆΊπ‘– αˆΊπ‘– VERB cana-5232 94 20 βˆ’ βˆ’ PROPN cana-5232 94 21 1ሻth 1ሻth NUM cana-5232 94 22 copy copy NOUN cana-5232 94 23 of of ADP cana-5232 94 24 𝐾𝑛,αˆΊπ‘–αˆ»th 𝐾𝑛,αˆΊπ‘–αˆ»th PROPN cana-5232 94 25 copy copy NOUN cana-5232 94 26 of of ADP cana-5232 94 27 𝐾𝑛,αˆΊπ‘– 𝐾𝑛,αˆΊπ‘– PUNCT cana-5232 95 1 + + CCONJ cana-5232 95 2 1ሻth 1ሻth NUM cana-5232 95 3 copy copy NOUN cana-5232 95 4 of of ADP cana-5232 95 5 𝐾𝑛,then 𝐾𝑛,then PROPN cana-5232 95 6 |π‘πΆπΆαˆΊπ‘’π‘–αˆ»| |π‘πΆπΆαˆΊπ‘’π‘–αˆ»| NOUN cana-5232 95 7 = = SYM cana-5232 95 8 3𝑛 3𝑛 NUM cana-5232 95 9 + + CCONJ cana-5232 95 10 2 2 X cana-5232 95 11 . . X cana-5232 95 12 then then ADV cana-5232 95 13 the the DET cana-5232 95 14 set set NOUN cana-5232 95 15 𝐷 𝐷 PROPN cana-5232 95 16 = = SYM cana-5232 95 17 { { PUNCT cana-5232 95 18 𝑒2 𝑒2 PROPN cana-5232 95 19 , , PUNCT cana-5232 95 20 𝑒5 𝑒5 PROPN cana-5232 95 21 , , PUNCT cana-5232 95 22 β‹― β‹― PROPN cana-5232 95 23 , , PUNCT cana-5232 95 24 π‘’π‘šβˆ’1 π‘’π‘šβˆ’1 PROPN cana-5232 95 25 } } PUNCT cana-5232 95 26 is be AUX cana-5232 95 27 ccdominating ccdominate VERB cana-5232 95 28 set set NOUN cana-5232 95 29 , , PUNCT cana-5232 95 30 on on ADP cana-5232 95 31 removing remove VERB cana-5232 95 32 one one NUM cana-5232 95 33 vertex vertex NOUN cana-5232 95 34 in in ADP cana-5232 95 35 𝐷 𝐷 NOUN cana-5232 95 36 it it PRON cana-5232 95 37 is be AUX cana-5232 95 38 not not PART cana-5232 95 39 𝐢𝐢-dominating 𝐢𝐢-dominate VERB cana-5232 95 40 set set NOUN cana-5232 95 41 . . PUNCT cana-5232 96 1 hence hence ADV cana-5232 96 2 of of ADP cana-5232 96 3 𝐷 𝐷 PROPN cana-5232 96 4 is be AUX cana-5232 96 5 the the DET cana-5232 96 6 minimal minimal ADJ cana-5232 96 7 cc cc ADJ cana-5232 96 8 - - ADJ cana-5232 96 9 dominating dominating ADJ cana-5232 96 10 set set NOUN cana-5232 96 11 and and CCONJ cana-5232 96 12 |𝐷| |𝐷| NOUN cana-5232 96 13 = = SYM cana-5232 96 14 ⌈ ⌈ SYM cana-5232 96 15 m m VERB cana-5232 96 16 3 3 NUM cana-5232 96 17 βŒ‰. βŒ‰. ADV cana-5232 96 18 so so ADV cana-5232 96 19 , , PUNCT cana-5232 96 20 we we PRON cana-5232 96 21 conclude conclude VERB cana-5232 96 22 π›Ύπ‘π‘αˆΊπΊαˆ» π›Ύπ‘π‘αˆΊπΊαˆ» PROPN cana-5232 96 23 = = PUNCT cana-5232 96 24 ⌈ ⌈ NOUN cana-5232 96 25 π‘š π‘š ADP cana-5232 96 26 3 3 NUM cana-5232 96 27 βŒ‰. βŒ‰. ADJ cana-5232 96 28 communications communication NOUN cana-5232 96 29 on on ADP cana-5232 96 30 applied apply VERB cana-5232 96 31 nonlinear nonlinear ADJ cana-5232 96 32 analysis analysis NOUN cana-5232 96 33 issn issn NOUN cana-5232 96 34 : : PUNCT cana-5232 96 35 1074 1074 NUM cana-5232 96 36 - - PUNCT cana-5232 96 37 133x 133x NUM cana-5232 96 38 vol vol VERB cana-5232 96 39 32 32 NUM cana-5232 96 40 no no NOUN cana-5232 96 41 . . PUNCT cana-5232 97 1 10s 10 NOUN cana-5232 97 2 ( ( PUNCT cana-5232 97 3 2025 2025 NUM cana-5232 97 4 ) ) PUNCT cana-5232 97 5 1300 1300 NUM cana-5232 97 6 https://internationalpubls.com https://internationalpubls.com X cana-5232 97 7 theorem theorem VERB cana-5232 97 8 3.10 3.10 NUM cana-5232 97 9 . . PUNCT cana-5232 98 1 let let VERB cana-5232 98 2 𝐻1 𝐻1 NOUN cana-5232 98 3 , , PUNCT cana-5232 98 4 𝐻2 𝐻2 VERB cana-5232 98 5 be be AUX cana-5232 98 6 two two NUM cana-5232 98 7 connected connected ADJ cana-5232 98 8 graphs graph NOUN cana-5232 98 9 and and CCONJ cana-5232 98 10 𝐻2 𝐻2 VERB cana-5232 98 11 β‰  β‰  PROPN cana-5232 98 12 𝐾1 𝐾1 PROPN cana-5232 98 13 , , PUNCT cana-5232 98 14 then then ADV cana-5232 98 15 π›Ύπ‘π‘αˆΊπ»1 π›Ύπ‘π‘αˆΊπ»1 PROPN cana-5232 98 16 ∘ ∘ PROPN cana-5232 98 17 𝐻2ሻ 𝐻2ሻ PROPN cana-5232 98 18 = = SYM cana-5232 98 19 π›Ύπ‘π‘αˆΊπ»1ሻ π›Ύπ‘π‘αˆΊπ»1ሻ PROPN cana-5232 98 20 . . PUNCT cana-5232 99 1 proof proof NOUN cana-5232 99 2 . . PUNCT cana-5232 100 1 let let VERB cana-5232 100 2 𝐺 𝐺 PROPN cana-5232 100 3 = = NOUN cana-5232 100 4 𝐻1 𝐻1 PROPN cana-5232 100 5 ∘ ∘ PROPN cana-5232 100 6 𝐻2 𝐻2 PROPN cana-5232 100 7 . . PUNCT cana-5232 101 1 in in ADP cana-5232 101 2 𝐺 𝐺 PROPN cana-5232 101 3 , , PUNCT cana-5232 101 4 the the DET cana-5232 101 5 𝑗th 𝑗th NUM cana-5232 101 6 vertex vertex NOUN cana-5232 101 7 of of ADP cana-5232 101 8 𝐻1 𝐻1 PROPN cana-5232 101 9 is be AUX cana-5232 101 10 adjacent adjacent ADJ cana-5232 101 11 to to ADP cana-5232 101 12 every every DET cana-5232 101 13 vertex vertex NOUN cana-5232 101 14 in in ADP cana-5232 101 15 the the DET cana-5232 101 16 𝑗th 𝑗th PROPN cana-5232 101 17 copy copy NOUN cana-5232 101 18 of of ADP cana-5232 101 19 𝐻2 𝐻2 NOUN cana-5232 101 20 . . PUNCT cana-5232 102 1 𝐻2 𝐻2 PROPN cana-5232 102 2 is be AUX cana-5232 102 3 non non ADJ cana-5232 102 4 - - ADJ cana-5232 102 5 trivial trivial ADJ cana-5232 102 6 connected connected ADJ cana-5232 102 7 graph graph NOUN cana-5232 102 8 , , PUNCT cana-5232 102 9 and and CCONJ cana-5232 102 10 every every DET cana-5232 102 11 pair pair NOUN cana-5232 102 12 of of ADP cana-5232 102 13 vertices vertex NOUN cana-5232 102 14 of of ADP cana-5232 102 15 𝐻2 𝐻2 NOUN cana-5232 102 16 in in ADP cana-5232 102 17 the the DET cana-5232 102 18 𝑖th 𝑖th PROPN cana-5232 102 19 copy copy NOUN cana-5232 102 20 of of ADP cana-5232 102 21 𝐻2 𝐻2 NOUN cana-5232 102 22 are be AUX cana-5232 102 23 closely closely ADV cana-5232 102 24 connected connect VERB cana-5232 102 25 . . PUNCT cana-5232 103 1 so so ADV cana-5232 103 2 , , PUNCT cana-5232 103 3 every every DET cana-5232 103 4 vertex vertex NOUN cana-5232 103 5 in in ADP cana-5232 103 6 the the DET cana-5232 103 7 𝑖th 𝑖th PROPN cana-5232 103 8 copy copy NOUN cana-5232 103 9 of of ADP cana-5232 103 10 𝐻2 𝐻2 PROPN cana-5232 103 11 is be AUX cana-5232 103 12 closely closely ADV cana-5232 103 13 connected connect VERB cana-5232 103 14 with with ADP cana-5232 103 15 the the DET cana-5232 103 16 𝑗th 𝑗th PROPN cana-5232 103 17 vertex vertex NOUN cana-5232 103 18 of of ADP cana-5232 103 19 𝐻1 𝐻1 PROPN cana-5232 103 20 . . PUNCT cana-5232 104 1 suppose suppose VERB cana-5232 104 2 𝐷 𝐷 NOUN cana-5232 104 3 is be AUX cana-5232 104 4 the the DET cana-5232 104 5 minimum minimum ADJ cana-5232 104 6 cc cc NOUN cana-5232 104 7 - - ADJ cana-5232 104 8 dominating dominating ADJ cana-5232 104 9 set set NOUN cana-5232 104 10 of of ADP cana-5232 104 11 𝐻1 𝐻1 PROPN cana-5232 104 12 , , PUNCT cana-5232 104 13 every every DET cana-5232 104 14 vertex vertex NOUN cana-5232 104 15 in in ADP cana-5232 104 16 π‘‰αˆΊπ»1ሻ π‘‰αˆΊπ»1ሻ PROPN cana-5232 104 17 βˆ’ βˆ’ PROPN cana-5232 104 18 𝐷 𝐷 PROPN cana-5232 104 19 is be AUX cana-5232 104 20 closely closely ADV cana-5232 104 21 connected connect VERB cana-5232 104 22 with with ADP cana-5232 104 23 at at ADP cana-5232 104 24 lest lest ADP cana-5232 104 25 one one NUM cana-5232 104 26 vertex vertex NOUN cana-5232 104 27 in in ADP cana-5232 104 28 𝐷 𝐷 NOUN cana-5232 104 29 and and CCONJ cana-5232 104 30 also also ADV cana-5232 104 31 every every DET cana-5232 104 32 vertex vertex NOUN cana-5232 104 33 in in ADP cana-5232 104 34 π‘‰αˆΊπΊαˆ» π‘‰αˆΊπΊαˆ» PROPN cana-5232 104 35 βˆ’ βˆ’ PROPN cana-5232 104 36 𝐷 𝐷 PROPN cana-5232 104 37 is be AUX cana-5232 104 38 closely closely ADV cana-5232 104 39 connected connect VERB cana-5232 104 40 with with ADP cana-5232 104 41 atleast atleast ADJ cana-5232 104 42 one one NUM cana-5232 104 43 vertex vertex NOUN cana-5232 104 44 in in ADP cana-5232 104 45 𝐷. 𝐷. PROPN cana-5232 104 46 hence hence ADV cana-5232 104 47 , , PUNCT cana-5232 104 48 we we PRON cana-5232 104 49 conclude conclude VERB cana-5232 104 50 that that SCONJ cana-5232 104 51 π›Ύπ‘π‘αˆΊπΊαˆ» π›Ύπ‘π‘αˆΊπΊαˆ» PROPN cana-5232 104 52 = = SYM cana-5232 104 53 |𝐷| |𝐷| X cana-5232 104 54 = = SYM cana-5232 104 55 π›Ύπ‘π‘αˆΊπ»1ሻ π›Ύπ‘π‘αˆΊπ»1ሻ PROPN cana-5232 104 56 . . PUNCT cana-5232 104 57 theorem theorem VERB cana-5232 104 58 3.11 3.11 NUM cana-5232 104 59 . . PUNCT cana-5232 105 1 let let VERB cana-5232 105 2 𝐻1 𝐻1 NOUN cana-5232 105 3 be be AUX cana-5232 105 4 a a DET cana-5232 105 5 connected connected ADJ cana-5232 105 6 graph graph NOUN cana-5232 105 7 and and CCONJ cana-5232 105 8 𝐻2 𝐻2 VERB cana-5232 105 9 = = SYM cana-5232 105 10 𝐾‾𝑛 𝐾‾𝑛 NOUN cana-5232 105 11 ( ( PUNCT cana-5232 105 12 complement complement NOUN cana-5232 105 13 of of ADP cana-5232 105 14 complete complete ADJ cana-5232 105 15 graph graph NOUN cana-5232 105 16 𝐾𝑛 𝐾𝑛 PROPN cana-5232 105 17 ) ) PUNCT cana-5232 105 18 . . PUNCT cana-5232 106 1 then then ADV cana-5232 106 2 π›Ύπ‘π‘αˆΊπ»1 π›Ύπ‘π‘αˆΊπ»1 PROPN cana-5232 106 3 ∘ ∘ PROPN cana-5232 107 1 𝐻2ሻ 𝐻2ሻ PROPN cana-5232 107 2 = = PRON cana-5232 107 3 π›Ύπ‘π‘αˆΊπ»1ሻ π›Ύπ‘π‘αˆΊπ»1ሻ X cana-5232 107 4 + + CCONJ cana-5232 107 5 |π‘‰αˆΊπ»1ሻ||π‘‰αˆΊπ»2ሻ| |π‘‰αˆΊπ»1ሻ||π‘‰αˆΊπ»2ሻ| PROPN cana-5232 107 6 . . PUNCT cana-5232 108 1 proof proof NOUN cana-5232 108 2 . . PUNCT cana-5232 109 1 let let VERB cana-5232 109 2 𝐺1 𝐺1 PROPN cana-5232 109 3 = = PUNCT cana-5232 109 4 𝐻1 𝐻1 VERB cana-5232 109 5 ∘ ∘ PROPN cana-5232 109 6 𝐻2 𝐻2 PROPN cana-5232 109 7 . . PUNCT cana-5232 110 1 in in ADP cana-5232 110 2 𝐺 𝐺 PROPN cana-5232 110 3 all all DET cana-5232 110 4 vertices vertex NOUN cana-5232 110 5 of of ADP cana-5232 110 6 𝐻2 𝐻2 NOUN cana-5232 110 7 are be AUX cana-5232 110 8 pendent pendent ADJ cana-5232 110 9 vertices vertex NOUN cana-5232 110 10 and and CCONJ cana-5232 110 11 also also ADV cana-5232 110 12 cc cc VERB cana-5232 110 13 - - ADJ cana-5232 110 14 isolated isolate VERB cana-5232 110 15 vertices vertex NOUN cana-5232 110 16 . . PUNCT cana-5232 111 1 ie ie X cana-5232 111 2 , , PUNCT cana-5232 111 3 the the DET cana-5232 111 4 𝑗th 𝑗th NUM cana-5232 111 5 vertex vertex NOUN cana-5232 111 6 of of ADP cana-5232 111 7 𝐻1 𝐻1 PROPN cana-5232 111 8 is be AUX cana-5232 111 9 adjacent adjacent ADJ cana-5232 111 10 to to ADP cana-5232 111 11 every every DET cana-5232 111 12 vertex vertex NOUN cana-5232 111 13 in in ADP cana-5232 111 14 the the DET cana-5232 111 15 𝑖th 𝑖th PROPN cana-5232 111 16 copy copy NOUN cana-5232 111 17 of of ADP cana-5232 111 18 𝐻2 𝐻2 PROPN cana-5232 111 19 . . PUNCT cana-5232 112 1 since since SCONJ cana-5232 112 2 𝐻2 𝐻2 PROPN cana-5232 112 3 is be AUX cana-5232 112 4 isolated isolate VERB cana-5232 112 5 graph graph NOUN cana-5232 112 6 , , PUNCT cana-5232 112 7 so so SCONJ cana-5232 112 8 we we PRON cana-5232 112 9 remove remove VERB cana-5232 112 10 any any DET cana-5232 112 11 edge edge NOUN cana-5232 112 12 between between ADP cana-5232 112 13 𝐻1 𝐻1 NOUN cana-5232 112 14 and and CCONJ cana-5232 112 15 𝐻2 𝐻2 PROPN cana-5232 112 16 , , PUNCT cana-5232 112 17 then then ADV cana-5232 112 18 the the DET cana-5232 112 19 graph graph NOUN cana-5232 112 20 𝐺 𝐺 NOUN cana-5232 112 21 is be AUX cana-5232 112 22 disconnected disconnect VERB cana-5232 112 23 . . PUNCT cana-5232 113 1 hence hence ADV cana-5232 113 2 , , PUNCT cana-5232 113 3 we we PRON cana-5232 113 4 have have VERB cana-5232 113 5 |π‘‰αˆΊπ»1ሻ||π‘‰αˆΊπ»2ሻ| |π‘‰αˆΊπ»1ሻ||π‘‰αˆΊπ»2ሻ| ADJ cana-5232 113 6 number number NOUN cana-5232 113 7 of of ADP cana-5232 113 8 isolated isolated ADJ cana-5232 113 9 vertices vertex NOUN cana-5232 113 10 and and CCONJ cana-5232 113 11 also also ADV cana-5232 113 12 𝐻2 𝐻2 PROPN cana-5232 113 13 does do AUX cana-5232 113 14 not not PART cana-5232 113 15 alter alter VERB cana-5232 113 16 the the DET cana-5232 113 17 𝐢𝐢-domination 𝐢𝐢-domination NOUN cana-5232 113 18 of of ADP cana-5232 113 19 𝐻1 𝐻1 PROPN cana-5232 113 20 . . PUNCT cana-5232 114 1 by by ADP cana-5232 114 2 theorem theorem NOUN cana-5232 114 3 1.1 1.1 NUM cana-5232 114 4 , , PUNCT cana-5232 114 5 every every DET cana-5232 114 6 cc cc NOUN cana-5232 114 7 - - ADJ cana-5232 114 8 isolated isolate VERB cana-5232 114 9 vertex vertex NOUN cana-5232 114 10 must must AUX cana-5232 114 11 belong belong VERB cana-5232 114 12 to to ADP cana-5232 114 13 every every DET cana-5232 114 14 𝐢𝐢-dominating 𝐢𝐢-dominate VERB cana-5232 114 15 set set NOUN cana-5232 114 16 . . PUNCT cana-5232 115 1 hence hence ADV cana-5232 115 2 , , PUNCT cana-5232 115 3 we we PRON cana-5232 115 4 have have VERB cana-5232 115 5 π›Ύπ‘π‘αˆΊπ»1 π›Ύπ‘π‘αˆΊπ»1 NUM cana-5232 115 6 ∘ ∘ NOUN cana-5232 116 1 𝐻2ሻ 𝐻2ሻ PROPN cana-5232 116 2 = = PUNCT cana-5232 116 3 π›Ύπ‘π‘αˆΊπ»1ሻ π›Ύπ‘π‘αˆΊπ»1ሻ X cana-5232 116 4 + + CCONJ cana-5232 116 5 |π‘‰αˆΊπ»1ሻ||π‘‰αˆΊπ»2ሻ| |π‘‰αˆΊπ»1ሻ||π‘‰αˆΊπ»2ሻ| PROPN cana-5232 116 6 . . PUNCT cana-5232 116 7 corollary corollary ADJ cana-5232 116 8 3.12 3.12 NUM cana-5232 116 9 . . PUNCT cana-5232 117 1 the the DET cana-5232 117 2 following follow VERB cana-5232 117 3 results result NOUN cana-5232 117 4 are be AUX cana-5232 117 5 direct direct ADJ cana-5232 117 6 computation computation NOUN cana-5232 117 7 from from ADP cana-5232 117 8 theorem theorem ADJ cana-5232 117 9 12 12 NUM cana-5232 117 10 . . NOUN cana-5232 118 1 1 1 NUM cana-5232 118 2 . . PUNCT cana-5232 119 1 π›Ύπ‘π‘αˆΊπΆπ‘š π›Ύπ‘π‘αˆΊπΆπ‘š NOUN cana-5232 119 2 ∘ ∘ ADJ cana-5232 120 1 π‘ƒπ‘›αˆ» π‘ƒπ‘›αˆ» NOUN cana-5232 120 2 = = SYM cana-5232 120 3 ⌈ ⌈ NOUN cana-5232 120 4 π‘š π‘š ADP cana-5232 120 5 3 3 NUM cana-5232 120 6 βŒ‰ βŒ‰ NOUN cana-5232 120 7 = = SYM cana-5232 120 8 π›Ύπ‘π‘αˆΊπΆπ‘šαˆ» π›Ύπ‘π‘αˆΊπΆπ‘šαˆ» NUM cana-5232 120 9 2 2 X cana-5232 120 10 . . PUNCT cana-5232 120 11 π›Ύπ‘π‘αˆΊπ‘ƒπ‘› π›Ύπ‘π‘αˆΊπ‘ƒπ‘› PROPN cana-5232 120 12 ∘ ∘ NOUN cana-5232 120 13 π‘ƒπ‘šαˆ» π‘ƒπ‘šαˆ» PROPN cana-5232 120 14 = = SYM cana-5232 120 15 𝑛 𝑛 PROPN cana-5232 120 16 = = PUNCT cana-5232 120 17 π›Ύπ‘π‘αˆΊπ‘ƒπ‘›αˆ» π›Ύπ‘π‘αˆΊπ‘ƒπ‘›αˆ» PROPN cana-5232 120 18 3 3 NUM cana-5232 120 19 . . PUNCT cana-5232 121 1 π›Ύπ‘π‘αˆΊπΆπ‘› π›Ύπ‘π‘αˆΊπΆπ‘› NOUN cana-5232 121 2 ∘ ∘ PUNCT cana-5232 122 1 πΆπ‘šαˆ» πΆπ‘šαˆ» NOUN cana-5232 122 2 = = SYM cana-5232 122 3 ⌈ ⌈ ADP cana-5232 122 4 𝑛 𝑛 PRON cana-5232 122 5 3 3 NUM cana-5232 122 6 βŒ‰ βŒ‰ NOUN cana-5232 122 7 = = PUNCT cana-5232 122 8 π›Ύπ‘π‘αˆΊπΆπ‘›αˆ» π›Ύπ‘π‘αˆΊπΆπ‘›αˆ» ADV cana-5232 122 9 4 4 X cana-5232 122 10 . . PUNCT cana-5232 122 11 π›Ύπ‘π‘αˆΊπ‘ƒπ‘› π›Ύπ‘π‘αˆΊπ‘ƒπ‘› NOUN cana-5232 122 12 ∘ ∘ PART cana-5232 122 13 πΆπ‘šαˆ» πΆπ‘šαˆ» NOUN cana-5232 122 14 = = SYM cana-5232 122 15 𝑛 𝑛 NOUN cana-5232 122 16 = = PUNCT cana-5232 122 17 π›Ύπ‘π‘αˆΊπ‘ƒπ‘›αˆ» π›Ύπ‘π‘αˆΊπ‘ƒπ‘›αˆ» PROPN cana-5232 122 18 4 4 NUM cana-5232 122 19 . . PUNCT cana-5232 122 20 applications application NOUN cana-5232 122 21 in in ADP cana-5232 122 22 traditional traditional ADJ cana-5232 122 23 network network NOUN cana-5232 122 24 is be AUX cana-5232 122 25 operated operate VERB cana-5232 122 26 by by ADP cana-5232 122 27 all all DET cana-5232 122 28 hosts host NOUN cana-5232 122 29 for for ADP cana-5232 122 30 two two NUM cana-5232 122 31 - - PUNCT cana-5232 122 32 way way NOUN cana-5232 122 33 communications communication NOUN cana-5232 122 34 . . PUNCT cana-5232 123 1 every every DET cana-5232 123 2 host host NOUN cana-5232 123 3 has have VERB cana-5232 123 4 a a DET cana-5232 123 5 service service NOUN cana-5232 123 6 area area NOUN cana-5232 123 7 and and CCONJ cana-5232 123 8 within within ADP cana-5232 123 9 this this DET cana-5232 123 10 close close ADJ cana-5232 123 11 - - PUNCT cana-5232 123 12 transmission transmission NOUN cana-5232 123 13 covers cover VERB cana-5232 123 14 the the DET cana-5232 123 15 range range NOUN cana-5232 123 16 , , PUNCT cana-5232 123 17 link link VERB cana-5232 123 18 failure failure NOUN cana-5232 123 19 of of ADP cana-5232 123 20 the the DET cana-5232 123 21 network network NOUN cana-5232 123 22 does do AUX cana-5232 123 23 not not PART cana-5232 123 24 disconnect disconnect VERB cana-5232 123 25 or or CCONJ cana-5232 123 26 disrupt disrupt VERB cana-5232 123 27 the the DET cana-5232 123 28 communication communication NOUN cana-5232 123 29 of of ADP cana-5232 123 30 the the DET cana-5232 123 31 entire entire ADJ cana-5232 123 32 network network NOUN cana-5232 123 33 . . PUNCT cana-5232 124 1 a a DET cana-5232 124 2 pair pair NOUN cana-5232 124 3 of of ADP cana-5232 124 4 such such ADJ cana-5232 124 5 hosts host NOUN cana-5232 124 6 that that PRON cana-5232 124 7 are be AUX cana-5232 124 8 communication communication NOUN cana-5232 124 9 with with ADP cana-5232 124 10 one one NUM cana-5232 124 11 another another PRON cana-5232 124 12 are be AUX cana-5232 124 13 called call VERB cana-5232 124 14 as as ADV cana-5232 124 15 closely closely ADV cana-5232 124 16 connected connected ADJ cana-5232 124 17 neighbors neighbor NOUN cana-5232 124 18 . . PUNCT cana-5232 125 1 thus thus ADV cana-5232 125 2 , , PUNCT cana-5232 125 3 closely closely ADV cana-5232 125 4 connected connected ADJ cana-5232 125 5 neighbors neighbor NOUN cana-5232 125 6 serves serve VERB cana-5232 125 7 as as ADP cana-5232 125 8 a a DET cana-5232 125 9 new new ADJ cana-5232 125 10 approach approach NOUN cana-5232 125 11 to to ADP cana-5232 125 12 design design NOUN cana-5232 125 13 networks network NOUN cana-5232 125 14 in in ADP cana-5232 125 15 [ [ X cana-5232 125 16 5 5 NUM cana-5232 125 17 ] ] PUNCT cana-5232 125 18 . . PUNCT cana-5232 126 1 5 5 X cana-5232 126 2 . . X cana-5232 126 3 conclusion conclusion NOUN cana-5232 126 4 in in ADP cana-5232 126 5 this this DET cana-5232 126 6 paper paper NOUN cana-5232 126 7 , , PUNCT cana-5232 126 8 the the DET cana-5232 126 9 concept concept NOUN cana-5232 126 10 of of ADP cana-5232 126 11 cc cc NOUN cana-5232 126 12 - - NOUN cana-5232 126 13 domination domination NOUN cana-5232 126 14 number number NOUN cana-5232 126 15 is be AUX cana-5232 126 16 studied study VERB cana-5232 126 17 using use VERB cana-5232 126 18 closely closely ADV cana-5232 126 19 connected connect VERB cana-5232 126 20 vertices vertex NOUN cana-5232 126 21 of of ADP cana-5232 126 22 graphs graph NOUN cana-5232 126 23 for for ADP cana-5232 126 24 corona corona NOUN cana-5232 126 25 product product NOUN cana-5232 126 26 of of ADP cana-5232 126 27 graphs graph NOUN cana-5232 126 28 . . PUNCT cana-5232 127 1 investigation investigation NOUN cana-5232 127 2 of of ADP cana-5232 127 3 𝐢𝐢-domination 𝐢𝐢-domination NOUN cana-5232 127 4 of of ADP cana-5232 127 5 various various ADJ cana-5232 127 6 other other ADJ cana-5232 127 7 products product NOUN cana-5232 127 8 is be AUX cana-5232 127 9 a a DET cana-5232 127 10 significant significant ADJ cana-5232 127 11 direction direction NOUN cana-5232 127 12 for for ADP cana-5232 127 13 further further ADJ cana-5232 127 14 research research NOUN cana-5232 127 15 . . PUNCT cana-5232 128 1 also also ADV cana-5232 128 2 determine determine VERB cana-5232 128 3 π›Ύπ‘π‘αˆΊπΊαˆ» π›Ύπ‘π‘αˆΊπΊαˆ» PROPN cana-5232 128 4 for for ADP cana-5232 128 5 the the DET cana-5232 128 6 family family NOUN cana-5232 128 7 of of ADP cana-5232 128 8 graphs graph NOUN cana-5232 128 9 . . PUNCT cana-5232 129 1 communications communication NOUN cana-5232 129 2 on on ADP cana-5232 129 3 applied apply VERB cana-5232 129 4 nonlinear nonlinear ADJ cana-5232 129 5 analysis analysis NOUN cana-5232 129 6 issn issn NOUN cana-5232 129 7 : : PUNCT cana-5232 129 8 1074 1074 NUM cana-5232 129 9 - - PUNCT cana-5232 129 10 133x 133x NUM cana-5232 129 11 vol vol VERB cana-5232 129 12 32 32 NUM cana-5232 129 13 no no NOUN cana-5232 129 14 . . PUNCT cana-5232 130 1 10s 10 NOUN cana-5232 130 2 ( ( PUNCT cana-5232 130 3 2025 2025 NUM cana-5232 130 4 ) ) PUNCT cana-5232 130 5 1301 1301 NUM cana-5232 131 1 https://internationalpubls.com https://internationalpubls.com X cana-5232 131 2 references reference NOUN cana-5232 131 3 [ [ X cana-5232 131 4 1 1 NUM cana-5232 131 5 ] ] X cana-5232 131 6 frucht frucht NOUN cana-5232 131 7 , , PUNCT cana-5232 131 8 r. r. PROPN cana-5232 131 9 , , PUNCT cana-5232 131 10 and and CCONJ cana-5232 131 11 harary harary NOUN cana-5232 131 12 , , PUNCT cana-5232 131 13 f. f. PROPN cana-5232 131 14 β€œ " PUNCT cana-5232 131 15 on on ADP cana-5232 131 16 the the DET cana-5232 131 17 corona corona NOUN cana-5232 131 18 of of ADP cana-5232 131 19 two two NUM cana-5232 131 20 graphs graph NOUN cana-5232 131 21 . . PUNCT cana-5232 131 22 ” " PUNCT cana-5232 132 1 aeq aeq PROPN cana-5232 132 2 . . PUNCT cana-5232 132 3 math math NOUN cana-5232 132 4 . . PUNCT cana-5232 133 1 4 4 NUM cana-5232 133 2 , , PUNCT cana-5232 133 3 322 322 NUM cana-5232 133 4 - - SYM cana-5232 133 5 325 325 NUM cana-5232 133 6 ( ( PUNCT cana-5232 133 7 1970 1970 NUM cana-5232 133 8 ) ) PUNCT cana-5232 133 9 . . PUNCT cana-5232 134 1 [ [ X cana-5232 134 2 2 2 NUM cana-5232 134 3 ] ] PUNCT cana-5232 134 4 haynes hayne NOUN cana-5232 134 5 , , PUNCT cana-5232 134 6 teresa teresa PROPN cana-5232 134 7 w. w. PROPN cana-5232 134 8 , , PUNCT cana-5232 134 9 stephen stephen PROPN cana-5232 134 10 hedetniemi hedetniemi PROPN cana-5232 134 11 , , PUNCT cana-5232 134 12 and and CCONJ cana-5232 134 13 peter peter PROPN cana-5232 134 14 slater slater PROPN cana-5232 134 15 . . PUNCT cana-5232 135 1 fundamentals fundamental NOUN cana-5232 135 2 of of ADP cana-5232 135 3 domination domination NOUN cana-5232 135 4 in in ADP cana-5232 135 5 graphs graph NOUN cana-5232 135 6 . . PUNCT cana-5232 136 1 crc crc PROPN cana-5232 136 2 press press PROPN cana-5232 136 3 , , PUNCT cana-5232 136 4 2013 2013 NUM cana-5232 136 5 . . PUNCT cana-5232 137 1 [ [ X cana-5232 137 2 3 3 NUM cana-5232 137 3 ] ] X cana-5232 137 4 haynes hayne NOUN cana-5232 137 5 , , PUNCT cana-5232 137 6 teresa teresa PROPN cana-5232 137 7 w. w. PROPN cana-5232 137 8 domination domination PROPN cana-5232 137 9 in in ADP cana-5232 137 10 graphs graph NOUN cana-5232 137 11 : : PUNCT cana-5232 137 12 volume volume NOUN cana-5232 137 13 2 2 NUM cana-5232 137 14 : : PUNCT cana-5232 137 15 advanced advanced ADJ cana-5232 137 16 topics topic NOUN cana-5232 137 17 . . PUNCT cana-5232 138 1 routledge routledge PROPN cana-5232 138 2 , , PUNCT cana-5232 138 3 2017 2017 NUM cana-5232 138 4 . . PUNCT cana-5232 139 1 [ [ X cana-5232 139 2 4 4 NUM cana-5232 139 3 ] ] X cana-5232 139 4 priya priya PROPN cana-5232 139 5 , , PUNCT cana-5232 139 6 k. k. PROPN cana-5232 139 7 , , PUNCT cana-5232 139 8 and and CCONJ cana-5232 139 9 anil anil PROPN cana-5232 139 10 kumar kumar PROPN cana-5232 139 11 . . PROPN cana-5232 139 12 cc cc PROPN cana-5232 139 13 - - NOUN cana-5232 139 14 domination domination NOUN cana-5232 139 15 in in ADP cana-5232 139 16 graphs graph NOUN cana-5232 139 17 , , PUNCT cana-5232 139 18 palestine palestine PROPN cana-5232 139 19 journal journal PROPN cana-5232 139 20 of of ADP cana-5232 139 21 mathematics mathematic NOUN cana-5232 139 22 , , PUNCT cana-5232 139 23 vol vol NOUN cana-5232 139 24 12(3 12(3 NUM cana-5232 139 25 ) ) PUNCT cana-5232 139 26 , , PUNCT cana-5232 139 27 2023 2023 NUM cana-5232 139 28 . . PUNCT cana-5232 140 1 [ [ X cana-5232 140 2 5 5 NUM cana-5232 140 3 ] ] X cana-5232 140 4 dekker dekker NOUN cana-5232 140 5 , , PUNCT cana-5232 140 6 a.h a.h PROPN cana-5232 140 7 , , PUNCT cana-5232 140 8 and and CCONJ cana-5232 140 9 colbert colbert NOUN cana-5232 140 10 , , PUNCT cana-5232 140 11 networks network NOUN cana-5232 140 12 and and CCONJ cana-5232 140 13 robustness robustness NOUN cana-5232 140 14 and and CCONJ cana-5232 140 15 graph graph NOUN cana-5232 140 16 topology topology NOUN cana-5232 140 17 , , PUNCT cana-5232 140 18 proceedings proceeding NOUN cana-5232 140 19 of of ADP cana-5232 140 20 the the DET cana-5232 140 21 27th 27th ADJ cana-5232 140 22 australasian australasian ADJ cana-5232 140 23 conference conference NOUN cana-5232 140 24 on on ADP cana-5232 140 25 computer computer NOUN cana-5232 140 26 science science NOUN cana-5232 140 27 , , PUNCT cana-5232 140 28 26(2004 26(2004 PROPN cana-5232 140 29 ) ) PUNCT cana-5232 140 30 , , PUNCT cana-5232 140 31 359 359 NUM cana-5232 140 32 368 368 NUM cana-5232 140 33 . . PUNCT