id sid tid token lemma pos cana-1301 1 1 communications communication NOUN cana-1301 1 2 on on ADP cana-1301 1 3 applied apply VERB cana-1301 1 4 nonlinear nonlinear ADJ cana-1301 1 5 analysis analysis NOUN cana-1301 1 6 issn issn NOUN cana-1301 1 7 : : PUNCT cana-1301 1 8 1074 1074 NUM cana-1301 1 9 - - PUNCT cana-1301 1 10 133x 133x NUM cana-1301 1 11 vol vol NOUN cana-1301 1 12 31 31 NUM cana-1301 1 13 no no NOUN cana-1301 1 14 . . PUNCT cana-1301 2 1 7s 7 NOUN cana-1301 2 2 ( ( PUNCT cana-1301 2 3 2024 2024 NUM cana-1301 2 4 ) ) PUNCT cana-1301 2 5 261 261 NUM cana-1301 2 6 https://internationalpubls.com https://internationalpubls.com X cana-1301 2 7 multi multi ADJ cana-1301 2 8 - - ADJ cana-1301 2 9 bipolar bipolar ADJ cana-1301 2 10 fuzzy fuzzy ADJ cana-1301 2 11 planar planar ADJ cana-1301 2 12 graphs graph NOUN cana-1301 2 13 k. k. PROPN cana-1301 2 14 rama rama PROPN cana-1301 3 1 kishore kishore PROPN cana-1301 3 2 a a PROPN cana-1301 3 3 , , PUNCT cana-1301 3 4 b b PROPN cana-1301 3 5 , , PUNCT cana-1301 3 6 ch ch NOUN cana-1301 3 7 . . PUNCT cana-1301 3 8 ramprasad ramprasad PROPN cana-1301 3 9 b b PROPN cana-1301 3 10 , , PUNCT cana-1301 3 11 * * PROPN cana-1301 3 12 , , PUNCT cana-1301 4 1 p. p. PROPN cana-1301 4 2 l. l. PROPN cana-1301 4 3 n. n. PROPN cana-1301 4 4 varma varma PROPN cana-1301 4 5 c c PROPN cana-1301 5 1 a a PRON cana-1301 5 2 , , PUNCT cana-1301 5 3 c c PROPN cana-1301 5 4 department department PROPN cana-1301 5 5 of of ADP cana-1301 5 6 mathematics mathematics PROPN cana-1301 5 7 and and CCONJ cana-1301 5 8 statistics statistic NOUN cana-1301 5 9 , , PUNCT cana-1301 5 10 school school NOUN cana-1301 5 11 of of ADP cana-1301 5 12 applied apply VERB cana-1301 5 13 sciences science NOUN cana-1301 5 14 , , PUNCT cana-1301 5 15 vfstr vfstr NOUN cana-1301 5 16 ( ( PUNCT cana-1301 5 17 deemed deem VERB cana-1301 5 18 to to PART cana-1301 5 19 be be AUX cana-1301 5 20 university university NOUN cana-1301 5 21 ) ) PUNCT cana-1301 5 22 , , PUNCT cana-1301 5 23 vadlamudi vadlamudi NOUN cana-1301 5 24 , , PUNCT cana-1301 5 25 522 522 NUM cana-1301 5 26 213 213 NUM cana-1301 5 27 , , PUNCT cana-1301 5 28 india india PROPN cana-1301 5 29 a a PRON cana-1301 5 30 , , PUNCT cana-1301 5 31 b b PROPN cana-1301 5 32 , , PUNCT cana-1301 5 33 * * PUNCT cana-1301 5 34 department department NOUN cana-1301 5 35 of of ADP cana-1301 5 36 mathematics mathematic NOUN cana-1301 5 37 , , PUNCT cana-1301 5 38 vasireddy vasireddy NOUN cana-1301 5 39 venkatadri venkatadri PROPN cana-1301 5 40 institute institute PROPN cana-1301 5 41 of of ADP cana-1301 5 42 technology technology PROPN cana-1301 5 43 , , PUNCT cana-1301 5 44 namburu namburu ADV cana-1301 5 45 , , PUNCT cana-1301 5 46 522 522 NUM cana-1301 5 47 508 508 NUM cana-1301 5 48 , , PUNCT cana-1301 5 49 india india PROPN cana-1301 5 50 email email NOUN cana-1301 5 51 ids id NOUN cana-1301 5 52 : : PUNCT cana-1301 5 53 aitskrk987@gmail.com aitskrk987@gmail.com NUM cana-1301 5 54 , , PUNCT cana-1301 5 55 b,*ramprasadchegu1984@gmail.com b,*ramprasadchegu1984@gmail.com X cana-1301 5 56 cplnvarma@gmail.com cplnvarma@gmail.com X cana-1301 6 1 article article NOUN cana-1301 6 2 history history NOUN cana-1301 6 3 : : PUNCT cana-1301 6 4 received receive VERB cana-1301 6 5 : : PUNCT cana-1301 6 6 01 01 NUM cana-1301 6 7 - - PUNCT cana-1301 6 8 06 06 NUM cana-1301 6 9 - - PUNCT cana-1301 6 10 2024 2024 NUM cana-1301 6 11 revised revise VERB cana-1301 6 12 : : PUNCT cana-1301 6 13 03 03 NUM cana-1301 6 14 - - PUNCT cana-1301 6 15 07 07 NUM cana-1301 6 16 - - PUNCT cana-1301 6 17 2024 2024 NUM cana-1301 6 18 accepted accept VERB cana-1301 6 19 : : PUNCT cana-1301 6 20 29 29 NUM cana-1301 6 21 - - SYM cana-1301 6 22 07 07 NUM cana-1301 6 23 - - PUNCT cana-1301 6 24 2024 2024 NUM cana-1301 6 25 abstract abstract NOUN cana-1301 6 26 : : PUNCT cana-1301 6 27 generalization generalization NOUN cana-1301 6 28 of of ADP cana-1301 6 29 a a DET cana-1301 6 30 bipolar bipolar ADJ cana-1301 6 31 fuzzy fuzzy ADJ cana-1301 6 32 set set NOUN cana-1301 6 33 helps help VERB cana-1301 6 34 to to PART cana-1301 6 35 arrive arrive VERB cana-1301 6 36 at at ADP cana-1301 6 37 an an DET cana-1301 6 38 m m NOUN cana-1301 6 39 - - PUNCT cana-1301 6 40 bpf bpf NOUN cana-1301 6 41 set set NOUN cana-1301 6 42 that that PRON cana-1301 6 43 can can AUX cana-1301 6 44 be be AUX cana-1301 6 45 extended extend VERB cana-1301 6 46 to to PART cana-1301 6 47 understand understand VERB cana-1301 6 48 important important ADJ cana-1301 6 49 properties property NOUN cana-1301 6 50 of of ADP cana-1301 6 51 planar planar ADJ cana-1301 6 52 and and CCONJ cana-1301 6 53 multigraph multigraph NOUN cana-1301 6 54 . . PUNCT cana-1301 7 1 the the DET cana-1301 7 2 definition definition NOUN cana-1301 7 3 of of ADP cana-1301 7 4 m m PROPN cana-1301 7 5 - - PUNCT cana-1301 7 6 bpfmgs bpfmgs NOUN cana-1301 7 7 , , PUNCT cana-1301 7 8 mbpfpgs mbpfpgs NOUN cana-1301 7 9 and and CCONJ cana-1301 7 10 m m NOUN cana-1301 7 11 - - NOUN cana-1301 7 12 bpfdgs bpfdgs NOUN cana-1301 7 13 are be AUX cana-1301 7 14 introduced introduce VERB cana-1301 7 15 , , PUNCT cana-1301 7 16 and and CCONJ cana-1301 7 17 some some PRON cana-1301 7 18 of of ADP cana-1301 7 19 their their PRON cana-1301 7 20 intriguing intriguing ADJ cana-1301 7 21 aspects aspect NOUN cana-1301 7 22 are be AUX cana-1301 7 23 studied study VERB cana-1301 7 24 . . PUNCT cana-1301 8 1 an an DET cana-1301 8 2 m m ADJ cana-1301 8 3 - - PUNCT cana-1301 8 4 bpfpgs bpfpgs ADJ cana-1301 8 5 level level NOUN cana-1301 8 6 of of ADP cana-1301 8 7 planarity planarity NOUN cana-1301 8 8 is be AUX cana-1301 8 9 measured measure VERB cana-1301 8 10 in in ADP cana-1301 8 11 this this DET cana-1301 8 12 case case NOUN cana-1301 8 13 using use VERB cana-1301 8 14 the the DET cana-1301 8 15 term term NOUN cana-1301 8 16 " " PUNCT cana-1301 8 17 degree degree NOUN cana-1301 8 18 of of ADP cana-1301 8 19 planarity planarity NOUN cana-1301 8 20 . . PUNCT cana-1301 8 21 " " PUNCT cana-1301 9 1 on on ADP cana-1301 9 2 the the DET cana-1301 9 3 subject subject NOUN cana-1301 9 4 of of ADP cana-1301 9 5 planarity planarity NOUN cana-1301 9 6 , , PUNCT cana-1301 9 7 some some DET cana-1301 9 8 theorems theorem NOUN cana-1301 9 9 have have AUX cana-1301 9 10 been be AUX cana-1301 9 11 proved prove VERB cana-1301 9 12 . . PUNCT cana-1301 10 1 additionally additionally ADV cana-1301 10 2 , , PUNCT cana-1301 10 3 we we PRON cana-1301 10 4 investigate investigate VERB cana-1301 10 5 isomorphism isomorphism NOUN cana-1301 10 6 across across ADP cana-1301 10 7 m m NOUN cana-1301 10 8 - - NOUN cana-1301 10 9 bpfpgs bpfpgs NOUN cana-1301 10 10 . . PUNCT cana-1301 11 1 keywords keyword NOUN cana-1301 11 2 : : PUNCT cana-1301 11 3 graphs graph NOUN cana-1301 11 4 , , PUNCT cana-1301 11 5 multigraph multigraph NOUN cana-1301 11 6 , , PUNCT cana-1301 11 7 planar planar ADJ cana-1301 11 8 graph graph NOUN cana-1301 11 9 , , PUNCT cana-1301 11 10 fuzzygraphs fuzzygraph NOUN cana-1301 11 11 . . PUNCT cana-1301 12 1 nomenclature nomenclature ADJ cana-1301 12 2 m m ADJ cana-1301 12 3 - - ADJ cana-1301 12 4 bipolar bipolar ADJ cana-1301 12 5 fuzzy fuzzy ADJ cana-1301 12 6 m m NOUN cana-1301 12 7 - - PUNCT cana-1301 12 8 bpf bpf NOUN cana-1301 12 9 m m ADJ cana-1301 12 10 - - ADJ cana-1301 12 11 bipolar bipolar ADJ cana-1301 12 12 fuzzy fuzzy ADJ cana-1301 12 13 graph graph NOUN cana-1301 12 14 m m NOUN cana-1301 12 15 - - PUNCT cana-1301 12 16 bpfg bpfg ADJ cana-1301 12 17 m m ADJ cana-1301 12 18 - - ADJ cana-1301 12 19 bipolar bipolar ADJ cana-1301 12 20 fuzzy fuzzy ADJ cana-1301 12 21 multigraph multigraph NOUN cana-1301 12 22 m m PROPN cana-1301 12 23 - - PUNCT cana-1301 12 24 bpfmg bpfmg ADJ cana-1301 12 25 mbipolar mbipolar ADJ cana-1301 12 26 fuzzy fuzzy ADJ cana-1301 12 27 multi multi ADJ cana-1301 12 28 line line NOUN cana-1301 12 29 m m NOUN cana-1301 12 30 - - PUNCT cana-1301 12 31 bpfml bpfml NOUN cana-1301 12 32 m m ADJ cana-1301 12 33 - - ADJ cana-1301 12 34 bipolar bipolar ADJ cana-1301 12 35 fuzzy fuzzy ADJ cana-1301 12 36 planar planar ADJ cana-1301 12 37 graph graph NOUN cana-1301 12 38 m m NOUN cana-1301 12 39 - - NOUN cana-1301 12 40 bpfpg bpfpg ADJ cana-1301 12 41 m m ADJ cana-1301 12 42 - - ADJ cana-1301 12 43 bipolar bipolar ADJ cana-1301 12 44 fuzzy fuzzy ADJ cana-1301 12 45 complete complete ADJ cana-1301 12 46 multi multi ADJ cana-1301 12 47 graph graph NOUN cana-1301 12 48 m m NOUN cana-1301 12 49 - - PUNCT cana-1301 12 50 bpfcmg bpfcmg NOUN cana-1301 12 51 m m ADJ cana-1301 12 52 - - ADJ cana-1301 12 53 bipolar bipolar ADJ cana-1301 12 54 fuzzy fuzzy ADJ cana-1301 12 55 dual dual ADJ cana-1301 12 56 graph graph NOUN cana-1301 12 57 m m PROPN cana-1301 12 58 - - PUNCT cana-1301 12 59 bpfdg bpfdg ADJ cana-1301 12 60 1 1 NUM cana-1301 12 61 . . PUNCT cana-1301 12 62 introduction introduction NOUN cana-1301 12 63 ease ease NOUN cana-1301 12 64 of of ADP cana-1301 12 65 use use NOUN cana-1301 12 66 and and CCONJ cana-1301 12 67 flexibility flexibility NOUN cana-1301 12 68 has have AUX cana-1301 12 69 made make VERB cana-1301 12 70 it it PRON cana-1301 12 71 possible possible ADJ cana-1301 12 72 to to PART cana-1301 12 73 apply apply VERB cana-1301 12 74 graph graph NOUN cana-1301 12 75 theory theory NOUN cana-1301 12 76 in in ADP cana-1301 12 77 real real ADJ cana-1301 12 78 life life NOUN cana-1301 12 79 situations situation NOUN cana-1301 12 80 requiring require VERB cana-1301 12 81 further further ADJ cana-1301 12 82 research research NOUN cana-1301 12 83 in in ADP cana-1301 12 84 this this DET cana-1301 12 85 field field NOUN cana-1301 12 86 . . PUNCT cana-1301 13 1 ideally ideally ADV cana-1301 13 2 , , PUNCT cana-1301 13 3 problems problem NOUN cana-1301 13 4 related relate VERB cana-1301 13 5 to to ADP cana-1301 13 6 traffic traffic NOUN cana-1301 13 7 and and CCONJ cana-1301 13 8 power power NOUN cana-1301 13 9 line line NOUN cana-1301 13 10 management management NOUN cana-1301 13 11 provide provide VERB cana-1301 13 12 suitable suitable ADJ cana-1301 13 13 areas area NOUN cana-1301 13 14 of of ADP cana-1301 13 15 applications application NOUN cana-1301 13 16 of of ADP cana-1301 13 17 graph graph NOUN cana-1301 13 18 theory theory NOUN cana-1301 13 19 . . PUNCT cana-1301 14 1 however however ADV cana-1301 14 2 , , PUNCT cana-1301 14 3 the the DET cana-1301 14 4 complex complex ADJ cana-1301 14 5 nature nature NOUN cana-1301 14 6 of of ADP cana-1301 14 7 space space NOUN cana-1301 14 8 organization organization NOUN cana-1301 14 9 and and CCONJ cana-1301 14 10 management management NOUN cana-1301 14 11 through through ADP cana-1301 14 12 cost cost NOUN cana-1301 14 13 effective effective ADJ cana-1301 14 14 procedures procedure NOUN cana-1301 14 15 is be AUX cana-1301 14 16 a a DET cana-1301 14 17 dominant dominant ADJ cana-1301 14 18 impediment impediment NOUN cana-1301 14 19 to to PART cana-1301 14 20 find find VERB cana-1301 14 21 a a DET cana-1301 14 22 tangible tangible ADJ cana-1301 14 23 solution solution NOUN cana-1301 14 24 , , PUNCT cana-1301 14 25 even even ADV cana-1301 14 26 to to ADP cana-1301 14 27 this this DET cana-1301 14 28 date date NOUN cana-1301 14 29 . . PUNCT cana-1301 15 1 for for ADP cana-1301 15 2 example example NOUN cana-1301 15 3 , , PUNCT cana-1301 15 4 the the DET cana-1301 15 5 problems problem NOUN cana-1301 15 6 of of ADP cana-1301 15 7 over over ADP cana-1301 15 8 lapping lapping NOUN cana-1301 15 9 and and CCONJ cana-1301 15 10 cross cross ADJ cana-1301 15 11 - - ADJ cana-1301 15 12 crossing crossing ADJ cana-1301 15 13 power power NOUN cana-1301 15 14 lines line NOUN cana-1301 15 15 have have VERB cana-1301 15 16 traditional traditional ADJ cana-1301 15 17 solutions solution NOUN cana-1301 15 18 of of ADP cana-1301 15 19 separating separate VERB cana-1301 15 20 them they PRON cana-1301 15 21 in in ADP cana-1301 15 22 space space NOUN cana-1301 15 23 and and CCONJ cana-1301 15 24 height height NOUN cana-1301 15 25 . . PUNCT cana-1301 16 1 in in ADP cana-1301 16 2 printed print VERB cana-1301 16 3 circuit circuit NOUN cana-1301 16 4 boards board NOUN cana-1301 16 5 , , PUNCT cana-1301 16 6 this this DET cana-1301 16 7 problem problem NOUN cana-1301 16 8 is be AUX cana-1301 16 9 solved solve VERB cana-1301 16 10 by by ADP cana-1301 16 11 using use VERB cana-1301 16 12 layers layer NOUN cana-1301 16 13 of of ADP cana-1301 16 14 separation separation NOUN cana-1301 16 15 . . PUNCT cana-1301 17 1 we we PRON cana-1301 17 2 can can AUX cana-1301 17 3 try try VERB cana-1301 17 4 to to PART cana-1301 17 5 minimize minimize VERB cana-1301 17 6 the the DET cana-1301 17 7 height height NOUN cana-1301 17 8 parameters parameter NOUN cana-1301 17 9 and and CCONJ cana-1301 17 10 decrease decrease VERB cana-1301 17 11 the the DET cana-1301 17 12 layers layer NOUN cana-1301 17 13 using use VERB cana-1301 17 14 the the DET cana-1301 17 15 concept concept NOUN cana-1301 17 16 of of ADP cana-1301 17 17 planar planar ADJ cana-1301 17 18 graphs graph NOUN cana-1301 17 19 . . PUNCT cana-1301 18 1 when when SCONJ cana-1301 18 2 it it PRON cana-1301 18 3 comes come VERB cana-1301 18 4 to to ADP cana-1301 18 5 traffic traffic NOUN cana-1301 18 6 management management NOUN cana-1301 18 7 , , PUNCT cana-1301 18 8 routes route NOUN cana-1301 18 9 with with ADP cana-1301 18 10 various various ADJ cana-1301 18 11 degrees degree NOUN cana-1301 18 12 of of ADP cana-1301 18 13 congestion congestion NOUN cana-1301 18 14 pose pose VERB cana-1301 18 15 a a DET cana-1301 18 16 common common ADJ cana-1301 18 17 problem problem NOUN cana-1301 18 18 in in ADP cana-1301 18 19 several several ADJ cana-1301 18 20 developed develop VERB cana-1301 18 21 cities city NOUN cana-1301 18 22 . . PUNCT cana-1301 19 1 with with ADP cana-1301 19 2 the the DET cana-1301 19 3 help help NOUN cana-1301 19 4 of of ADP cana-1301 19 5 the the DET cana-1301 19 6 graph graph NOUN cana-1301 19 7 theory theory NOUN cana-1301 19 8 an an DET cana-1301 19 9 application application NOUN cana-1301 19 10 for for ADP cana-1301 19 11 an an DET cana-1301 19 12 information information NOUN cana-1301 19 13 system system NOUN cana-1301 19 14 that that PRON cana-1301 19 15 informs inform VERB cana-1301 19 16 road road NOUN cana-1301 19 17 users user NOUN cana-1301 19 18 and and CCONJ cana-1301 19 19 drivers driver NOUN cana-1301 19 20 about about ADP cana-1301 19 21 levels level NOUN cana-1301 19 22 of of ADP cana-1301 19 23 congestion congestion NOUN cana-1301 19 24 on on ADP cana-1301 19 25 roads road NOUN cana-1301 19 26 can can AUX cana-1301 19 27 be be AUX cana-1301 19 28 developed develop VERB cana-1301 19 29 . . PUNCT cana-1301 20 1 mailto:itskrk987@gmail.com mailto:itskrk987@gmail.com PROPN cana-1301 20 2 mailto:ramprasadchegu1984@gmail.com mailto:ramprasadchegu1984@gmail.com PROPN cana-1301 20 3 communications communication NOUN cana-1301 20 4 on on ADP cana-1301 20 5 applied apply VERB cana-1301 20 6 nonlinear nonlinear ADJ cana-1301 20 7 analysis analysis NOUN cana-1301 20 8 issn issn NOUN cana-1301 20 9 : : PUNCT cana-1301 20 10 1074 1074 NUM cana-1301 20 11 - - PUNCT cana-1301 20 12 133x 133x NUM cana-1301 20 13 vol vol NOUN cana-1301 20 14 31 31 NUM cana-1301 20 15 no no NOUN cana-1301 20 16 . . PUNCT cana-1301 21 1 7s 7 NOUN cana-1301 21 2 ( ( PUNCT cana-1301 21 3 2024 2024 NUM cana-1301 21 4 ) ) PUNCT cana-1301 21 5 262 262 NUM cana-1301 21 6 https://internationalpubls.com https://internationalpubls.com X cana-1301 21 7 electrical electrical ADJ cana-1301 21 8 circuits circuit NOUN cana-1301 21 9 can can AUX cana-1301 21 10 be be AUX cana-1301 21 11 represented represent VERB cana-1301 21 12 using use VERB cana-1301 21 13 a a DET cana-1301 21 14 graph graph NOUN cana-1301 21 15 model model NOUN cana-1301 21 16 . . PUNCT cana-1301 22 1 in in ADP cana-1301 22 2 this this DET cana-1301 22 3 system system NOUN cana-1301 22 4 , , PUNCT cana-1301 22 5 minimizing minimize VERB cana-1301 22 6 no no DET cana-1301 22 7 overlapping overlap VERB cana-1301 22 8 circuits circuit NOUN cana-1301 22 9 is be AUX cana-1301 22 10 the the DET cana-1301 22 11 primary primary ADJ cana-1301 22 12 goal goal NOUN cana-1301 22 13 . . PUNCT cana-1301 23 1 rail rail NOUN cana-1301 23 2 lines line NOUN cana-1301 23 3 , , PUNCT cana-1301 23 4 transmission transmission NOUN cana-1301 23 5 lines line NOUN cana-1301 23 6 , , PUNCT cana-1301 23 7 underground underground ADJ cana-1301 23 8 tunnels tunnel NOUN cana-1301 23 9 , , PUNCT cana-1301 23 10 etc etc X cana-1301 23 11 . . X cana-1301 23 12 are be AUX cana-1301 23 13 all all PRON cana-1301 23 14 crucial crucial ADJ cana-1301 23 15 components component NOUN cana-1301 23 16 of of ADP cana-1301 23 17 city city NOUN cana-1301 23 18 development development NOUN cana-1301 23 19 . . PUNCT cana-1301 24 1 accidents accident NOUN cana-1301 24 2 can can AUX cana-1301 24 3 happen happen VERB cana-1301 24 4 when when SCONJ cana-1301 24 5 people people NOUN cana-1301 24 6 are be AUX cana-1301 24 7 crossing cross VERB cana-1301 24 8 . . PUNCT cana-1301 25 1 despite despite SCONJ cana-1301 25 2 the the DET cana-1301 25 3 preference preference NOUN cana-1301 25 4 for for ADP cana-1301 25 5 routes route NOUN cana-1301 25 6 without without ADP cana-1301 25 7 crossings crossing NOUN cana-1301 25 8 , , PUNCT cana-1301 25 9 space space NOUN cana-1301 25 10 constraints constraint NOUN cana-1301 25 11 sometimes sometimes ADV cana-1301 25 12 make make VERB cana-1301 25 13 them they PRON cana-1301 25 14 necessary necessary ADJ cana-1301 25 15 . . PUNCT cana-1301 26 1 it it PRON cana-1301 26 2 is be AUX cana-1301 26 3 advisable advisable ADJ cana-1301 26 4 to to PART cana-1301 26 5 cross cross VERB cana-1301 26 6 between between ADP cana-1301 26 7 congested congested ADJ cana-1301 26 8 and and CCONJ cana-1301 26 9 less less ADV cana-1301 26 10 congested congested ADJ cana-1301 26 11 routes route NOUN cana-1301 26 12 rather rather ADV cana-1301 26 13 than than ADP cana-1301 26 14 two two NUM cana-1301 26 15 congested congested ADJ cana-1301 26 16 ones one NOUN cana-1301 26 17 . . PUNCT cana-1301 27 1 the the DET cana-1301 27 2 definition definition NOUN cana-1301 27 3 of of ADP cana-1301 27 4 " " PUNCT cana-1301 27 5 congested congested ADJ cana-1301 27 6 " " PUNCT cana-1301 27 7 is be AUX cana-1301 27 8 ambiguous ambiguous ADJ cana-1301 27 9 . . PUNCT cana-1301 28 1 we we PRON cana-1301 28 2 frequently frequently ADV cana-1301 28 3 use use VERB cana-1301 28 4 terms term NOUN cana-1301 28 5 like like ADP cana-1301 28 6 " " PUNCT cana-1301 28 7 congested,""extremely congested,""extremely ADV cana-1301 28 8 congested,""very congested,""very NOUN cana-1301 28 9 congested congested ADJ cana-1301 28 10 , , PUNCT cana-1301 28 11 " " PUNCT cana-1301 28 12 etc etc X cana-1301 28 13 . . X cana-1301 28 14 linguistic linguistic ADJ cana-1301 28 15 terms term NOUN cana-1301 28 16 are be AUX cana-1301 28 17 what what PRON cana-1301 28 18 they they PRON cana-1301 28 19 are be AUX cana-1301 28 20 , , PUNCT cana-1301 28 21 and and CCONJ cana-1301 28 22 they they PRON cana-1301 28 23 have have VERB cana-1301 28 24 certain certain ADJ cana-1301 28 25 relationship relationship NOUN cana-1301 28 26 values value NOUN cana-1301 28 27 . . PUNCT cana-1301 29 1 strong strong ADJ cana-1301 29 2 routes route NOUN cana-1301 29 3 are be AUX cana-1301 29 4 those those PRON cana-1301 29 5 that that PRON cana-1301 29 6 are be AUX cana-1301 29 7 congested congest VERB cana-1301 29 8 , , PUNCT cana-1301 29 9 whereas whereas SCONJ cana-1301 29 10 weak weak ADJ cana-1301 29 11 routes route NOUN cana-1301 29 12 are be AUX cana-1301 29 13 those those PRON cana-1301 29 14 that that PRON cana-1301 29 15 are be AUX cana-1301 29 16 less less ADV cana-1301 29 17 congested congested ADJ cana-1301 29 18 . . PUNCT cana-1301 30 1 thus thus ADV cana-1301 30 2 , , PUNCT cana-1301 30 3 in in ADP cana-1301 30 4 a a DET cana-1301 30 5 city city NOUN cana-1301 30 6 plan plan NOUN cana-1301 30 7 , , PUNCT cana-1301 30 8 crossing cross VERB cana-1301 30 9 across across ADP cana-1301 30 10 strong strong ADJ cana-1301 30 11 and and CCONJ cana-1301 30 12 weak weak ADJ cana-1301 30 13 pathways pathway NOUN cana-1301 30 14 may may AUX cana-1301 30 15 be be AUX cana-1301 30 16 allowed allow VERB cana-1301 30 17 with with ADP cana-1301 30 18 a a DET cana-1301 30 19 certain certain ADJ cana-1301 30 20 level level NOUN cana-1301 30 21 of of ADP cana-1301 30 22 safety safety NOUN cana-1301 30 23 . . PUNCT cana-1301 31 1 strong strong ADJ cana-1301 31 2 line line NOUN cana-1301 31 3 and and CCONJ cana-1301 31 4 weak weak ADJ cana-1301 31 5 line line NOUN cana-1301 31 6 of of ADP cana-1301 31 7 a a DET cana-1301 31 8 fuzzy fuzzy ADJ cana-1301 31 9 planar planar ADJ cana-1301 31 10 graph graph NOUN cana-1301 31 11 are be AUX cana-1301 31 12 denoted denote VERB cana-1301 31 13 by by ADP cana-1301 31 14 the the DET cana-1301 31 15 words word NOUN cana-1301 31 16 " " PUNCT cana-1301 31 17 strong strong ADJ cana-1301 31 18 route route NOUN cana-1301 31 19 " " PUNCT cana-1301 31 20 and and CCONJ cana-1301 31 21 " " PUNCT cana-1301 31 22 weak weak ADJ cana-1301 31 23 route route NOUN cana-1301 31 24 , , PUNCT cana-1301 31 25 " " PUNCT cana-1301 31 26 respectively respectively ADV cana-1301 31 27 , , PUNCT cana-1301 31 28 and and CCONJ cana-1301 31 29 the the DET cana-1301 31 30 idea idea NOUN cana-1301 31 31 of of ADP cana-1301 31 32 fuzzy fuzzy ADJ cana-1301 31 33 planar planar ADJ cana-1301 31 34 graph graph NOUN cana-1301 31 35 is be AUX cana-1301 31 36 denoted denote VERB cana-1301 31 37 by by ADP cana-1301 31 38 the the DET cana-1301 31 39 ability ability NOUN cana-1301 31 40 to to PART cana-1301 31 41 traverse traverse VERB cana-1301 31 42 between between ADP cana-1301 31 43 strong strong ADJ cana-1301 31 44 and and CCONJ cana-1301 31 45 weak weak ADJ cana-1301 31 46 lines line NOUN cana-1301 32 1 [ [ X cana-1301 32 2 1 1 NUM cana-1301 32 3 , , PUNCT cana-1301 32 4 19 19 NUM cana-1301 32 5 , , PUNCT cana-1301 32 6 23 23 NUM cana-1301 32 7 ] ] PUNCT cana-1301 32 8 . . PUNCT cana-1301 33 1 zhang zhang PROPN cana-1301 34 1 [ [ X cana-1301 34 2 32 32 NUM cana-1301 34 3 ] ] PUNCT cana-1301 34 4 originally originally ADV cana-1301 34 5 put put VERB cana-1301 34 6 forward forward ADV cana-1301 34 7 the the DET cana-1301 34 8 concept concept NOUN cana-1301 34 9 of of ADP cana-1301 34 10 bipolar bipolar ADJ cana-1301 34 11 fuzzy fuzzy ADJ cana-1301 34 12 sets set NOUN cana-1301 34 13 in in ADP cana-1301 34 14 1994 1994 NUM cana-1301 34 15 as as ADP cana-1301 34 16 an an DET cana-1301 34 17 extension extension NOUN cana-1301 34 18 of of ADP cana-1301 34 19 fuzzy fuzzy ADJ cana-1301 34 20 sets set NOUN cana-1301 34 21 [ [ X cana-1301 34 22 30 30 NUM cana-1301 34 23 ] ] PUNCT cana-1301 34 24 . . PUNCT cana-1301 35 1 fuzzy fuzzy ADJ cana-1301 35 2 sets set NOUN cana-1301 35 3 with with ADP cana-1301 35 4 a a DET cana-1301 35 5 connection connection NOUN cana-1301 35 6 degree degree NOUN cana-1301 35 7 range range NOUN cana-1301 35 8 of of ADP cana-1301 35 9 [ [ X cana-1301 35 10 -1 -1 X cana-1301 35 11 , , PUNCT cana-1301 35 12 1 1 NUM cana-1301 35 13 ] ] PUNCT cana-1301 35 14 are be AUX cana-1301 35 15 extended extend VERB cana-1301 35 16 in in ADP cana-1301 35 17 bipolar bipolar ADJ cana-1301 35 18 fuzzy fuzzy ADJ cana-1301 35 19 sets set NOUN cana-1301 35 20 [ [ X cana-1301 35 21 30 30 NUM cana-1301 35 22 ] ] PUNCT cana-1301 35 23 . . PUNCT cana-1301 36 1 in in ADP cana-1301 36 2 a a DET cana-1301 36 3 bipolar bipolar ADJ cana-1301 36 4 fuzzy fuzzy ADJ cana-1301 36 5 set set NOUN cana-1301 36 6 , , PUNCT cana-1301 36 7 an an DET cana-1301 36 8 element element NOUN cana-1301 36 9 's 's PART cana-1301 36 10 connection connection NOUN cana-1301 36 11 degree degree NOUN cana-1301 36 12 of of ADP cana-1301 36 13 0 0 NUM cana-1301 36 14 indicates indicate VERB cana-1301 36 15 that that SCONJ cana-1301 36 16 it it PRON cana-1301 36 17 has have VERB cana-1301 36 18 no no DET cana-1301 36 19 bearing bearing NOUN cana-1301 36 20 on on ADP cana-1301 36 21 the the DET cana-1301 36 22 consequent consequent ADJ cana-1301 36 23 property property NOUN cana-1301 36 24 , , PUNCT cana-1301 36 25 its its PRON cana-1301 36 26 relationship relationship NOUN cana-1301 36 27 degree degree NOUN cana-1301 36 28 of of ADP cana-1301 36 29 ( ( PUNCT cana-1301 36 30 0 0 NUM cana-1301 36 31 , , PUNCT cana-1301 36 32 1 1 NUM cana-1301 36 33 ] ] PUNCT cana-1301 36 34 that that SCONJ cana-1301 36 35 it it PRON cana-1301 36 36 somewhat somewhat ADV cana-1301 36 37 fulfills fulfill VERB cana-1301 36 38 the the DET cana-1301 36 39 property property NOUN cana-1301 36 40 , , PUNCT cana-1301 36 41 and and CCONJ cana-1301 36 42 its its PRON cana-1301 36 43 relationship relationship NOUN cana-1301 36 44 degree degree NOUN cana-1301 36 45 of of ADP cana-1301 36 46 [ [ X cana-1301 36 47 -1 -1 X cana-1301 36 48 , , PUNCT cana-1301 36 49 0 0 NUM cana-1301 36 50 ) ) PUNCT cana-1301 36 51 that that SCONJ cana-1301 36 52 it it PRON cana-1301 36 53 slightly slightly ADV cana-1301 36 54 satisfies satisfy VERB cana-1301 36 55 the the DET cana-1301 36 56 implied imply VERB cana-1301 36 57 counter counter NOUN cana-1301 36 58 - - NOUN cana-1301 36 59 property property NOUN cana-1301 36 60 . . PUNCT cana-1301 37 1 despite despite SCONJ cana-1301 37 2 having have VERB cana-1301 37 3 a a DET cana-1301 37 4 similar similar ADJ cana-1301 37 5 appearance appearance NOUN cana-1301 37 6 , , PUNCT cana-1301 37 7 bipolar bipolar ADJ cana-1301 37 8 fuzzy fuzzy ADJ cana-1301 37 9 sets set NOUN cana-1301 37 10 and and CCONJ cana-1301 37 11 intuitionistic intuitionistic ADJ cana-1301 37 12 fuzzy fuzzy ADJ cana-1301 37 13 sets set NOUN cana-1301 37 14 are be AUX cana-1301 37 15 fundamentally fundamentally ADV cana-1301 37 16 distinct distinct ADJ cana-1301 37 17 sets set NOUN cana-1301 37 18 . . PUNCT cana-1301 38 1 it it PRON cana-1301 38 2 's be AUX cana-1301 38 3 crucial crucial ADJ cana-1301 38 4 to to PART cana-1301 38 5 have have VERB cana-1301 38 6 the the DET cana-1301 38 7 ability ability NOUN cana-1301 38 8 to to PART cana-1301 38 9 handle handle VERB cana-1301 38 10 bipolar bipolar ADJ cana-1301 38 11 information information NOUN cana-1301 38 12 in in ADP cana-1301 38 13 various various ADJ cana-1301 38 14 fields field NOUN cana-1301 38 15 . . PUNCT cana-1301 39 1 positive positive ADJ cana-1301 39 2 information information NOUN cana-1301 39 3 is be AUX cana-1301 39 4 known know VERB cana-1301 39 5 to to PART cana-1301 39 6 reflect reflect VERB cana-1301 39 7 what what PRON cana-1301 39 8 is be AUX cana-1301 39 9 seen see VERB cana-1301 39 10 as as ADP cana-1301 39 11 possible possible ADJ cana-1301 39 12 , , PUNCT cana-1301 39 13 even even ADV cana-1301 39 14 as as ADP cana-1301 39 15 negative negative ADJ cana-1301 39 16 in in SCONJ cana-1301 39 17 order order NOUN cana-1301 39 18 is be AUX cana-1301 39 19 known know VERB cana-1301 39 20 to to PART cana-1301 39 21 stand stand VERB cana-1301 39 22 for for ADP cana-1301 39 23 what what PRON cana-1301 39 24 is be AUX cana-1301 39 25 thought think VERB cana-1301 39 26 to to PART cana-1301 39 27 be be AUX cana-1301 39 28 unfeasible unfeasible ADJ cana-1301 39 29 . . PUNCT cana-1301 40 1 recent recent ADJ cana-1301 40 2 fresh fresh ADJ cana-1301 40 3 studies study NOUN cana-1301 40 4 have have AUX cana-1301 40 5 been be AUX cana-1301 40 6 inspired inspire VERB cana-1301 40 7 by by ADP cana-1301 40 8 this this DET cana-1301 40 9 area area NOUN cana-1301 40 10 in in ADP cana-1301 40 11 various various ADJ cana-1301 40 12 different different ADJ cana-1301 40 13 areas area NOUN cana-1301 40 14 . . PUNCT cana-1301 41 1 bipolarity bipolarity NOUN cana-1301 41 2 also also ADV cana-1301 41 3 occurs occur VERB cana-1301 41 4 because because SCONJ cana-1301 41 5 we we PRON cana-1301 41 6 work work VERB cana-1301 41 7 with with ADP cana-1301 41 8 spatial spatial ADJ cana-1301 41 9 information information NOUN cana-1301 41 10 in in ADP cana-1301 41 11 applications application NOUN cana-1301 41 12 such such ADJ cana-1301 41 13 as as ADP cana-1301 41 14 image image NOUN cana-1301 41 15 processing processing NOUN cana-1301 41 16 and and CCONJ cana-1301 41 17 spatial spatial ADJ cana-1301 41 18 reasoning reasoning NOUN cana-1301 41 19 ; ; PUNCT cana-1301 41 20 as as ADP cana-1301 41 21 a a DET cana-1301 41 22 result result NOUN cana-1301 41 23 , , PUNCT cana-1301 41 24 fuzzy fuzzy ADJ cana-1301 41 25 and and CCONJ cana-1301 41 26 possibility possibility NOUN cana-1301 41 27 formalisms formalism NOUN cana-1301 41 28 for for ADP cana-1301 41 29 bipolar bipolar ADJ cana-1301 41 30 information information NOUN cana-1301 41 31 have have AUX cana-1301 41 32 been be AUX cana-1301 41 33 created create VERB cana-1301 41 34 [ [ PUNCT cana-1301 41 35 12 12 NUM cana-1301 41 36 ] ] PUNCT cana-1301 41 37 . . PUNCT cana-1301 42 1 for for ADP cana-1301 42 2 instance instance NOUN cana-1301 42 3 , , PUNCT cana-1301 42 4 positive positive ADJ cana-1301 42 5 information information NOUN cana-1301 42 6 may may AUX cana-1301 42 7 be be AUX cana-1301 42 8 stated state VERB cana-1301 42 9 as as ADP cana-1301 42 10 a a DET cana-1301 42 11 set set NOUN cana-1301 42 12 of of ADP cana-1301 42 13 potential potential ADJ cana-1301 42 14 places place NOUN cana-1301 42 15 and and CCONJ cana-1301 42 16 negative negative ADJ cana-1301 42 17 in in ADP cana-1301 42 18 order order NOUN cana-1301 42 19 may may AUX cana-1301 42 20 be be AUX cana-1301 42 21 expressed express VERB cana-1301 42 22 as as ADP cana-1301 42 23 a a DET cana-1301 42 24 set set NOUN cana-1301 42 25 of of ADP cana-1301 42 26 unfeasible unfeasible ADJ cana-1301 42 27 places place NOUN cana-1301 42 28 when when SCONJ cana-1301 42 29 evaluating evaluate VERB cana-1301 42 30 the the DET cana-1301 42 31 position position NOUN cana-1301 42 32 of of ADP cana-1301 42 33 an an DET cana-1301 42 34 item item NOUN cana-1301 42 35 in in ADP cana-1301 42 36 a a DET cana-1301 42 37 space space NOUN cana-1301 42 38 . . PUNCT cana-1301 43 1 the the DET cana-1301 43 2 modeling modeling NOUN cana-1301 43 3 of of ADP cana-1301 43 4 real real ADJ cana-1301 43 5 - - PUNCT cana-1301 43 6 time time NOUN cana-1301 43 7 systems system NOUN cana-1301 43 8 where where SCONJ cana-1301 43 9 the the DET cana-1301 43 10 underlying underlie VERB cana-1301 43 11 amount amount NOUN cana-1301 43 12 of of ADP cana-1301 43 13 information information NOUN cana-1301 43 14 fluctuates fluctuate NOUN cana-1301 43 15 with with ADP cana-1301 43 16 different different ADJ cana-1301 43 17 degrees degree NOUN cana-1301 43 18 of of ADP cana-1301 43 19 precision precision NOUN cana-1301 43 20 is be AUX cana-1301 43 21 increasingly increasingly ADV cana-1301 43 22 being be AUX cana-1301 43 23 done do VERB cana-1301 43 24 using use VERB cana-1301 43 25 fuzzy fuzzy ADJ cana-1301 43 26 graph graph NOUN cana-1301 43 27 theory theory NOUN cana-1301 43 28 . . PUNCT cana-1301 44 1 fuzzy fuzzy ADJ cana-1301 44 2 models model NOUN cana-1301 44 3 are be AUX cana-1301 44 4 beginning begin VERB cana-1301 44 5 to to PART cana-1301 44 6 show show VERB cana-1301 44 7 promise promise NOUN cana-1301 44 8 because because SCONJ cana-1301 44 9 they they PRON cana-1301 44 10 aim aim VERB cana-1301 44 11 to to PART cana-1301 44 12 reduce reduce VERB cana-1301 44 13 the the DET cana-1301 44 14 differences difference NOUN cana-1301 44 15 between between ADP cana-1301 44 16 the the DET cana-1301 44 17 symbolic symbolic ADJ cana-1301 44 18 models model NOUN cana-1301 44 19 used use VERB cana-1301 44 20 in in ADP cana-1301 44 21 expert expert NOUN cana-1301 44 22 systems system NOUN cana-1301 44 23 and and CCONJ cana-1301 44 24 the the DET cana-1301 44 25 conventional conventional ADJ cana-1301 44 26 numerical numerical ADJ cana-1301 44 27 models model NOUN cana-1301 44 28 used use VERB cana-1301 44 29 in in ADP cana-1301 44 30 research research NOUN cana-1301 44 31 and and CCONJ cana-1301 44 32 engineering engineering NOUN cana-1301 44 33 . . PUNCT cana-1301 45 1 kaufmann kaufmann PROPN cana-1301 45 2 proposed propose VERB cana-1301 45 3 the the DET cana-1301 45 4 first first ADJ cana-1301 45 5 notion notion NOUN cana-1301 45 6 of of ADP cana-1301 45 7 a a DET cana-1301 45 8 fuzzy fuzzy ADJ cana-1301 45 9 graph graph NOUN cana-1301 45 10 [ [ X cana-1301 45 11 15 15 NUM cana-1301 45 12 ] ] PUNCT cana-1301 45 13 based base VERB cana-1301 45 14 on on ADP cana-1301 45 15 zadeh zadeh PROPN cana-1301 45 16 's 's PART cana-1301 45 17 fuzzy fuzzy ADJ cana-1301 45 18 relations relation NOUN cana-1301 45 19 [ [ X cana-1301 45 20 31 31 NUM cana-1301 45 21 ] ] PUNCT cana-1301 45 22 . . PUNCT cana-1301 46 1 cutnodes cutnode NOUN cana-1301 46 2 , , PUNCT cana-1301 46 3 roadways roadway NOUN cana-1301 46 4 , , PUNCT cana-1301 46 5 connectivity connectivity NOUN cana-1301 46 6 , , PUNCT cana-1301 46 7 trees tree NOUN cana-1301 46 8 , , PUNCT cana-1301 46 9 and and CCONJ cana-1301 46 10 cycles cycle NOUN cana-1301 46 11 are be AUX cana-1301 46 12 only only ADV cana-1301 46 13 a a DET cana-1301 46 14 few few ADJ cana-1301 46 15 of of ADP cana-1301 46 16 the the DET cana-1301 46 17 fundamental fundamental ADJ cana-1301 46 18 graph graph NOUN cana-1301 46 19 - - PUNCT cana-1301 46 20 theoretic theoretic ADJ cana-1301 46 21 concepts concept NOUN cana-1301 46 22 that that PRON cana-1301 46 23 rosenfeld rosenfeld PROPN cana-1301 47 1 [ [ X cana-1301 47 2 22 22 NUM cana-1301 47 3 ] ] PUNCT cana-1301 47 4 proposed propose VERB cana-1301 47 5 as as ADP cana-1301 47 6 having have VERB cana-1301 47 7 a a DET cana-1301 47 8 fuzzy fuzzy ADJ cana-1301 47 9 equivalent equivalent NOUN cana-1301 47 10 . . PUNCT cana-1301 48 1 sunitha sunitha PROPN cana-1301 48 2 and and CCONJ cana-1301 48 3 vijaya vijaya PROPN cana-1301 48 4 kumar kumar PROPN cana-1301 49 1 [ [ X cana-1301 49 2 27 27 NUM cana-1301 49 3 ] ] PUNCT cana-1301 49 4 described describe VERB cana-1301 49 5 fuzzy fuzzy ADJ cana-1301 49 6 trees tree NOUN cana-1301 49 7 , , PUNCT cana-1301 49 8 while while SCONJ cana-1301 49 9 bhattacharya bhattacharya PROPN cana-1301 49 10 [ [ X cana-1301 49 11 10 10 NUM cana-1301 49 12 ] ] PUNCT cana-1301 49 13 made make VERB cana-1301 49 14 some some DET cana-1301 49 15 comments comment NOUN cana-1301 49 16 on on ADP cana-1301 49 17 fuzzy fuzzy ADJ cana-1301 49 18 graphs graph NOUN cana-1301 49 19 . . PUNCT cana-1301 50 1 research research NOUN cana-1301 50 2 in in ADP cana-1301 50 3 artificial artificial ADJ cana-1301 50 4 intelligence intelligence NOUN cana-1301 50 5 and and CCONJ cana-1301 50 6 quantum quantum NOUN cana-1301 50 7 computing computing NOUN cana-1301 50 8 makes make VERB cana-1301 50 9 it it PRON cana-1301 50 10 necessary necessary ADJ cana-1301 50 11 to to PART cana-1301 50 12 introduce introduce VERB cana-1301 50 13 concepts concept NOUN cana-1301 50 14 related relate VERB cana-1301 50 15 to to ADP cana-1301 50 16 logical logical ADJ cana-1301 50 17 causality causality NOUN cana-1301 50 18 and and CCONJ cana-1301 50 19 decision decision NOUN cana-1301 50 20 - - PUNCT cana-1301 50 21 making making NOUN cana-1301 50 22 that that PRON cana-1301 50 23 are be AUX cana-1301 50 24 integral integral ADJ cana-1301 50 25 to to ADP cana-1301 50 26 technical technical ADJ cana-1301 50 27 applications application NOUN cana-1301 50 28 . . PUNCT cana-1301 51 1 this this PRON cana-1301 51 2 can can AUX cana-1301 51 3 be be AUX cana-1301 51 4 to to ADP cana-1301 51 5 some some DET cana-1301 51 6 extent extent NOUN cana-1301 51 7 captured capture VERB cana-1301 51 8 by by ADP cana-1301 51 9 the the DET cana-1301 51 10 m m PROPN cana-1301 51 11 - - PUNCT cana-1301 51 12 bpfg bpfg ADJ cana-1301 51 13 as as SCONJ cana-1301 51 14 this this DET cana-1301 51 15 theory theory NOUN cana-1301 51 16 provides provide VERB cana-1301 51 17 possibility possibility NOUN cana-1301 51 18 for for ADP cana-1301 51 19 a a DET cana-1301 51 20 ternary ternary ADJ cana-1301 51 21 classification classification NOUN cana-1301 51 22 in in ADP cana-1301 51 23 the the DET cana-1301 51 24 place place NOUN cana-1301 51 25 at at ADP cana-1301 51 26 conventional conventional ADJ cana-1301 51 27 binary binary ADJ cana-1301 51 28 classification classification NOUN cana-1301 51 29 . . PUNCT cana-1301 52 1 we we PRON cana-1301 52 2 know know VERB cana-1301 52 3 that that SCONJ cana-1301 52 4 we we PRON cana-1301 52 5 have have VERB cana-1301 52 6 positive positive ADJ cana-1301 52 7 ( ( PUNCT cana-1301 52 8 + + NOUN cana-1301 52 9 ) ) PUNCT cana-1301 52 10 and and CCONJ cana-1301 52 11 negative negative ADJ cana-1301 52 12 ( ( PUNCT cana-1301 52 13 - - INTJ cana-1301 52 14 ) ) PUNCT cana-1301 52 15 as as ADV cana-1301 52 16 well well ADV cana-1301 52 17 - - PUNCT cana-1301 52 18 established establish VERB cana-1301 52 19 bipolar bipolar ADJ cana-1301 52 20 coordinates coordinate NOUN cana-1301 52 21 for for ADP cana-1301 52 22 traditional traditional ADJ cana-1301 52 23 graph graph NOUN cana-1301 52 24 analysis analysis NOUN cana-1301 52 25 . . PUNCT cana-1301 53 1 however however ADV cana-1301 53 2 , , PUNCT cana-1301 53 3 complex complex ADJ cana-1301 53 4 developments development NOUN cana-1301 53 5 in in ADP cana-1301 53 6 the the DET cana-1301 53 7 field field NOUN cana-1301 53 8 of of ADP cana-1301 53 9 science science NOUN cana-1301 53 10 tell tell VERB cana-1301 53 11 us we PRON cana-1301 53 12 that that SCONJ cana-1301 53 13 there there PRON cana-1301 53 14 are be VERB cana-1301 53 15 categories category NOUN cana-1301 53 16 which which PRON cana-1301 53 17 are be AUX cana-1301 53 18 both both ADV cana-1301 53 19 positive positive ADJ cana-1301 53 20 and and CCONJ cana-1301 53 21 negative negative ADJ cana-1301 53 22 at at ADP cana-1301 53 23 the the DET cana-1301 53 24 given give VERB cana-1301 53 25 point point NOUN cana-1301 53 26 of of ADP cana-1301 53 27 time time NOUN cana-1301 53 28 [ [ X cana-1301 53 29 2 2 NUM cana-1301 53 30 - - SYM cana-1301 53 31 6 6 NUM cana-1301 53 32 ] ] PUNCT cana-1301 53 33 . . PUNCT cana-1301 54 1 communications communication NOUN cana-1301 54 2 on on ADP cana-1301 54 3 applied apply VERB cana-1301 54 4 nonlinear nonlinear ADJ cana-1301 54 5 analysis analysis NOUN cana-1301 54 6 issn issn NOUN cana-1301 54 7 : : PUNCT cana-1301 54 8 1074 1074 NUM cana-1301 54 9 - - PUNCT cana-1301 54 10 133x 133x NUM cana-1301 54 11 vol vol NOUN cana-1301 54 12 31 31 NUM cana-1301 54 13 no no NOUN cana-1301 54 14 . . PUNCT cana-1301 55 1 7s 7 NOUN cana-1301 55 2 ( ( PUNCT cana-1301 55 3 2024 2024 NUM cana-1301 55 4 ) ) PUNCT cana-1301 55 5 263 263 NUM cana-1301 55 6 https://internationalpubls.com https://internationalpubls.com X cana-1301 55 7 this this PRON cana-1301 55 8 can can AUX cana-1301 55 9 be be AUX cana-1301 55 10 demonstrated demonstrate VERB cana-1301 55 11 with with ADP cana-1301 55 12 the the DET cana-1301 55 13 example example NOUN cana-1301 55 14 of of ADP cana-1301 55 15 opinion opinion NOUN cana-1301 55 16 making make VERB cana-1301 55 17 behavior behavior NOUN cana-1301 55 18 of of ADP cana-1301 55 19 a a DET cana-1301 55 20 country country NOUN cana-1301 55 21 like like ADP cana-1301 55 22 india india PROPN cana-1301 55 23 . . PUNCT cana-1301 56 1 though though SCONJ cana-1301 56 2 india india PROPN cana-1301 56 3 has have VERB cana-1301 56 4 a a DET cana-1301 56 5 democratically democratically ADV cana-1301 56 6 elected elect VERB cana-1301 56 7 government government NOUN cana-1301 56 8 at at ADP cana-1301 56 9 the the DET cana-1301 56 10 center center NOUN cana-1301 56 11 , , PUNCT cana-1301 56 12 each each DET cana-1301 56 13 state state NOUN cana-1301 56 14 has have VERB cana-1301 56 15 its its PRON cana-1301 56 16 own own ADJ cana-1301 56 17 democratically democratically ADV cana-1301 56 18 elected elect VERB cana-1301 56 19 state state NOUN cana-1301 56 20 governments government NOUN cana-1301 56 21 . . PUNCT cana-1301 57 1 while while SCONJ cana-1301 57 2 opinions opinion NOUN cana-1301 57 3 regarding regard VERB cana-1301 57 4 many many ADJ cana-1301 57 5 central central ADJ cana-1301 57 6 government government NOUN cana-1301 57 7 policies policy NOUN cana-1301 57 8 are be AUX cana-1301 57 9 expressed express VERB cana-1301 57 10 as as ADP cana-1301 57 11 agreements agreement NOUN cana-1301 57 12 or or CCONJ cana-1301 57 13 oppositions opposition NOUN cana-1301 57 14 by by ADP cana-1301 57 15 state state NOUN cana-1301 57 16 governments government NOUN cana-1301 57 17 , , PUNCT cana-1301 57 18 there there PRON cana-1301 57 19 are be VERB cana-1301 57 20 some some DET cana-1301 57 21 central central ADJ cana-1301 57 22 policies policy NOUN cana-1301 57 23 where where SCONJ cana-1301 57 24 there there PRON cana-1301 57 25 is be VERB cana-1301 57 26 a a DET cana-1301 57 27 degree degree NOUN cana-1301 57 28 of of ADP cana-1301 57 29 both both DET cana-1301 57 30 agreement agreement NOUN cana-1301 57 31 and and CCONJ cana-1301 57 32 opposition opposition NOUN cana-1301 57 33 . . PUNCT cana-1301 58 1 let let VERB cana-1301 58 2 us we PRON cana-1301 58 3 explain explain VERB cana-1301 58 4 an an DET cana-1301 58 5 example example NOUN cana-1301 58 6 from from ADP cana-1301 58 7 latest late ADJ cana-1301 58 8 farm farm NOUN cana-1301 58 9 law law NOUN cana-1301 58 10 proposed propose VERB cana-1301 58 11 by by ADP cana-1301 58 12 the the DET cana-1301 58 13 government government NOUN cana-1301 58 14 of of ADP cana-1301 58 15 india india PROPN cana-1301 58 16 with with ADP cana-1301 58 17 all all DET cana-1301 58 18 29 29 NUM cana-1301 58 19 states state NOUN cana-1301 58 20 taken take VERB cana-1301 58 21 as as ADP cana-1301 58 22 an an DET cana-1301 58 23 m m NOUN cana-1301 58 24 - - NOUN cana-1301 58 25 block block NOUN cana-1301 58 26 . . PUNCT cana-1301 59 1 in in ADP cana-1301 59 2 principle principle NOUN cana-1301 59 3 , , PUNCT cana-1301 59 4 this this DET cana-1301 59 5 law law NOUN cana-1301 59 6 was be AUX cana-1301 59 7 directly directly ADV cana-1301 59 8 accepted accept VERB cana-1301 59 9 by by ADP cana-1301 59 10 20 20 NUM cana-1301 59 11 states state NOUN cana-1301 59 12 while while SCONJ cana-1301 59 13 six six NUM cana-1301 59 14 states state NOUN cana-1301 59 15 opposed oppose VERB cana-1301 59 16 it it PRON cana-1301 59 17 . . PUNCT cana-1301 60 1 there there PRON cana-1301 60 2 are be VERB cana-1301 60 3 three three NUM cana-1301 60 4 states state NOUN cana-1301 60 5 which which PRON cana-1301 60 6 expressed express VERB cana-1301 60 7 modifications modification NOUN cana-1301 60 8 in in ADP cana-1301 60 9 the the DET cana-1301 60 10 policy policy NOUN cana-1301 60 11 as as SCONJ cana-1301 60 12 it it PRON cana-1301 60 13 evolved evolve VERB cana-1301 60 14 a a DET cana-1301 60 15 mixed mixed ADJ cana-1301 60 16 response response NOUN cana-1301 60 17 . . PUNCT cana-1301 61 1 this this PRON cana-1301 61 2 is be AUX cana-1301 61 3 important important ADJ cana-1301 61 4 as as SCONJ cana-1301 61 5 it it PRON cana-1301 61 6 influences influence VERB cana-1301 61 7 the the DET cana-1301 61 8 political political ADJ cana-1301 61 9 system system NOUN cana-1301 61 10 and and CCONJ cana-1301 61 11 patterns pattern NOUN cana-1301 61 12 of of ADP cana-1301 61 13 voting voting NOUN cana-1301 61 14 in in ADP cana-1301 61 15 india india PROPN cana-1301 61 16 . . PUNCT cana-1301 62 1 however however ADV cana-1301 62 2 , , PUNCT cana-1301 62 3 in in ADP cana-1301 62 4 each each PRON cana-1301 62 5 of of ADP cana-1301 62 6 the the DET cana-1301 62 7 states state NOUN cana-1301 62 8 that that PRON cana-1301 62 9 have have AUX cana-1301 62 10 accepted accept VERB cana-1301 62 11 the the DET cana-1301 62 12 policy policy NOUN cana-1301 62 13 some some DET cana-1301 62 14 sections section NOUN cana-1301 62 15 of of ADP cana-1301 62 16 the the DET cana-1301 62 17 farmers farmer NOUN cana-1301 62 18 opposed oppose VERB cana-1301 62 19 the the DET cana-1301 62 20 policy policy NOUN cana-1301 62 21 ; ; PUNCT cana-1301 62 22 and and CCONJ cana-1301 62 23 in in ADP cana-1301 62 24 the the DET cana-1301 62 25 states state NOUN cana-1301 62 26 which which PRON cana-1301 62 27 opposed oppose VERB cana-1301 62 28 the the DET cana-1301 62 29 policy policy NOUN cana-1301 62 30 there there PRON cana-1301 62 31 was be VERB cana-1301 62 32 a a DET cana-1301 62 33 call call NOUN cana-1301 62 34 for for ADP cana-1301 62 35 agreement agreement NOUN cana-1301 62 36 from from ADP cana-1301 62 37 some some DET cana-1301 62 38 sections section NOUN cana-1301 62 39 of of ADP cana-1301 62 40 the the DET cana-1301 62 41 farmers farmer NOUN cana-1301 62 42 . . PUNCT cana-1301 63 1 this this PRON cana-1301 63 2 provides provide VERB cana-1301 63 3 a a DET cana-1301 63 4 suitable suitable ADJ cana-1301 63 5 a a DET cana-1301 63 6 set set NOUN cana-1301 63 7 for for ADP cana-1301 63 8 applying apply VERB cana-1301 63 9 the the DET cana-1301 63 10 concept concept NOUN cana-1301 63 11 of of ADP cana-1301 63 12 m m PROPN cana-1301 63 13 - - PUNCT cana-1301 63 14 bpfgs bpfgs PROPN cana-1301 63 15 . . PUNCT cana-1301 64 1 m m NOUN cana-1301 64 2 - - PUNCT cana-1301 64 3 bpfmgs bpfmgs NOUN cana-1301 64 4 , , PUNCT cana-1301 64 5 mbpfpgs mbpfpgs NOUN cana-1301 64 6 , , PUNCT cana-1301 64 7 and and CCONJ cana-1301 64 8 m m NOUN cana-1301 64 9 - - NOUN cana-1301 64 10 bpfdgs bpfdgs NOUN cana-1301 64 11 are be AUX cana-1301 64 12 introduced introduce VERB cana-1301 64 13 , , PUNCT cana-1301 64 14 and and CCONJ cana-1301 64 15 some some PRON cana-1301 64 16 of of ADP cana-1301 64 17 their their PRON cana-1301 64 18 interesting interesting ADJ cana-1301 64 19 aspects aspect NOUN cana-1301 64 20 are be AUX cana-1301 64 21 studied study VERB cana-1301 64 22 . . PUNCT cana-1301 65 1 additionally additionally ADV cana-1301 65 2 , , PUNCT cana-1301 65 3 we we PRON cana-1301 65 4 discuss discuss VERB cana-1301 65 5 isomorphism isomorphism NOUN cana-1301 65 6 across across ADP cana-1301 65 7 m m NOUN cana-1301 65 8 - - NOUN cana-1301 65 9 bpfpgs bpfpgs NOUN cana-1301 65 10 [ [ X cana-1301 65 11 20 20 NUM cana-1301 65 12 , , PUNCT cana-1301 65 13 33 33 NUM cana-1301 65 14 - - SYM cana-1301 65 15 35 35 NUM cana-1301 65 16 ] ] PUNCT cana-1301 65 17 . . PUNCT cana-1301 66 1 in in ADP cana-1301 66 2 this this DET cana-1301 66 3 work work NOUN cana-1301 66 4 , , PUNCT cana-1301 66 5 we we PRON cana-1301 66 6 have have AUX cana-1301 66 7 only only ADV cana-1301 66 8 utilized utilize VERB cana-1301 66 9 accepted accept VERB cana-1301 66 10 terminology terminology NOUN cana-1301 66 11 and and CCONJ cana-1301 66 12 definitions definition NOUN cana-1301 66 13 . . PUNCT cana-1301 67 1 the the DET cana-1301 67 2 readers reader NOUN cana-1301 67 3 are be AUX cana-1301 67 4 directed direct VERB cana-1301 67 5 to to ADP cana-1301 67 6 [ [ X cana-1301 67 7 7 7 NUM cana-1301 67 8 , , PUNCT cana-1301 67 9 8 8 NUM cana-1301 67 10 , , PUNCT cana-1301 67 11 11 11 NUM cana-1301 67 12 , , PUNCT cana-1301 67 13 13 13 NUM cana-1301 67 14 , , PUNCT cana-1301 67 15 14 14 NUM cana-1301 67 16 , , PUNCT cana-1301 67 17 16 16 NUM cana-1301 67 18 - - SYM cana-1301 67 19 18 18 NUM cana-1301 67 20 , , PUNCT cana-1301 67 21 21 21 NUM cana-1301 67 22 , , PUNCT cana-1301 67 23 24 24 NUM cana-1301 67 24 - - SYM cana-1301 67 25 26 26 NUM cana-1301 67 26 , , PUNCT cana-1301 67 27 28 28 NUM cana-1301 67 28 , , PUNCT cana-1301 67 29 29 29 NUM cana-1301 67 30 ] ] PUNCT cana-1301 67 31 extra extra ADJ cana-1301 67 32 symbols symbol NOUN cana-1301 67 33 , , PUNCT cana-1301 67 34 jargon jargon PROPN cana-1301 67 35 , , PUNCT cana-1301 67 36 and and CCONJ cana-1301 67 37 uses uses AUX cana-1301 67 38 not not PART cana-1301 67 39 addressed address VERB cana-1301 67 40 in in ADP cana-1301 67 41 the the DET cana-1301 67 42 research research NOUN cana-1301 67 43 . . PUNCT cana-1301 68 1 2 2 X cana-1301 68 2 . . X cana-1301 68 3 preliminaries preliminary NOUN cana-1301 68 4 the the DET cana-1301 68 5 terms term NOUN cana-1301 68 6 " " PUNCT cana-1301 68 7 m m NOUN cana-1301 68 8 - - NOUN cana-1301 68 9 bpfs bpfs NOUN cana-1301 68 10 , , PUNCT cana-1301 68 11 " " PUNCT cana-1301 68 12 " " PUNCT cana-1301 68 13 m m NOUN cana-1301 68 14 - - PUNCT cana-1301 68 15 bpfg bpfg ADJ cana-1301 68 16 , , PUNCT cana-1301 68 17 " " PUNCT cana-1301 68 18 are be AUX cana-1301 68 19 defined define VERB cana-1301 68 20 in in ADP cana-1301 68 21 this this DET cana-1301 68 22 section section NOUN cana-1301 68 23 . . PUNCT cana-1301 69 1 for for SCONJ cana-1301 69 2 the the DET cana-1301 69 3 m m PROPN cana-1301 69 4 - - PUNCT cana-1301 69 5 bpfg bpfg ADJ cana-1301 69 6 to to PART cana-1301 69 7 be be AUX cana-1301 69 8 generalized generalize VERB cana-1301 69 9 , , PUNCT cana-1301 69 10 an an DET cana-1301 69 11 equivalence equivalence NOUN cana-1301 69 12 criterion criterion NOUN cana-1301 69 13 was be AUX cana-1301 69 14 established establish VERB cana-1301 69 15 . . PUNCT cana-1301 70 1 create create VERB cana-1301 70 2 a a DET cana-1301 70 3 relationship relationship NOUN cana-1301 70 4 of of ADP cana-1301 70 5 equivalence equivalence NOUN cana-1301 70 6 from from ADP cana-1301 70 7 the the DET cana-1301 70 8 given give VERB cana-1301 70 9 set set NOUN cana-1301 70 10 v v NOUN cana-1301 70 11 , , PUNCT cana-1301 70 12 ↔ ↔ PROPN cana-1301 70 13 on on ADP cana-1301 70 14 𝑉 𝑉 PROPN cana-1301 70 15 Γ— Γ— NOUN cana-1301 70 16 𝑉 𝑉 PROPN cana-1301 70 17 βˆ’ βˆ’ PROPN cana-1301 70 18 { { PUNCT cana-1301 70 19 ( ( PUNCT cana-1301 70 20 πœ„ πœ„ PROPN cana-1301 70 21 , , PUNCT cana-1301 70 22 πœ„ πœ„ PROPN cana-1301 70 23 ): ): PUNCT cana-1301 70 24 πœ„ πœ„ PROPN cana-1301 70 25 ∈ ∈ PROPN cana-1301 70 26 𝑉 𝑉 PROPN cana-1301 70 27 } } PUNCT cana-1301 70 28 as as SCONJ cana-1301 70 29 follows follow VERB cana-1301 70 30 : : PUNCT cana-1301 70 31 ( ( PUNCT cana-1301 71 1 πœ„1,πœ…1 πœ„1,πœ…1 PROPN cana-1301 71 2 ) ) PUNCT cana-1301 71 3 ↔ ↔ PROPN cana-1301 71 4 ( ( PUNCT cana-1301 71 5 πœ„2,πœ…2 πœ„2,πœ…2 PROPN cana-1301 71 6 ) ) PUNCT cana-1301 71 7 if if SCONJ cana-1301 71 8 and and CCONJ cana-1301 71 9 only only ADV cana-1301 71 10 if if SCONJ cana-1301 71 11 either either PRON cana-1301 71 12 ( ( PUNCT cana-1301 71 13 πœ„1,πœ…1 πœ„1,πœ…1 PROPN cana-1301 71 14 ) ) PUNCT cana-1301 71 15 = = PRON cana-1301 71 16 ( ( PUNCT cana-1301 71 17 πœ„2,πœ…2 πœ„2,πœ…2 PROPN cana-1301 71 18 ) ) PUNCT cana-1301 71 19 or or CCONJ cana-1301 71 20 πœ„1 πœ„1 NOUN cana-1301 71 21 , , PUNCT cana-1301 71 22 = = SYM cana-1301 71 23 πœ…2 πœ…2 NOUN cana-1301 71 24 , , PUNCT cana-1301 71 25 πœ…1 πœ…1 NOUN cana-1301 71 26 = = SYM cana-1301 71 27 πœ„2 πœ„2 PROPN cana-1301 71 28 , , PUNCT cana-1301 71 29 . . PUNCT cana-1301 72 1 the the DET cana-1301 72 2 equivalence equivalence NOUN cana-1301 72 3 class class NOUN cana-1301 72 4 containing contain VERB cana-1301 72 5 the the DET cana-1301 72 6 components component NOUN cana-1301 72 7 ( ( PUNCT cana-1301 72 8 πœ„ πœ„ PROPN cana-1301 72 9 , , PUNCT cana-1301 72 10 πœ… πœ… NUM cana-1301 72 11 ) ) PUNCT cana-1301 72 12 is be AUX cana-1301 72 13 represented represent VERB cana-1301 72 14 by by ADP cana-1301 72 15 πœ„πœ… πœ„πœ… NOUN cana-1301 72 16 or or CCONJ cana-1301 72 17 πœ…πœ„ πœ…πœ„ NOUN cana-1301 72 18 , , PUNCT cana-1301 72 19 while while SCONJ cana-1301 72 20 the the DET cana-1301 72 21 quotient quotient NOUN cana-1301 72 22 set set NOUN cana-1301 72 23 is be AUX cana-1301 72 24 denoted denote VERB cana-1301 72 25 by by ADP cana-1301 72 26 𝑉2 𝑉2 PROPN cana-1301 72 27 ⃑ ⃑ PROPN cana-1301 72 28 . . PUNCT cana-1301 73 1 definition definition NOUN cana-1301 73 2 2.1 2.1 NUM cana-1301 73 3 : : PUNCT cana-1301 73 4 an an DET cana-1301 73 5 m m NOUN cana-1301 73 6 - - PUNCT cana-1301 73 7 bpfg bpfg ADJ cana-1301 73 8 of of ADP cana-1301 73 9 a a DET cana-1301 73 10 graph graph NOUN cana-1301 73 11 πΊβˆ— πΊβˆ— NOUN cana-1301 73 12 = = SYM cana-1301 73 13 ( ( PUNCT cana-1301 73 14 𝑉 𝑉 PROPN cana-1301 73 15 , , PUNCT cana-1301 73 16 𝐸 𝐸 PROPN cana-1301 73 17 ) ) PUNCT cana-1301 73 18 is be AUX cana-1301 73 19 a a DET cana-1301 73 20 pair pair NOUN cana-1301 73 21 𝐺 𝐺 NOUN cana-1301 73 22 = = SYM cana-1301 73 23 ( ( PUNCT cana-1301 73 24 𝑉 𝑉 PROPN cana-1301 73 25 , , PUNCT cana-1301 73 26 𝑄 𝑄 PROPN cana-1301 73 27 , , PUNCT cana-1301 73 28 𝑅 𝑅 PROPN cana-1301 73 29 ) ) PUNCT cana-1301 73 30 where where SCONJ cana-1301 73 31 𝑄 𝑄 PROPN cana-1301 73 32 = = SYM cana-1301 73 33 〈 〈 PROPN cana-1301 73 34 [ [ X cana-1301 73 35 π‘ƒβ„Žoψq π‘ƒβ„Žoψq PROPN cana-1301 73 36 + + PROPN cana-1301 73 37 , , PUNCT cana-1301 73 38 π‘ƒβ„Žoψq π‘ƒβ„Žoψq PROPN cana-1301 73 39 βˆ’ βˆ’ PROPN cana-1301 73 40 ] ] PUNCT cana-1301 73 41 h=1 h=1 VERB cana-1301 73 42 m m VERB cana-1301 73 43 βŒͺ βŒͺ NOUN cana-1301 73 44 , , PUNCT cana-1301 73 45 π‘ƒβ„Žoψq π‘ƒβ„Žoψq PROPN cana-1301 73 46 + + PROPN cana-1301 73 47 : : PUNCT cana-1301 73 48 v v NOUN cana-1301 73 49 β†’ β†’ SYM cana-1301 73 50 [ [ X cana-1301 73 51 0 0 NUM cana-1301 73 52 , , PUNCT cana-1301 73 53 1 1 NUM cana-1301 73 54 ] ] PUNCT cana-1301 73 55 and and CCONJ cana-1301 73 56 π‘ƒβ„Žoψq π‘ƒβ„Žoψq PROPN cana-1301 73 57 βˆ’ βˆ’ PROPN cana-1301 73 58 : : PUNCT cana-1301 73 59 v v NOUN cana-1301 73 60 β†’ β†’ SYM cana-1301 73 61 [ [ X cana-1301 73 62 βˆ’1 βˆ’1 NOUN cana-1301 73 63 , , PUNCT cana-1301 73 64 0 0 NUM cana-1301 73 65 ] ] X cana-1301 73 66 an an DET cana-1301 73 67 m m NOUN cana-1301 73 68 - - PUNCT cana-1301 73 69 bpfs bpfs NOUN cana-1301 73 70 is be AUX cana-1301 73 71 an an DET cana-1301 73 72 m m NOUN cana-1301 73 73 - - PUNCT cana-1301 73 74 bpfs bpfs NOUN cana-1301 73 75 on on ADP cana-1301 73 76 𝑉 𝑉 PROPN cana-1301 73 77 and and CCONJ cana-1301 73 78 𝑅 𝑅 PROPN cana-1301 73 79 = = PUNCT cana-1301 73 80 〈 〈 PROPN cana-1301 73 81 [ [ X cana-1301 73 82 π‘ƒβ„Žoψr π‘ƒβ„Žoψr PROPN cana-1301 73 83 + + PROPN cana-1301 73 84 , , PUNCT cana-1301 73 85 π‘ƒβ„Žoψr π‘ƒβ„Žoψr PROPN cana-1301 73 86 βˆ’]h=1 βˆ’]h=1 PROPN cana-1301 73 87 m m VERB cana-1301 73 88 βŒͺ βŒͺ NOUN cana-1301 73 89 , , PUNCT cana-1301 73 90 π‘ƒβ„Žoψr π‘ƒβ„Žoψr PROPN cana-1301 73 91 + + PROPN cana-1301 73 92 : : PUNCT cana-1301 73 93 𝑉2 𝑉2 PROPN cana-1301 73 94 ⃑ ⃑ PROPN cana-1301 74 1 β†’ β†’ PUNCT cana-1301 74 2 [ [ X cana-1301 74 3 0 0 NUM cana-1301 74 4 , , PUNCT cana-1301 74 5 1 1 NUM cana-1301 74 6 ] ] PUNCT cana-1301 74 7 and and CCONJ cana-1301 74 8 π‘ƒβ„Žoψq π‘ƒβ„Žoψq PROPN cana-1301 74 9 βˆ’ βˆ’ PROPN cana-1301 74 10 : : PUNCT cana-1301 74 11 𝑉2 𝑉2 PROPN cana-1301 74 12 ⃑ ⃑ PROPN cana-1301 75 1 β†’ β†’ PUNCT cana-1301 75 2 [ [ X cana-1301 75 3 βˆ’1 βˆ’1 NOUN cana-1301 75 4 , , PUNCT cana-1301 75 5 0 0 NUM cana-1301 75 6 ] ] PUNCT cana-1301 75 7 in in ADP cana-1301 75 8 an an DET cana-1301 75 9 m m NOUN cana-1301 75 10 - - PUNCT cana-1301 75 11 bpfs bpfs NOUN cana-1301 75 12 in in ADP cana-1301 75 13 𝑉2 𝑉2 PROPN cana-1301 75 14 ⃑ ⃑ NOUN cana-1301 75 15 such such ADJ cana-1301 75 16 that that SCONJ cana-1301 75 17 π‘ƒβ„Žoψr π‘ƒβ„Žoψr PROPN cana-1301 75 18 + + PROPN cana-1301 75 19 ( ( PUNCT cana-1301 75 20 πœ„πœ… πœ„πœ… NOUN cana-1301 75 21 ) ) PUNCT cana-1301 75 22 ≀ ≀ NOUN cana-1301 75 23 min{π‘ƒβ„Žoψq min{π‘ƒβ„Žoψq PUNCT cana-1301 76 1 + + ADJ cana-1301 76 2 ( ( PUNCT cana-1301 76 3 πœ„ πœ„ NOUN cana-1301 76 4 ) ) PUNCT cana-1301 76 5 , , PUNCT cana-1301 76 6 π‘ƒβ„Žoψq π‘ƒβ„Žoψq PROPN cana-1301 76 7 + + PROPN cana-1301 76 8 ( ( PUNCT cana-1301 76 9 πœ… πœ… NOUN cana-1301 76 10 ) ) PUNCT cana-1301 76 11 } } PUNCT cana-1301 76 12 , , PUNCT cana-1301 76 13 π‘ƒβ„Žoψr π‘ƒβ„Žoψr PROPN cana-1301 76 14 βˆ’(πœ„πœ… βˆ’(πœ„πœ… PROPN cana-1301 76 15 ) ) PUNCT cana-1301 76 16 β‰₯ β‰₯ NOUN cana-1301 76 17 max{π‘ƒβ„Žoψq max{π‘ƒβ„Žoψq PROPN cana-1301 76 18 βˆ’(πœ„ βˆ’(πœ„ PROPN cana-1301 76 19 ) ) PUNCT cana-1301 76 20 , , PUNCT cana-1301 76 21 π‘ƒβ„Žoψq π‘ƒβ„Žoψq PROPN cana-1301 76 22 βˆ’(πœ… βˆ’(πœ… NOUN cana-1301 76 23 ) ) PUNCT cana-1301 76 24 } } PUNCT cana-1301 76 25 for for ADP cana-1301 76 26 all all DET cana-1301 76 27 πœ„πœ… πœ„πœ… PRON cana-1301 76 28 ∈ ∈ PROPN cana-1301 76 29 𝑉2 𝑉2 PROPN cana-1301 76 30 ⃑ ⃑ PROPN cana-1301 76 31 , , PUNCT cana-1301 76 32 β„Ž β„Ž PROPN cana-1301 76 33 = = SYM cana-1301 76 34 1 1 NUM cana-1301 76 35 , , PUNCT cana-1301 76 36 2 2 NUM cana-1301 76 37 , , PUNCT cana-1301 76 38 β‹― β‹― PROPN cana-1301 76 39 , , PUNCT cana-1301 76 40 π‘š π‘š PROPN cana-1301 76 41 and and CCONJ cana-1301 76 42 π‘ƒβ„Žoψr π‘ƒβ„Žoψr PROPN cana-1301 76 43 + + PROPN cana-1301 76 44 ( ( PUNCT cana-1301 76 45 πœ„πœ… πœ„πœ… NOUN cana-1301 76 46 ) ) PUNCT cana-1301 76 47 = = PROPN cana-1301 77 1 π‘ƒβ„Žoψr π‘ƒβ„Žoψr PROPN cana-1301 77 2 βˆ’(πœ„πœ… βˆ’(πœ„πœ… PROPN cana-1301 77 3 ) ) PUNCT cana-1301 77 4 = = SYM cana-1301 77 5 0 0 NUM cana-1301 78 1 for for ADP cana-1301 78 2 all all DET cana-1301 78 3 πœ„πœ… πœ„πœ… PRON cana-1301 78 4 ∈ ∈ PROPN cana-1301 78 5 𝑉2 𝑉2 PROPN cana-1301 78 6 ⃑ ⃑ PROPN cana-1301 78 7 βˆ’ βˆ’ PROPN cana-1301 78 8 𝐸. 𝐸. PROPN cana-1301 78 9 definition definition NOUN cana-1301 78 10 2.2 2.2 NUM cana-1301 78 11 : : PUNCT cana-1301 78 12 let let VERB cana-1301 78 13 𝑍 𝑍 PROPN cana-1301 78 14 β‰  β‰  ADJ cana-1301 78 15 πœ™. πœ™. NOUN cana-1301 78 16 two two NUM cana-1301 78 17 functions function NOUN cana-1301 78 18 , , PUNCT cana-1301 78 19 " " PUNCT cana-1301 78 20 count count VERB cana-1301 78 21 positive positive ADJ cana-1301 78 22 relationship relationship NOUN cana-1301 78 23 " " PUNCT cana-1301 78 24 of of ADP cana-1301 78 25 𝐻(𝐢𝐻 𝐻(𝐢𝐻 PROPN cana-1301 78 26 + + PROPN cana-1301 78 27 ) ) PUNCT cana-1301 78 28 and and CCONJ cana-1301 78 29 " " PUNCT cana-1301 78 30 count count VERB cana-1301 78 31 negative negative ADJ cana-1301 78 32 relationship relationship NOUN cana-1301 78 33 " " PUNCT cana-1301 78 34 of of ADP cana-1301 78 35 𝐻(𝐢𝐻 𝐻(𝐢𝐻 ADP cana-1301 78 36 βˆ’),are βˆ’),are NOUN cana-1301 78 37 used use VERB cana-1301 78 38 to to PART cana-1301 78 39 describe describe VERB cana-1301 78 40 an an DET cana-1301 78 41 m m ADJ cana-1301 78 42 - - ADJ cana-1301 78 43 bipolar bipolar ADJ cana-1301 78 44 fuzzy fuzzy ADJ cana-1301 78 45 multiset multiset NOUN cana-1301 78 46 ( ( PUNCT cana-1301 78 47 m m NOUN cana-1301 78 48 - - ADJ cana-1301 78 49 bpfms bpfms ADJ cana-1301 78 50 ) ) PUNCT cana-1301 78 51 h h NOUN cana-1301 78 52 generated generate VERB cana-1301 78 53 from from ADP cana-1301 78 54 z. z. PROPN cana-1301 78 55 here𝐢𝐻 here𝐢𝐻 PROPN cana-1301 79 1 + + ADV cana-1301 79 2 : : PUNCT cana-1301 79 3 𝑍 𝑍 PROPN cana-1301 79 4 β†’ β†’ SYM cana-1301 79 5 𝑅1and 𝑅1and NUM cana-1301 79 6 𝐢𝐻 𝐢𝐻 PROPN cana-1301 79 7 βˆ’ βˆ’ NOUN cana-1301 79 8 : : PUNCT cana-1301 79 9 𝑍 𝑍 PROPN cana-1301 79 10 β†’ β†’ SYM cana-1301 79 11 𝑅2 𝑅2 NOUN cana-1301 79 12 , , PUNCT cana-1301 79 13 [ [ X cana-1301 79 14 0,1 0,1 NUM cana-1301 79 15 ] ] PUNCT cana-1301 79 16 and and CCONJ cana-1301 79 17 [ [ X cana-1301 79 18 -1,0 -1,0 X cana-1301 79 19 ] ] X cana-1301 79 20 are be AUX cana-1301 79 21 the the DET cana-1301 79 22 intervals interval NOUN cana-1301 79 23 from from ADP cana-1301 79 24 which which PRON cana-1301 79 25 the the DET cana-1301 79 26 collections collection NOUN cana-1301 79 27 of of ADP cana-1301 79 28 every every DET cana-1301 79 29 crisp crisp ADJ cana-1301 79 30 multisite multisite NOUN cana-1301 79 31 , , PUNCT cana-1301 79 32 𝑅1 𝑅1 PROPN cana-1301 79 33 and and CCONJ cana-1301 79 34 𝑅2are 𝑅2are PROPN cana-1301 79 35 chosen choose VERB cana-1301 79 36 . . PUNCT cana-1301 80 1 the the DET cana-1301 80 2 definition definition NOUN cana-1301 80 3 of of ADP cana-1301 80 4 the the DET cana-1301 80 5 positive positive ADJ cana-1301 80 6 sponsorship sponsorship NOUN cana-1301 80 7 sequence sequence NOUN cana-1301 80 8 is be AUX cana-1301 80 9 as as SCONJ cana-1301 80 10 follows follow VERB cana-1301 80 11 : : PUNCT cana-1301 80 12 〈 〈 X cana-1301 80 13 𝑃1oψh 𝑃1oψh ADJ cana-1301 80 14 + + NOUN cana-1301 80 15 ( ( PUNCT cana-1301 80 16 πœ„ πœ„ PROPN cana-1301 80 17 ) ) PUNCT cana-1301 80 18 , , PUNCT cana-1301 80 19 𝑃2oψh 𝑃2oψh PROPN cana-1301 80 20 + + PROPN cana-1301 80 21 ( ( PUNCT cana-1301 80 22 πœ„ πœ„ PROPN cana-1301 80 23 ) ) PUNCT cana-1301 80 24 , , PUNCT cana-1301 80 25 β‹― β‹― PROPN cana-1301 80 26 , , PUNCT cana-1301 80 27 π‘ƒπ‘šoψh π‘ƒπ‘šoψh PROPN cana-1301 80 28 + + PROPN cana-1301 80 29 ( ( PUNCT cana-1301 80 30 πœ„)βŒͺand πœ„)βŒͺand NUM cana-1301 80 31 〈 〈 NOUN cana-1301 80 32 𝑃1oψh 𝑃1oψh NOUN cana-1301 80 33 βˆ’(πœ„ βˆ’(πœ„ NOUN cana-1301 80 34 ) ) PUNCT cana-1301 80 35 , , PUNCT cana-1301 80 36 𝑃2oψh 𝑃2oψh PROPN cana-1301 80 37 βˆ’(πœ„ βˆ’(πœ„ VERB cana-1301 80 38 ) ) PUNCT cana-1301 80 39 , , PUNCT cana-1301 80 40 β‹― β‹― PROPN cana-1301 80 41 , , PUNCT cana-1301 80 42 π‘ƒπ‘šoψh π‘ƒπ‘šoψh PROPN cana-1301 80 43 βˆ’(πœ„)βŒͺwill βˆ’(πœ„)βŒͺwill PUNCT cana-1301 80 44 be be AUX cana-1301 80 45 used use VERB cana-1301 80 46 to to PART cana-1301 80 47 represent represent VERB cana-1301 80 48 the the DET cana-1301 80 49 negative negative ADJ cana-1301 80 50 relationship relationship NOUN cana-1301 80 51 sequence sequence NOUN cana-1301 80 52 . . PUNCT cana-1301 81 1 the the DET cana-1301 81 2 following follow VERB cana-1301 81 3 symbols symbol NOUN cana-1301 81 4 stand stand VERB cana-1301 81 5 for for ADP cana-1301 81 6 an an DET cana-1301 81 7 m m NOUN cana-1301 81 8 - - NOUN cana-1301 81 9 bpfms bpfms ADJ cana-1301 81 10 : : PUNCT cana-1301 81 11 communications communication NOUN cana-1301 81 12 on on ADP cana-1301 81 13 applied apply VERB cana-1301 81 14 nonlinear nonlinear ADJ cana-1301 81 15 analysis analysis NOUN cana-1301 81 16 issn issn NOUN cana-1301 81 17 : : PUNCT cana-1301 81 18 1074 1074 NUM cana-1301 81 19 - - PUNCT cana-1301 81 20 133x 133x NUM cana-1301 81 21 vol vol NOUN cana-1301 81 22 31 31 NUM cana-1301 81 23 no no NOUN cana-1301 81 24 . . PUNCT cana-1301 82 1 7s 7 NOUN cana-1301 82 2 ( ( PUNCT cana-1301 82 3 2024 2024 NUM cana-1301 82 4 ) ) PUNCT cana-1301 82 5 264 264 NUM cana-1301 83 1 https://internationalpubls.com https://internationalpubls.com X cana-1301 83 2 { { PUNCT cana-1301 83 3 πœ„ πœ„ NOUN cana-1301 83 4 : : PUNCT cana-1301 83 5 〈 〈 ADP cana-1301 83 6 [ [ X cana-1301 83 7 𝑃1oψh 𝑃1oψh X cana-1301 83 8 + + ADJ cana-1301 83 9 ( ( PUNCT cana-1301 83 10 πœ„ πœ„ NOUN cana-1301 83 11 ) ) PUNCT cana-1301 83 12 , , PUNCT cana-1301 83 13 𝑃1oψh 𝑃1oψh NOUN cana-1301 83 14 βˆ’(πœ„ βˆ’(πœ„ VERB cana-1301 83 15 ) ) PUNCT cana-1301 83 16 ] ] PUNCT cana-1301 83 17 , , PUNCT cana-1301 83 18 [ [ X cana-1301 83 19 𝑃2oψh 𝑃2oψh X cana-1301 83 20 + + PROPN cana-1301 83 21 ( ( PUNCT cana-1301 83 22 πœ„ πœ„ PROPN cana-1301 83 23 ) ) PUNCT cana-1301 83 24 , , PUNCT cana-1301 83 25 𝑃2oψh 𝑃2oψh PROPN cana-1301 83 26 βˆ’(πœ„ βˆ’(πœ„ VERB cana-1301 83 27 ) ) PUNCT cana-1301 83 28 ] ] PUNCT cana-1301 83 29 , , PUNCT cana-1301 83 30 β‹― β‹― PROPN cana-1301 83 31 , , PUNCT cana-1301 84 1 [ [ X cana-1301 84 2 π‘ƒπ‘šoψh π‘ƒπ‘šoψh NOUN cana-1301 84 3 + + PROPN cana-1301 84 4 ( ( PUNCT cana-1301 84 5 πœ„ πœ„ PROPN cana-1301 84 6 ) ) PUNCT cana-1301 84 7 , , PUNCT cana-1301 84 8 π‘ƒπ‘šoψh π‘ƒπ‘šoψh NOUN cana-1301 84 9 βˆ’(πœ„ βˆ’(πœ„ VERB cana-1301 84 10 ) ) PUNCT cana-1301 84 11 ] ] SYM cana-1301 84 12 βŒͺ βŒͺ NOUN cana-1301 84 13 : : PUNCT cana-1301 84 14 πœ„ πœ„ PROPN cana-1301 84 15 ∈ ∈ PROPN cana-1301 84 16 𝑍 𝑍 PROPN cana-1301 84 17 } } PUNCT cana-1301 84 18 . . PUNCT cana-1301 85 1 3 3 X cana-1301 85 2 . . X cana-1301 85 3 m m ADJ cana-1301 85 4 - - ADJ cana-1301 85 5 bipolar bipolar ADJ cana-1301 85 6 fuzzy fuzzy ADJ cana-1301 85 7 planar planar ADJ cana-1301 85 8 graphs graph NOUN cana-1301 85 9 ( ( PUNCT cana-1301 85 10 m m NOUN cana-1301 85 11 - - PUNCT cana-1301 85 12 bpfpgs bpfpgs NOUN cana-1301 85 13 ) ) PUNCT cana-1301 85 14 using use VERB cana-1301 85 15 the the DET cana-1301 85 16 idea idea NOUN cana-1301 85 17 of of ADP cana-1301 85 18 an an DET cana-1301 85 19 m m NOUN cana-1301 85 20 - - NOUN cana-1301 85 21 bpfms bpfms ADJ cana-1301 85 22 , , PUNCT cana-1301 85 23 we we PRON cana-1301 85 24 first first ADV cana-1301 85 25 propose propose VERB cana-1301 85 26 the the DET cana-1301 85 27 idea idea NOUN cana-1301 85 28 of of ADP cana-1301 85 29 an an DET cana-1301 85 30 m m NOUN cana-1301 85 31 - - NOUN cana-1301 85 32 bpfpg bpfpg NOUN cana-1301 85 33 . . PUNCT cana-1301 86 1 definition definition NOUN cana-1301 86 2 3.1 3.1 NUM cana-1301 86 3 : : PUNCT cana-1301 86 4 let let VERB cana-1301 86 5 𝑉 𝑉 PROPN cana-1301 86 6 β‰  β‰  PROPN cana-1301 86 7 πœ™ πœ™ NOUN cana-1301 86 8 and and CCONJ cana-1301 86 9 𝑄 𝑄 PROPN cana-1301 86 10 = = SYM cana-1301 86 11 〈 〈 PROPN cana-1301 86 12 [ [ X cana-1301 86 13 π‘ƒβ„Žoψq π‘ƒβ„Žoψq PROPN cana-1301 86 14 + + PROPN cana-1301 86 15 , , PUNCT cana-1301 86 16 π‘ƒβ„Žoψq π‘ƒβ„Žoψq PROPN cana-1301 86 17 βˆ’ βˆ’ PROPN cana-1301 86 18 ] ] PUNCT cana-1301 86 19 h=1 h=1 VERB cana-1301 86 20 m m VERB cana-1301 86 21 βŒͺ βŒͺ NOUN cana-1301 86 22 be be VERB cana-1301 86 23 an an DET cana-1301 86 24 m m NOUN cana-1301 86 25 - - NOUN cana-1301 86 26 bpfms bpfms ADJ cana-1301 86 27 on on ADP cana-1301 86 28 𝑉 𝑉 PROPN cana-1301 86 29 . . PUNCT cana-1301 87 1 let let VERB cana-1301 87 2 𝑅 𝑅 PROPN cana-1301 87 3 = = PROPN cana-1301 87 4 { { PUNCT cana-1301 87 5 πœ„πœ… πœ„πœ… NOUN cana-1301 87 6 〈 〈 PROPN cana-1301 87 7 [ [ X cana-1301 87 8 π‘ƒβ„Žoψr π‘ƒβ„Žoψr PROPN cana-1301 87 9 + + PROPN cana-1301 87 10 ( ( PUNCT cana-1301 87 11 πœ„πœ…)𝑒 πœ„πœ…)𝑒 NOUN cana-1301 87 12 , , PUNCT cana-1301 87 13 π‘ƒβ„Žoψr π‘ƒβ„Žoψr PROPN cana-1301 87 14 βˆ’(πœ„πœ…)𝑒 βˆ’(πœ„πœ…)𝑒 PROPN cana-1301 87 15 ] ] PUNCT cana-1301 87 16 h=1 h=1 NOUN cana-1301 87 17 m m VERB cana-1301 87 18 βŒͺ βŒͺ NOUN cana-1301 87 19 , , PUNCT cana-1301 87 20 𝑒 𝑒 X cana-1301 87 21 = = SYM cana-1301 87 22 1,2 1,2 NUM cana-1301 87 23 , , PUNCT cana-1301 87 24 β‹― β‹― VERB cana-1301 87 25 , , PUNCT cana-1301 87 26 𝑑|πœ„πœ… 𝑑|πœ„πœ… NOUN cana-1301 87 27 ∈ ∈ ADP cana-1301 87 28 𝑉 𝑉 PROPN cana-1301 87 29 Γ— Γ— NOUN cana-1301 87 30 𝑉 𝑉 PROPN cana-1301 87 31 } } PUNCT cana-1301 87 32 be be VERB cana-1301 87 33 an an DET cana-1301 87 34 m m NOUN cana-1301 87 35 - - NOUN cana-1301 87 36 bpfms bpfms NOUN cana-1301 87 37 of of ADP cana-1301 87 38 𝑉 𝑉 PROPN cana-1301 87 39 Γ— Γ— PROPN cana-1301 87 40 𝑉 𝑉 PROPN cana-1301 87 41 such such ADJ cana-1301 87 42 that that SCONJ cana-1301 87 43 π‘ƒβ„Žoψr π‘ƒβ„Žoψr PROPN cana-1301 87 44 + + PROPN cana-1301 87 45 ( ( PUNCT cana-1301 87 46 πœ„πœ…)𝑒 πœ„πœ…)𝑒 NOUN cana-1301 87 47 ≀ ≀ PUNCT cana-1301 87 48 min{π‘ƒβ„Žoψq min{π‘ƒβ„Žoψq X cana-1301 88 1 + + ADJ cana-1301 88 2 ( ( PUNCT cana-1301 88 3 πœ„ πœ„ NOUN cana-1301 88 4 ) ) PUNCT cana-1301 88 5 , , PUNCT cana-1301 88 6 π‘ƒβ„Žoψq π‘ƒβ„Žoψq PROPN cana-1301 88 7 + + PROPN cana-1301 88 8 ( ( PUNCT cana-1301 88 9 πœ… πœ… NOUN cana-1301 88 10 ) ) PUNCT cana-1301 88 11 } } PUNCT cana-1301 88 12 , , PUNCT cana-1301 88 13 π‘ƒβ„Žoψr π‘ƒβ„Žoψr PROPN cana-1301 88 14 βˆ’(πœ„πœ…)𝑒 βˆ’(πœ„πœ…)𝑒 PROPN cana-1301 88 15 β‰₯ β‰₯ NOUN cana-1301 88 16 max{π‘ƒβ„Žoψq max{π‘ƒβ„Žoψq PROPN cana-1301 88 17 βˆ’(πœ„ βˆ’(πœ„ PROPN cana-1301 88 18 ) ) PUNCT cana-1301 88 19 , , PUNCT cana-1301 88 20 π‘ƒβ„Žoψq π‘ƒβ„Žoψq PROPN cana-1301 88 21 βˆ’(πœ… βˆ’(πœ… NOUN cana-1301 88 22 ) ) PUNCT cana-1301 88 23 } } PUNCT cana-1301 88 24 for for ADP cana-1301 88 25 all all DET cana-1301 88 26 πœ„ πœ„ PROPN cana-1301 88 27 , , PUNCT cana-1301 88 28 πœ… πœ… PROPN cana-1301 88 29 ∈ ∈ PROPN cana-1301 88 30 𝑉 𝑉 PROPN cana-1301 88 31 , , PUNCT cana-1301 88 32 𝑒 𝑒 X cana-1301 88 33 = = SYM cana-1301 88 34 1,2 1,2 NUM cana-1301 88 35 , , PUNCT cana-1301 88 36 β‹― β‹― PROPN cana-1301 88 37 , , PUNCT cana-1301 88 38 𝑑 𝑑 PROPN cana-1301 88 39 and and CCONJ cana-1301 88 40 β„Ž β„Ž X cana-1301 88 41 = = SYM cana-1301 88 42 1 1 NUM cana-1301 88 43 , , PUNCT cana-1301 88 44 2 2 NUM cana-1301 88 45 , , PUNCT cana-1301 88 46 β‹― β‹― PROPN cana-1301 88 47 , , PUNCT cana-1301 88 48 π‘š.then𝐺is π‘š.then𝐺is PROPN cana-1301 88 49 called call VERB cana-1301 88 50 an an DET cana-1301 88 51 m m NOUN cana-1301 88 52 - - NOUN cana-1301 88 53 bpfmg bpfmg PROPN cana-1301 88 54 . . PUNCT cana-1301 89 1 remember remember VERB cana-1301 89 2 that that SCONJ cana-1301 89 3 there there PRON cana-1301 89 4 could could AUX cana-1301 89 5 be be AUX cana-1301 89 6 a a DET cana-1301 89 7 number number NOUN cana-1301 89 8 of of ADP cana-1301 89 9 lines line NOUN cana-1301 89 10 linking link VERB cana-1301 89 11 the the DET cana-1301 89 12 nodes node NOUN cana-1301 89 13 πœ„ πœ„ X cana-1301 89 14 and and CCONJ cana-1301 89 15 πœ… πœ… X cana-1301 89 16 . . PUNCT cana-1301 90 1 [ [ X cana-1301 90 2 π‘ƒβ„Žoψr π‘ƒβ„Žoψr PROPN cana-1301 90 3 + + PROPN cana-1301 90 4 ( ( PUNCT cana-1301 90 5 πœ„πœ…)𝑒 πœ„πœ…)𝑒 NOUN cana-1301 90 6 , , PUNCT cana-1301 90 7 π‘ƒβ„Žoψr π‘ƒβ„Žoψr PROPN cana-1301 90 8 βˆ’(πœ„πœ…)𝑒]β„Ž=1 βˆ’(πœ„πœ…)𝑒]β„Ž=1 PROPN cana-1301 90 9 π‘š π‘š PROPN cana-1301 90 10 represent represent VERB cana-1301 90 11 , , PUNCT cana-1301 90 12 respectively respectively ADV cana-1301 90 13 , , PUNCT cana-1301 90 14 the the DET cana-1301 90 15 positive positive ADJ cana-1301 90 16 and and CCONJ cana-1301 90 17 negative negative ADJ cana-1301 90 18 relationship relationship NOUN cana-1301 90 19 values value NOUN cana-1301 90 20 of of ADP cana-1301 90 21 the the DET cana-1301 90 22 line line NOUN cana-1301 90 23 πœ„πœ… πœ„πœ… NOUN cana-1301 90 24 in in ADP cana-1301 90 25 𝐺. 𝐺. PROPN cana-1301 90 26 𝑒 𝑒 NOUN cana-1301 90 27 stands stand VERB cana-1301 90 28 for for ADP cana-1301 90 29 the the DET cana-1301 90 30 quantity quantity NOUN cana-1301 90 31 of of ADP cana-1301 90 32 lines line NOUN cana-1301 90 33 connecting connect VERB cana-1301 90 34 the the DET cana-1301 90 35 nodes node NOUN cana-1301 90 36 . . PUNCT cana-1301 91 1 𝑅 𝑅 NOUN cana-1301 91 2 is be AUX cana-1301 91 3 referred refer VERB cana-1301 91 4 to to ADP cana-1301 91 5 as as ADP cana-1301 91 6 being be AUX cana-1301 91 7 mbpfml mbpfml NOUN cana-1301 91 8 set set VERB cana-1301 91 9 in in ADP cana-1301 91 10 m m NOUN cana-1301 91 11 - - ADJ cana-1301 91 12 bpfmg bpfmg ADJ cana-1301 91 13 𝐺 𝐺 NOUN cana-1301 91 14 . . PUNCT cana-1301 92 1 π‘ƒβ„Žoψr π‘ƒβ„Žoψr PROPN cana-1301 92 2 + + PROPN cana-1301 92 3 ( ( PUNCT cana-1301 92 4 πœ„πœ…)𝑒 πœ„πœ…)𝑒 NOUN cana-1301 92 5 = = SYM cana-1301 92 6 0 0 PUNCT cana-1301 92 7 = = SYM cana-1301 92 8 π‘ƒβ„Žoψr π‘ƒβ„Žoψr PROPN cana-1301 92 9 βˆ’(πœ„πœ…)𝑒 βˆ’(πœ„πœ…)𝑒 PROPN cana-1301 92 10 for for ADP cana-1301 92 11 all all DET cana-1301 92 12 πœ„πœ… πœ„πœ… DET cana-1301 92 13 ∈ ∈ PROPN cana-1301 92 14 𝑉 𝑉 PROPN cana-1301 92 15 Γ— Γ— NOUN cana-1301 92 16 𝑉 𝑉 PROPN cana-1301 92 17 βˆ’ βˆ’ PROPN cana-1301 92 18 𝐸 𝐸 PROPN cana-1301 92 19 , , PUNCT cana-1301 92 20 0 0 NUM cana-1301 92 21 ≀ ≀ NOUN cana-1301 93 1 π‘ƒβ„Žoψr π‘ƒβ„Žoψr PROPN cana-1301 93 2 + + PROPN cana-1301 93 3 ( ( PUNCT cana-1301 93 4 πœ„πœ…)𝑒 πœ„πœ…)𝑒 NOUN cana-1301 93 5 ≀ ≀ NUM cana-1301 93 6 1 1 NUM cana-1301 93 7 , , PUNCT cana-1301 93 8 βˆ’1 βˆ’1 NOUN cana-1301 93 9 ≀ ≀ NOUN cana-1301 94 1 π‘ƒβ„Žoψr π‘ƒβ„Žoψr PROPN cana-1301 94 2 βˆ’(πœ„πœ…)𝑒 βˆ’(πœ„πœ…)𝑒 PROPN cana-1301 94 3 ≀ ≀ NOUN cana-1301 94 4 0 0 NUM cana-1301 95 1 for for ADP cana-1301 95 2 β„Ž β„Ž PROPN cana-1301 95 3 = = SYM cana-1301 95 4 1 1 NUM cana-1301 95 5 , , PUNCT cana-1301 95 6 2 2 NUM cana-1301 95 7 , , PUNCT cana-1301 95 8 β‹― β‹― PROPN cana-1301 95 9 , , PUNCT cana-1301 95 10 π‘š π‘š PROPN cana-1301 95 11 . . PUNCT cana-1301 95 12 example example NOUN cana-1301 95 13 1 1 NUM cana-1301 95 14 : : PUNCT cana-1301 95 15 take take VERB cana-1301 95 16 a a DET cana-1301 95 17 multigraph multigraph NOUN cana-1301 95 18 of of ADP cana-1301 95 19 πΊβˆ— πΊβˆ— NOUN cana-1301 95 20 = = SYM cana-1301 95 21 ( ( PUNCT cana-1301 95 22 𝑉 𝑉 PROPN cana-1301 95 23 , , PUNCT cana-1301 95 24 𝐸 𝐸 PROPN cana-1301 95 25 ) ) PUNCT cana-1301 95 26 such such ADJ cana-1301 95 27 that that PRON cana-1301 95 28 𝑉 𝑉 PROPN cana-1301 95 29 = = PUNCT cana-1301 95 30 { { PUNCT cana-1301 95 31 πœ„ πœ„ PROPN cana-1301 95 32 , , PUNCT cana-1301 95 33 πœ… πœ… PROPN cana-1301 95 34 , , PUNCT cana-1301 95 35 𝜐 𝜐 NOUN cana-1301 95 36 , , PUNCT cana-1301 95 37 𝜏 𝜏 NOUN cana-1301 95 38 } } PUNCT cana-1301 95 39 , , PUNCT cana-1301 95 40 𝐸 𝐸 PROPN cana-1301 95 41 = = SYM cana-1301 95 42 { { PUNCT cana-1301 95 43 πœ„πœ… πœ„πœ… NOUN cana-1301 95 44 , , PUNCT cana-1301 95 45 , , PUNCT cana-1301 95 46 πœ…πœ πœ…πœ NOUN cana-1301 95 47 , , PUNCT cana-1301 95 48 𝜐𝜏 𝜐𝜏 ADP cana-1301 95 49 , , PUNCT cana-1301 95 50 πœ„πœ πœ„πœ ADP cana-1301 95 51 } } PUNCT cana-1301 95 52 . . PUNCT cana-1301 96 1 let let VERB cana-1301 96 2 𝑄 𝑄 PRON cana-1301 96 3 = = SYM cana-1301 96 4 〈 〈 PROPN cana-1301 96 5 [ [ X cana-1301 96 6 π‘ƒβ„Žoψq π‘ƒβ„Žoψq PROPN cana-1301 96 7 + + PROPN cana-1301 96 8 , , PUNCT cana-1301 96 9 π‘ƒβ„Žoψq π‘ƒβ„Žoψq PROPN cana-1301 96 10 βˆ’ βˆ’ PROPN cana-1301 96 11 ] ] PUNCT cana-1301 96 12 h=1 h=1 VERB cana-1301 96 13 m m VERB cana-1301 96 14 βŒͺ βŒͺ NOUN cana-1301 96 15 be be VERB cana-1301 96 16 an an DET cana-1301 96 17 m m NOUN cana-1301 96 18 - - PUNCT cana-1301 96 19 bpfs bpfs NOUN cana-1301 96 20 on on ADP cana-1301 96 21 𝑉 𝑉 PROPN cana-1301 96 22 and and CCONJ cana-1301 96 23 let let VERB cana-1301 96 24 𝑅 𝑅 PROPN cana-1301 96 25 = = PROPN cana-1301 96 26 { { PUNCT cana-1301 96 27 πœ„πœ… πœ„πœ… NOUN cana-1301 96 28 〈 〈 PROPN cana-1301 96 29 [ [ X cana-1301 96 30 π‘ƒβ„Žoψr π‘ƒβ„Žoψr PROPN cana-1301 96 31 + + PROPN cana-1301 96 32 ( ( PUNCT cana-1301 96 33 πœ„πœ…)𝑒 πœ„πœ…)𝑒 NOUN cana-1301 96 34 , , PUNCT cana-1301 96 35 π‘ƒβ„Žoψr π‘ƒβ„Žoψr PROPN cana-1301 96 36 βˆ’(πœ„πœ…)𝑒 βˆ’(πœ„πœ…)𝑒 PROPN cana-1301 96 37 ] ] PUNCT cana-1301 96 38 h=1 h=1 NOUN cana-1301 96 39 m m VERB cana-1301 96 40 βŒͺ βŒͺ NOUN cana-1301 96 41 , , PUNCT cana-1301 96 42 𝑒 𝑒 X cana-1301 96 43 = = SYM cana-1301 96 44 1,2 1,2 NUM cana-1301 96 45 , , PUNCT cana-1301 96 46 β‹― β‹― VERB cana-1301 96 47 , , PUNCT cana-1301 96 48 𝑑|πœ„πœ… 𝑑|πœ„πœ… NOUN cana-1301 96 49 ∈ ∈ ADP cana-1301 96 50 𝑉 𝑉 PROPN cana-1301 96 51 Γ— Γ— PROPN cana-1301 96 52 𝑉}be 𝑉}be NOUN cana-1301 96 53 an an DET cana-1301 96 54 m m NOUN cana-1301 96 55 - - PUNCT cana-1301 96 56 bpfml bpfml NOUN cana-1301 96 57 set set NOUN cana-1301 96 58 of of ADP cana-1301 96 59 𝑉 𝑉 PROPN cana-1301 96 60 Γ— Γ— PROPN cana-1301 96 61 𝑉 𝑉 PROPN cana-1301 96 62 defined define VERB cana-1301 96 63 by by ADP cana-1301 96 64 fig fig NOUN cana-1301 96 65 . . PUNCT cana-1301 97 1 1 1 NUM cana-1301 97 2 m m ADJ cana-1301 97 3 - - ADJ cana-1301 97 4 bipolar bipolar ADJ cana-1301 97 5 fuzzy fuzzy ADJ cana-1301 97 6 multigraph multigraph NOUN cana-1301 97 7 it it PRON cana-1301 97 8 is be AUX cana-1301 97 9 clear clear ADJ cana-1301 97 10 from from ADP cana-1301 97 11 fig fig NOUN cana-1301 97 12 . . PUNCT cana-1301 98 1 1 1 NUM cana-1301 98 2 through through ADP cana-1301 98 3 simple simple ADJ cana-1301 98 4 calculations calculation NOUN cana-1301 98 5 that that SCONJ cana-1301 98 6 it it PRON cana-1301 98 7 is be AUX cana-1301 98 8 an an DET cana-1301 98 9 m m NOUN cana-1301 98 10 - - NOUN cana-1301 98 11 bpfmg bpfmg NOUN cana-1301 98 12 . . PUNCT cana-1301 99 1 communications communication NOUN cana-1301 99 2 on on ADP cana-1301 99 3 applied apply VERB cana-1301 99 4 nonlinear nonlinear ADJ cana-1301 99 5 analysis analysis NOUN cana-1301 99 6 issn issn NOUN cana-1301 99 7 : : PUNCT cana-1301 99 8 1074 1074 NUM cana-1301 99 9 - - PUNCT cana-1301 99 10 133x 133x NUM cana-1301 99 11 vol vol NOUN cana-1301 99 12 31 31 NUM cana-1301 99 13 no no NOUN cana-1301 99 14 . . PUNCT cana-1301 100 1 7s 7 NOUN cana-1301 100 2 ( ( PUNCT cana-1301 100 3 2024 2024 NUM cana-1301 100 4 ) ) PUNCT cana-1301 100 5 265 265 NUM cana-1301 100 6 https://internationalpubls.com https://internationalpubls.com X cana-1301 100 7 here here ADV cana-1301 100 8 𝑄 𝑄 PROPN cana-1301 100 9 = = PUNCT cana-1301 100 10 { { PUNCT cana-1301 100 11 πœ„ πœ„ X cana-1301 100 12 〈 〈 PROPN cana-1301 100 13 [ [ X cana-1301 100 14 0.8,βˆ’0.3],[0.6,βˆ’0.8 0.8,βˆ’0.3],[0.6,βˆ’0.8 NUM cana-1301 100 15 ] ] X cana-1301 100 16 βŒͺ βŒͺ NOUN cana-1301 100 17 , , PUNCT cana-1301 100 18 πœ… πœ… NOUN cana-1301 100 19 〈 〈 PROPN cana-1301 100 20 [ [ X cana-1301 100 21 0.6,βˆ’0.7],[0.8,βˆ’0.9 0.6,βˆ’0.7],[0.8,βˆ’0.9 NUM cana-1301 100 22 ] ] X cana-1301 100 23 βŒͺ βŒͺ NOUN cana-1301 100 24 , , PUNCT cana-1301 100 25 𝜐 𝜐 PROPN cana-1301 100 26 〈 〈 PROPN cana-1301 100 27 [ [ X cana-1301 100 28 0.9,βˆ’0.3],[0.8,βˆ’0.4 0.9,βˆ’0.3],[0.8,βˆ’0.4 NUM cana-1301 100 29 ] ] X cana-1301 100 30 βŒͺ βŒͺ NOUN cana-1301 100 31 , , PUNCT cana-1301 100 32 𝜏 𝜏 NOUN cana-1301 100 33 〈 〈 PROPN cana-1301 100 34 [ [ X cana-1301 100 35 0.5,βˆ’0.8],[0.7,βˆ’0.3 0.5,βˆ’0.8],[0.7,βˆ’0.3 NUM cana-1301 100 36 ] ] X cana-1301 100 37 βŒͺ βŒͺ NOUN cana-1301 100 38 } } PUNCT cana-1301 100 39 , , PUNCT cana-1301 100 40 𝑅 𝑅 PROPN cana-1301 100 41 = = PROPN cana-1301 100 42 { { PUNCT cana-1301 100 43 πœ„πœ… πœ„πœ… X cana-1301 100 44 〈 〈 PROPN cana-1301 100 45 [ [ X cana-1301 100 46 0.4,βˆ’0.2],[0.5,βˆ’0.2 0.4,βˆ’0.2],[0.5,βˆ’0.2 NUM cana-1301 100 47 ] ] X cana-1301 100 48 βŒͺ βŒͺ NOUN cana-1301 100 49 , , PUNCT cana-1301 100 50 πœ„πœ… πœ„πœ… NOUN cana-1301 100 51 〈 〈 PROPN cana-1301 100 52 [ [ NOUN cana-1301 100 53 0.3,βˆ’0.1],[0.3,βˆ’0.4 0.3,βˆ’0.1],[0.3,βˆ’0.4 NOUN cana-1301 100 54 ] ] X cana-1301 100 55 βŒͺ βŒͺ NOUN cana-1301 100 56 , , PUNCT cana-1301 100 57 πœ…πœ πœ…πœ ADP cana-1301 100 58 〈 〈 PROPN cana-1301 100 59 [ [ X cana-1301 100 60 0.3,βˆ’0.2],[0.6,βˆ’0.1 0.3,βˆ’0.2],[0.6,βˆ’0.1 ADJ cana-1301 100 61 ] ] X cana-1301 100 62 βŒͺ βŒͺ NOUN cana-1301 100 63 , , PUNCT cana-1301 100 64 πœ…πœ πœ…πœ ADP cana-1301 100 65 〈 〈 PROPN cana-1301 100 66 [ [ NOUN cana-1301 100 67 0.2,βˆ’0.1],[0.4,βˆ’0.2 0.2,βˆ’0.1],[0.4,βˆ’0.2 NUM cana-1301 100 68 ] ] X cana-1301 100 69 βŒͺ βŒͺ NOUN cana-1301 100 70 , , PUNCT cana-1301 100 71 𝜐𝜏 𝜐𝜏 ADP cana-1301 100 72 〈 〈 PROPN cana-1301 100 73 [ [ X cana-1301 100 74 0.4,βˆ’0.2],[0.5,βˆ’0.1 0.4,βˆ’0.2],[0.5,βˆ’0.1 ADJ cana-1301 100 75 ] ] X cana-1301 100 76 βŒͺ βŒͺ NOUN cana-1301 100 77 , , PUNCT cana-1301 100 78 πœ„πœ πœ„πœ NOUN cana-1301 100 79 〈 〈 PROPN cana-1301 100 80 [ [ X cana-1301 100 81 0.1,βˆ’0.2],[0.5,βˆ’0.2 0.1,βˆ’0.2],[0.5,βˆ’0.2 NUM cana-1301 100 82 ] ] X cana-1301 100 83 βŒͺ βŒͺ NOUN cana-1301 100 84 } } PUNCT cana-1301 100 85 . . PUNCT cana-1301 101 1 definition definition NOUN cana-1301 101 2 3.2 3.2 NUM cana-1301 101 3 : : PUNCT cana-1301 101 4 let let VERB cana-1301 101 5 𝑅 𝑅 PROPN cana-1301 101 6 = = PROPN cana-1301 101 7 { { PUNCT cana-1301 101 8 πœ„πœ… πœ„πœ… NOUN cana-1301 101 9 〈 〈 PROPN cana-1301 101 10 [ [ X cana-1301 101 11 π‘ƒβ„Žoψr π‘ƒβ„Žoψr PROPN cana-1301 101 12 + + PROPN cana-1301 101 13 ( ( PUNCT cana-1301 101 14 πœ„πœ…)𝑒 πœ„πœ…)𝑒 NOUN cana-1301 101 15 , , PUNCT cana-1301 101 16 π‘ƒβ„Žoψr π‘ƒβ„Žoψr PROPN cana-1301 101 17 βˆ’(πœ„πœ…)𝑒 βˆ’(πœ„πœ…)𝑒 PROPN cana-1301 101 18 ] ] PUNCT cana-1301 101 19 h=1 h=1 NOUN cana-1301 101 20 m m VERB cana-1301 101 21 βŒͺ βŒͺ NOUN cana-1301 101 22 , , PUNCT cana-1301 101 23 𝑒 𝑒 X cana-1301 101 24 = = SYM cana-1301 101 25 1,2 1,2 NUM cana-1301 101 26 , , PUNCT cana-1301 101 27 β‹― β‹― VERB cana-1301 101 28 , , PUNCT cana-1301 101 29 𝑑|πœ„πœ… 𝑑|πœ„πœ… NOUN cana-1301 101 30 ∈ ∈ ADP cana-1301 101 31 𝑉 𝑉 PROPN cana-1301 101 32 Γ— Γ— PROPN cana-1301 101 33 𝑉}be 𝑉}be NOUN cana-1301 101 34 an an DET cana-1301 101 35 m m NOUN cana-1301 101 36 - - PUNCT cana-1301 101 37 bpfme bpfme NOUN cana-1301 101 38 set set NOUN cana-1301 101 39 in in ADP cana-1301 101 40 an an DET cana-1301 101 41 m m NOUN cana-1301 101 42 - - NOUN cana-1301 101 43 bpfmg bpfmg NOUN cana-1301 101 44 𝐺. 𝐺. NOUN cana-1301 101 45 a a DET cana-1301 101 46 multiline multiline ADJ cana-1301 101 47 πœ„πœ… πœ„πœ… NOUN cana-1301 101 48 of of ADP cana-1301 101 49 𝐺is 𝐺is PROPN cana-1301 101 50 strong strong ADJ cana-1301 101 51 if if SCONJ cana-1301 101 52 1 1 NUM cana-1301 101 53 2 2 NUM cana-1301 101 54 min{π‘ƒβ„Žoψq min{π‘ƒβ„Žoψq NOUN cana-1301 101 55 + + ADJ cana-1301 101 56 ( ( PUNCT cana-1301 101 57 πœ„ πœ„ NOUN cana-1301 101 58 ) ) PUNCT cana-1301 101 59 , , PUNCT cana-1301 101 60 π‘ƒβ„Žoψq π‘ƒβ„Žoψq PROPN cana-1301 101 61 + + PROPN cana-1301 101 62 ( ( PUNCT cana-1301 101 63 πœ… πœ… NOUN cana-1301 101 64 ) ) PUNCT cana-1301 101 65 } } PUNCT cana-1301 101 66 ≀ ≀ NOUN cana-1301 102 1 π‘ƒβ„Žoψr π‘ƒβ„Žoψr PROPN cana-1301 102 2 + + PROPN cana-1301 102 3 ( ( PUNCT cana-1301 102 4 πœ„πœ…)𝑒 πœ„πœ…)𝑒 NOUN cana-1301 102 5 , , PUNCT cana-1301 102 6 1 1 NUM cana-1301 102 7 2 2 NUM cana-1301 102 8 max{π‘ƒβ„Žoψq max{π‘ƒβ„Žoψq NOUN cana-1301 102 9 βˆ’(πœ„ βˆ’(πœ„ NOUN cana-1301 102 10 ) ) PUNCT cana-1301 102 11 , , PUNCT cana-1301 102 12 π‘ƒβ„Žoψq π‘ƒβ„Žoψq PROPN cana-1301 102 13 βˆ’(πœ… βˆ’(πœ… NOUN cana-1301 102 14 ) ) PUNCT cana-1301 102 15 } } PUNCT cana-1301 102 16 ≀ ≀ NOUN cana-1301 103 1 π‘ƒβ„Žoψr π‘ƒβ„Žoψr PROPN cana-1301 103 2 βˆ’(πœ„πœ…)𝑒 βˆ’(πœ„πœ…)𝑒 PROPN cana-1301 103 3 for for ADP cana-1301 103 4 all all PRON cana-1301 103 5 𝑒 𝑒 PART cana-1301 103 6 = = SYM cana-1301 103 7 1,2 1,2 NUM cana-1301 103 8 , , PUNCT cana-1301 103 9 β‹― β‹― PROPN cana-1301 103 10 , , PUNCT cana-1301 103 11 𝑑 𝑑 PROPN cana-1301 103 12 and and CCONJ cana-1301 103 13 β„Ž β„Ž X cana-1301 103 14 = = SYM cana-1301 103 15 1 1 NUM cana-1301 103 16 , , PUNCT cana-1301 103 17 2 2 NUM cana-1301 103 18 , , PUNCT cana-1301 103 19 β‹― β‹― NOUN cana-1301 103 20 , , PUNCT cana-1301 103 21 π‘š. π‘š. ADJ cana-1301 103 22 definition definition NOUN cana-1301 103 23 3.3 3.3 NUM cana-1301 103 24 : : PUNCT cana-1301 103 25 let let VERB cana-1301 103 26 𝑅 𝑅 PROPN cana-1301 103 27 = = PROPN cana-1301 103 28 { { PUNCT cana-1301 103 29 πœ„πœ… πœ„πœ… NOUN cana-1301 103 30 〈 〈 PROPN cana-1301 103 31 [ [ X cana-1301 103 32 π‘ƒβ„Žoψr π‘ƒβ„Žoψr PROPN cana-1301 103 33 + + PROPN cana-1301 103 34 ( ( PUNCT cana-1301 103 35 πœ„πœ…)𝑒 πœ„πœ…)𝑒 NOUN cana-1301 103 36 , , PUNCT cana-1301 103 37 π‘ƒβ„Žoψr π‘ƒβ„Žoψr PROPN cana-1301 103 38 βˆ’(πœ„πœ…)𝑒 βˆ’(πœ„πœ…)𝑒 PROPN cana-1301 103 39 ] ] PUNCT cana-1301 103 40 h=1 h=1 NOUN cana-1301 103 41 m m VERB cana-1301 103 42 βŒͺ βŒͺ NOUN cana-1301 103 43 , , PUNCT cana-1301 103 44 𝑒 𝑒 X cana-1301 103 45 = = SYM cana-1301 103 46 1,2 1,2 NUM cana-1301 103 47 , , PUNCT cana-1301 103 48 β‹― β‹― VERB cana-1301 103 49 , , PUNCT cana-1301 103 50 𝑑|πœ„πœ… 𝑑|πœ„πœ… NOUN cana-1301 103 51 ∈ ∈ ADP cana-1301 103 52 𝑉 𝑉 PROPN cana-1301 103 53 Γ— Γ— PROPN cana-1301 103 54 𝑉}be 𝑉}be NOUN cana-1301 103 55 an an DET cana-1301 103 56 m m NOUN cana-1301 103 57 - - PUNCT cana-1301 103 58 bpfme bpfme NOUN cana-1301 103 59 set set NOUN cana-1301 103 60 in in ADP cana-1301 103 61 an an DET cana-1301 103 62 m m NOUN cana-1301 103 63 - - ADJ cana-1301 103 64 bpfmg bpfmg ADJ cana-1301 103 65 𝐺 𝐺 NOUN cana-1301 103 66 . . PUNCT cana-1301 104 1 an an DET cana-1301 104 2 m m NOUN cana-1301 104 3 - - ADJ cana-1301 104 4 bpfmg bpfmg ADJ cana-1301 104 5 𝐺 𝐺 PROPN cana-1301 104 6 is be AUX cana-1301 104 7 complete complete ADJ cana-1301 104 8 if if SCONJ cana-1301 104 9 π‘ƒβ„Žoψr π‘ƒβ„Žoψr PROPN cana-1301 104 10 + + PROPN cana-1301 104 11 ( ( PUNCT cana-1301 104 12 πœ„πœ…)𝑒 πœ„πœ…)𝑒 NOUN cana-1301 104 13 = = PUNCT cana-1301 104 14 min{π‘ƒβ„Žoψq min{π‘ƒβ„Žoψq NOUN cana-1301 105 1 + + ADJ cana-1301 105 2 ( ( PUNCT cana-1301 105 3 πœ„ πœ„ NOUN cana-1301 105 4 ) ) PUNCT cana-1301 105 5 , , PUNCT cana-1301 105 6 π‘ƒβ„Žoψq π‘ƒβ„Žoψq PROPN cana-1301 105 7 + + PROPN cana-1301 105 8 ( ( PUNCT cana-1301 105 9 πœ… πœ… NOUN cana-1301 105 10 ) ) PUNCT cana-1301 105 11 } } PUNCT cana-1301 105 12 , , PUNCT cana-1301 105 13 π‘ƒβ„Žoψr π‘ƒβ„Žoψr PROPN cana-1301 105 14 βˆ’(πœ„πœ…)𝑒 βˆ’(πœ„πœ…)𝑒 PROPN cana-1301 105 15 = = ADJ cana-1301 105 16 max{π‘ƒβ„Žoψq max{π‘ƒβ„Žoψq PROPN cana-1301 105 17 βˆ’(πœ„ βˆ’(πœ„ PROPN cana-1301 105 18 ) ) PUNCT cana-1301 105 19 , , PUNCT cana-1301 105 20 π‘ƒβ„Žoψq π‘ƒβ„Žoψq PROPN cana-1301 105 21 βˆ’(πœ… βˆ’(πœ… NOUN cana-1301 105 22 ) ) PUNCT cana-1301 105 23 } } PUNCT cana-1301 105 24 for for ADP cana-1301 105 25 all all DET cana-1301 105 26 πœ„ πœ„ PROPN cana-1301 105 27 , , PUNCT cana-1301 105 28 πœ… πœ… PROPN cana-1301 105 29 ∈ ∈ PROPN cana-1301 105 30 𝑉 𝑉 PROPN cana-1301 105 31 , , PUNCT cana-1301 105 32 𝑒 𝑒 X cana-1301 105 33 = = SYM cana-1301 105 34 1,2 1,2 NUM cana-1301 105 35 , , PUNCT cana-1301 105 36 β‹― β‹― PROPN cana-1301 105 37 , , PUNCT cana-1301 105 38 𝑑 𝑑 PROPN cana-1301 105 39 and and CCONJ cana-1301 105 40 β„Ž β„Ž X cana-1301 105 41 = = SYM cana-1301 105 42 1 1 NUM cana-1301 105 43 , , PUNCT cana-1301 105 44 2 2 NUM cana-1301 105 45 , , PUNCT cana-1301 105 46 β‹― β‹― NOUN cana-1301 105 47 , , PUNCT cana-1301 105 48 π‘š. π‘š. ADJ cana-1301 105 49 example example NOUN cana-1301 105 50 2 2 NUM cana-1301 105 51 : : PUNCT cana-1301 105 52 consider consider VERB cana-1301 105 53 an an DET cana-1301 105 54 m m NOUN cana-1301 105 55 - - ADJ cana-1301 105 56 bpfmg bpfmg ADJ cana-1301 105 57 𝐺 𝐺 NOUN cana-1301 105 58 = = SYM cana-1301 105 59 ( ( PUNCT cana-1301 105 60 𝑉 𝑉 PROPN cana-1301 105 61 , , PUNCT cana-1301 105 62 𝑄 𝑄 PROPN cana-1301 105 63 , , PUNCT cana-1301 105 64 𝑅 𝑅 PROPN cana-1301 105 65 ) ) PUNCT cana-1301 105 66 as as SCONJ cana-1301 105 67 exposed expose VERB cana-1301 105 68 in in ADP cana-1301 105 69 fig fig NOUN cana-1301 105 70 . . PUNCT cana-1301 106 1 2 2 X cana-1301 106 2 . . X cana-1301 106 3 it it PRON cana-1301 106 4 is be AUX cana-1301 106 5 obvious obvious ADJ cana-1301 106 6 from from ADP cana-1301 106 7 fig fig NOUN cana-1301 106 8 . . PUNCT cana-1301 107 1 2 2 NUM cana-1301 107 2 by by ADP cana-1301 107 3 simple simple ADJ cana-1301 107 4 calculations calculation NOUN cana-1301 107 5 that that SCONJ cana-1301 107 6 it it PRON cana-1301 107 7 is be AUX cana-1301 107 8 an an DET cana-1301 107 9 m m NOUN cana-1301 107 10 - - NOUN cana-1301 107 11 bpfcmg bpfcmg NOUN cana-1301 107 12 . . PUNCT cana-1301 108 1 fig fig NOUN cana-1301 108 2 . . PUNCT cana-1301 109 1 2 2 NUM cana-1301 109 2 . . SYM cana-1301 109 3 2 2 NUM cana-1301 109 4 - - PUNCT cana-1301 109 5 bipolar bipolar ADJ cana-1301 109 6 fuzzy fuzzy ADJ cana-1301 109 7 complete complete ADJ cana-1301 109 8 multigraph multigraph NOUN cana-1301 109 9 here𝑄 here𝑄 PROPN cana-1301 110 1 = = PUNCT cana-1301 110 2 { { PUNCT cana-1301 110 3 𝜐 𝜐 X cana-1301 110 4 〈 〈 PROPN cana-1301 110 5 [ [ X cana-1301 110 6 0.9,βˆ’0.8],[0.6,βˆ’0.5 0.9,βˆ’0.8],[0.6,βˆ’0.5 NUM cana-1301 110 7 ] ] X cana-1301 110 8 βŒͺ βŒͺ NOUN cana-1301 110 9 , , PUNCT cana-1301 110 10 πœ„ πœ„ PROPN cana-1301 110 11 〈 〈 PROPN cana-1301 110 12 [ [ X cana-1301 110 13 0.7,βˆ’0.5],[0.5,βˆ’0.7 0.7,βˆ’0.5],[0.5,βˆ’0.7 NUM cana-1301 110 14 ] ] SYM cana-1301 110 15 βŒͺ βŒͺ NOUN cana-1301 110 16 , , PUNCT cana-1301 110 17 𝜏 𝜏 NOUN cana-1301 110 18 〈 〈 NOUN cana-1301 110 19 [ [ X cana-1301 110 20 0.9,βˆ’0.5],[0.7,βˆ’0.9 0.9,βˆ’0.5],[0.7,βˆ’0.9 NUM cana-1301 110 21 ] ] X cana-1301 110 22 βŒͺ βŒͺ NOUN cana-1301 110 23 } } PUNCT cana-1301 110 24 , , PUNCT cana-1301 110 25 𝑅 𝑅 PROPN cana-1301 110 26 = = NOUN cana-1301 110 27 { { PUNCT cana-1301 110 28 πœπœ„ πœπœ„ NOUN cana-1301 110 29 〈 〈 PROPN cana-1301 110 30 [ [ X cana-1301 110 31 0.7,βˆ’0.5],[0.5,βˆ’0.5 0.7,βˆ’0.5],[0.5,βˆ’0.5 NUM cana-1301 110 32 ] ] X cana-1301 110 33 βŒͺ βŒͺ NOUN cana-1301 110 34 , , PUNCT cana-1301 110 35 πœπœ„ πœπœ„ X cana-1301 110 36 〈 〈 PROPN cana-1301 110 37 [ [ X cana-1301 110 38 0.7,βˆ’0.5],[0.5,βˆ’0.5 0.7,βˆ’0.5],[0.5,βˆ’0.5 NUM cana-1301 110 39 ] ] X cana-1301 110 40 βŒͺ βŒͺ NOUN cana-1301 110 41 , , PUNCT cana-1301 110 42 𝜐𝜏 𝜐𝜏 ADP cana-1301 110 43 〈 〈 PROPN cana-1301 110 44 [ [ X cana-1301 110 45 0.9,βˆ’0.5],[0.6,βˆ’0.5 0.9,βˆ’0.5],[0.6,βˆ’0.5 NUM cana-1301 110 46 ] ] X cana-1301 110 47 βŒͺ βŒͺ NOUN cana-1301 110 48 , , PUNCT cana-1301 110 49 𝜐𝜏 𝜐𝜏 ADP cana-1301 110 50 〈 〈 PROPN cana-1301 110 51 [ [ X cana-1301 110 52 0.9,βˆ’0.5],[0.6,βˆ’0.5 0.9,βˆ’0.5],[0.6,βˆ’0.5 NUM cana-1301 110 53 ] ] X cana-1301 110 54 βŒͺ βŒͺ NOUN cana-1301 110 55 , , PUNCT cana-1301 110 56 πœ„πœ πœ„πœ NOUN cana-1301 110 57 〈 〈 PROPN cana-1301 110 58 [ [ X cana-1301 110 59 0.7,βˆ’0.5],[0.5,βˆ’0.7 0.7,βˆ’0.5],[0.5,βˆ’0.7 NUM cana-1301 110 60 ] ] SYM cana-1301 110 61 βŒͺ βŒͺ NOUN cana-1301 110 62 } } PUNCT cana-1301 110 63 . . PUNCT cana-1301 111 1 communications communication NOUN cana-1301 111 2 on on ADP cana-1301 111 3 applied apply VERB cana-1301 111 4 nonlinear nonlinear ADJ cana-1301 111 5 analysis analysis NOUN cana-1301 111 6 issn issn NOUN cana-1301 111 7 : : PUNCT cana-1301 111 8 1074 1074 NUM cana-1301 111 9 - - PUNCT cana-1301 111 10 133x 133x NUM cana-1301 111 11 vol vol NOUN cana-1301 111 12 31 31 NUM cana-1301 111 13 no no NOUN cana-1301 111 14 . . PUNCT cana-1301 112 1 7s 7 NOUN cana-1301 112 2 ( ( PUNCT cana-1301 112 3 2024 2024 NUM cana-1301 112 4 ) ) PUNCT cana-1301 112 5 266 266 NUM cana-1301 112 6 https://internationalpubls.com https://internationalpubls.com X cana-1301 112 7 definition definition NOUN cana-1301 112 8 3.4 3.4 NUM cana-1301 112 9 : : PUNCT cana-1301 112 10 energy energy NOUN cana-1301 112 11 of of ADP cana-1301 112 12 an an DET cana-1301 112 13 m m NOUN cana-1301 112 14 - - ADJ cana-1301 112 15 bpfe bpfe ADJ cana-1301 112 16 πœ„πœ… πœ„πœ… NOUN cana-1301 112 17 can can AUX cana-1301 112 18 be be AUX cana-1301 112 19 calculated calculate VERB cana-1301 112 20 by by ADP cana-1301 112 21 the the DET cana-1301 112 22 value value NOUN cana-1301 112 23 πΈπœ„πœ… πΈπœ„πœ… PROPN cana-1301 112 24 = = SYM cana-1301 112 25 〈 〈 PROPN cana-1301 112 26 [ [ X cana-1301 112 27 π‘ƒβ„ŽoπΈπœ„πœ… π‘ƒβ„ŽoπΈπœ„πœ… PROPN cana-1301 112 28 + + CCONJ cana-1301 112 29 , , PUNCT cana-1301 112 30 π‘ƒβ„ŽoπΈπœ„πœ… π‘ƒβ„ŽoπΈπœ„πœ… ADJ cana-1301 112 31 βˆ’ βˆ’ NOUN cana-1301 112 32 ] ] SYM cana-1301 112 33 β„Ž=1 β„Ž=1 PROPN cana-1301 112 34 π‘š π‘š ADP cana-1301 112 35 βŒͺ βŒͺ NOUN cana-1301 112 36 = = SYM cana-1301 112 37 〈 〈 PROPN cana-1301 112 38 [ [ PUNCT cana-1301 112 39 π‘ƒβ„ŽoΟˆπ‘… π‘ƒβ„ŽoΟˆπ‘… PROPN cana-1301 112 40 + + PROPN cana-1301 112 41 ( ( PUNCT cana-1301 112 42 πœ„πœ…)𝑒 πœ„πœ…)𝑒 NOUN cana-1301 112 43 min{π‘ƒβ„Žoψq min{π‘ƒβ„Žoψq NOUN cana-1301 113 1 + + ADJ cana-1301 113 2 ( ( PUNCT cana-1301 113 3 πœ„ πœ„ NOUN cana-1301 113 4 ) ) PUNCT cana-1301 113 5 , , PUNCT cana-1301 113 6 π‘ƒβ„Žoψq π‘ƒβ„Žoψq PROPN cana-1301 113 7 + + PROPN cana-1301 113 8 ( ( PUNCT cana-1301 113 9 πœ… πœ… NOUN cana-1301 113 10 ) ) PUNCT cana-1301 113 11 } } PUNCT cana-1301 113 12 , , PUNCT cana-1301 113 13 π‘ƒβ„ŽoΟˆπ‘… π‘ƒβ„ŽoΟˆπ‘… PROPN cana-1301 113 14 βˆ’(πœ„πœ…)𝑒 βˆ’(πœ„πœ…)𝑒 PROPN cana-1301 113 15 max{π‘ƒβ„Žoψq max{π‘ƒβ„Žoψq ADJ cana-1301 113 16 βˆ’(πœ„ βˆ’(πœ„ PROPN cana-1301 113 17 ) ) PUNCT cana-1301 113 18 , , PUNCT cana-1301 113 19 π‘ƒβ„Žoψq π‘ƒβ„Žoψq PROPN cana-1301 113 20 βˆ’(πœ… βˆ’(πœ… PROPN cana-1301 113 21 ) ) PUNCT cana-1301 113 22 } } PUNCT cana-1301 113 23 ] ] PUNCT cana-1301 113 24 β„Ž=1 β„Ž=1 PROPN cana-1301 113 25 π‘š π‘š X cana-1301 113 26 βŒͺ βŒͺ NOUN cana-1301 113 27 . . PUNCT cana-1301 114 1 definition definition NOUN cana-1301 114 2 3.5 3.5 NUM cana-1301 114 3 : : PUNCT cana-1301 114 4 let let VERB cana-1301 114 5 𝐺 𝐺 PRON cana-1301 114 6 be be AUX cana-1301 114 7 an an DET cana-1301 114 8 m m NOUN cana-1301 114 9 - - NOUN cana-1301 114 10 bpfmg bpfmg PROPN cana-1301 114 11 . . PUNCT cana-1301 115 1 a a DET cana-1301 115 2 line line NOUN cana-1301 115 3 πœ„πœ… πœ„πœ… NOUN cana-1301 115 4 should should AUX cana-1301 115 5 be be AUX cana-1301 115 6 an an DET cana-1301 115 7 m m NOUN cana-1301 115 8 - - PUNCT cana-1301 115 9 bpf bpf ADV cana-1301 115 10 strong strong ADJ cana-1301 115 11 if if SCONJ cana-1301 115 12 π‘ƒβ„ŽoπΈπœ„πœ… π‘ƒβ„ŽoπΈπœ„πœ… PROPN cana-1301 115 13 + + NUM cana-1301 115 14 β‰₯ β‰₯ NOUN cana-1301 115 15 0.5 0.5 NUM cana-1301 115 16 or or CCONJ cana-1301 115 17 π‘ƒβ„ŽoπΈπœ„πœ… π‘ƒβ„ŽoπΈπœ„πœ… ADJ cana-1301 115 18 βˆ’ βˆ’ NOUN cana-1301 115 19 ≀ ≀ NUM cana-1301 115 20 0.5 0.5 NUM cana-1301 115 21 , , PUNCT cana-1301 115 22 for for ADP cana-1301 115 23 β„Ž β„Ž PROPN cana-1301 115 24 = = SYM cana-1301 115 25 1 1 NUM cana-1301 115 26 , , PUNCT cana-1301 115 27 2 2 NUM cana-1301 115 28 , , PUNCT cana-1301 115 29 β‹― β‹― PROPN cana-1301 115 30 , , PUNCT cana-1301 115 31 π‘š π‘š X cana-1301 115 32 otherwise otherwise ADV cana-1301 115 33 weak weak ADJ cana-1301 115 34 . . PUNCT cana-1301 116 1 definition definition NOUN cana-1301 116 2 3.6 3.6 NUM cana-1301 116 3 : : PUNCT cana-1301 116 4 let let VERB cana-1301 116 5 𝐺 𝐺 PROPN cana-1301 116 6 be be AUX cana-1301 116 7 an an DET cana-1301 116 8 m m NOUN cana-1301 116 9 - - NOUN cana-1301 116 10 bpfmg bpfmg PROPN cana-1301 116 11 and and CCONJ cana-1301 116 12 let let VERB cana-1301 116 13 𝑅 𝑅 PROPN cana-1301 116 14 includes include VERB cana-1301 116 15 two two NUM cana-1301 116 16 lines line NOUN cana-1301 116 17 πœ„πœ… πœ„πœ… PRON cana-1301 116 18 〈 〈 PROPN cana-1301 116 19 [ [ X cana-1301 116 20 π‘ƒβ„Žoψr π‘ƒβ„Žoψr PROPN cana-1301 116 21 + + PROPN cana-1301 116 22 ( ( PUNCT cana-1301 116 23 πœ„πœ…)𝑒 πœ„πœ…)𝑒 NOUN cana-1301 116 24 , , PUNCT cana-1301 116 25 π‘ƒβ„Žoψr π‘ƒβ„Žoψr PROPN cana-1301 116 26 βˆ’(πœ„πœ…)𝑒 βˆ’(πœ„πœ…)𝑒 PROPN cana-1301 116 27 ] ] PUNCT cana-1301 116 28 h=1 h=1 VERB cana-1301 116 29 m m VERB cana-1301 116 30 βŒͺ βŒͺ NOUN cana-1301 116 31 and and CCONJ cana-1301 116 32 𝜐𝜏 𝜐𝜏 ADP cana-1301 116 33 〈 〈 PROPN cana-1301 116 34 [ [ X cana-1301 116 35 π‘ƒβ„Žoψr π‘ƒβ„Žoψr PROPN cana-1301 116 36 + + PROPN cana-1301 116 37 ( ( PUNCT cana-1301 116 38 𝜐𝜏)𝑠 𝜐𝜏)𝑠 PROPN cana-1301 116 39 , , PUNCT cana-1301 116 40 π‘ƒβ„Žoψr π‘ƒβ„Žoψr PROPN cana-1301 116 41 βˆ’(𝜐𝜏)𝑠 βˆ’(𝜐𝜏)𝑠 PROPN cana-1301 116 42 ] ] PUNCT cana-1301 116 43 h=1 h=1 VERB cana-1301 116 44 m m VERB cana-1301 116 45 βŒͺ βŒͺ NOUN cana-1301 116 46 which which PRON cana-1301 116 47 are be AUX cana-1301 116 48 intersect intersect ADJ cana-1301 116 49 at at ADP cana-1301 116 50 a a DET cana-1301 116 51 location location NOUN cana-1301 116 52 π‘Œ π‘Œ PROPN cana-1301 116 53 , , PUNCT cana-1301 116 54 here here ADV cana-1301 116 55 𝑒 𝑒 NOUN cana-1301 116 56 and and CCONJ cana-1301 116 57 𝑠 𝑠 PROPN cana-1301 116 58 are be AUX cana-1301 116 59 permanent permanent ADJ cana-1301 116 60 integers integer NOUN cana-1301 116 61 . . PUNCT cana-1301 117 1 the the DET cana-1301 117 2 intersecting intersecting ADJ cana-1301 117 3 value value NOUN cana-1301 117 4 at at ADP cana-1301 117 5 π‘Œ π‘Œ PROPN cana-1301 117 6 is be AUX cana-1301 117 7 defined define VERB cana-1301 117 8 as as ADP cana-1301 117 9 Ο‡π‘Œ Ο‡π‘Œ NOUN cana-1301 117 10 = = SYM cana-1301 117 11 〈 〈 PROPN cana-1301 117 12 [ [ X cana-1301 117 13 π‘ƒβ„ŽoΟ‡π‘Œ π‘ƒβ„ŽoΟ‡π‘Œ PROPN cana-1301 117 14 + + PROPN cana-1301 117 15 , , PUNCT cana-1301 117 16 π‘ƒβ„ŽoΟ‡π‘Œ π‘ƒβ„ŽoΟ‡π‘Œ PROPN cana-1301 117 17 βˆ’]β„Ž=1 βˆ’]β„Ž=1 PROPN cana-1301 117 18 π‘š π‘š ADP cana-1301 117 19 βŒͺ βŒͺ NOUN cana-1301 117 20 , , PUNCT cana-1301 117 21 where where SCONJ cana-1301 117 22 π‘ƒβ„ŽoΟ‡π‘Œ π‘ƒβ„ŽoΟ‡π‘Œ PROPN cana-1301 117 23 + + PROPN cana-1301 117 24 = = SYM cana-1301 117 25 π‘ƒβ„ŽoπΈπœ„πœ… π‘ƒβ„ŽoπΈπœ„πœ… PROPN cana-1301 117 26 + + CCONJ cana-1301 117 27 + + ADJ cana-1301 117 28 π‘ƒβ„Žo𝐸𝜐𝜏 π‘ƒβ„Žo𝐸𝜐𝜏 ADJ cana-1301 117 29 + + CCONJ cana-1301 117 30 2 2 NUM cana-1301 117 31 , , PUNCT cana-1301 117 32 π‘ƒβ„ŽoΟ‡π‘Œ π‘ƒβ„ŽoΟ‡π‘Œ PROPN cana-1301 117 33 βˆ’ βˆ’ PROPN cana-1301 117 34 = = SYM cana-1301 117 35 π‘ƒβ„ŽoπΈπœ„πœ… π‘ƒβ„ŽoπΈπœ„πœ… PROPN cana-1301 117 36 βˆ’ βˆ’ ADP cana-1301 117 37 + + ADJ cana-1301 117 38 π‘ƒβ„Žo𝐸𝜐𝜏 π‘ƒβ„Žo𝐸𝜐𝜏 PROPN cana-1301 117 39 βˆ’ βˆ’ ADJ cana-1301 117 40 2 2 NUM cana-1301 117 41 for for ADP cana-1301 117 42 β„Ž β„Ž PROPN cana-1301 117 43 = = SYM cana-1301 117 44 1 1 NUM cana-1301 117 45 , , PUNCT cana-1301 117 46 2 2 NUM cana-1301 117 47 , , PUNCT cana-1301 117 48 β‹― β‹― NOUN cana-1301 117 49 , , PUNCT cana-1301 117 50 π‘š. π‘š. ADJ cana-1301 117 51 planarity planarity NOUN cana-1301 117 52 falls fall VERB cana-1301 117 53 as as ADP cana-1301 117 54 the the DET cana-1301 117 55 amount amount NOUN cana-1301 117 56 of of ADP cana-1301 117 57 points point NOUN cana-1301 117 58 of of ADP cana-1301 117 59 junction junction NOUN cana-1301 117 60 in in ADP cana-1301 117 61 an an DET cana-1301 117 62 m m NOUN cana-1301 117 63 - - ADJ cana-1301 117 64 bpfmg bpfmg NOUN cana-1301 117 65 rises rise VERB cana-1301 117 66 . . PUNCT cana-1301 118 1 with with ADP cana-1301 118 2 this this PRON cana-1301 118 3 in in ADP cana-1301 118 4 consideration consideration NOUN cana-1301 118 5 , , PUNCT cana-1301 118 6 we we PRON cana-1301 118 7 now now ADV cana-1301 118 8 presented present VERB cana-1301 118 9 the the DET cana-1301 118 10 notion notion NOUN cana-1301 118 11 of of ADP cana-1301 118 12 an an DET cana-1301 118 13 m m NOUN cana-1301 118 14 - - NOUN cana-1301 118 15 bpfpg bpfpg NOUN cana-1301 118 16 below below ADV cana-1301 118 17 . . PUNCT cana-1301 119 1 definition definition NOUN cana-1301 119 2 3.7 3.7 NUM cana-1301 119 3 : : PUNCT cana-1301 119 4 let let VERB cana-1301 119 5 𝐺 𝐺 PRON cana-1301 119 6 be be AUX cana-1301 119 7 an an DET cana-1301 119 8 m m NOUN cana-1301 119 9 - - NOUN cana-1301 119 10 bpfmg bpfmg NOUN cana-1301 119 11 for for ADP cana-1301 119 12 a a DET cana-1301 119 13 particular particular ADJ cana-1301 119 14 geometrical geometrical ADJ cana-1301 119 15 representation representation NOUN cana-1301 119 16 and and CCONJ cana-1301 119 17 defineπ‘Œ1 defineπ‘Œ1 ADJ cana-1301 119 18 , , PUNCT cana-1301 119 19 π‘Œ2 π‘Œ2 NOUN cana-1301 119 20 , , PUNCT cana-1301 119 21 β‹― β‹― PROPN cana-1301 119 22 , , PUNCT cana-1301 119 23 π‘Œπ‘Ÿ π‘Œπ‘Ÿ PROPN cana-1301 119 24 as as ADP cana-1301 119 25 the the DET cana-1301 119 26 points point NOUN cana-1301 119 27 where where SCONJ cana-1301 119 28 the the DET cana-1301 119 29 lines line NOUN cana-1301 119 30 are be AUX cana-1301 119 31 intersect intersect ADJ cana-1301 119 32 . . PUNCT cana-1301 120 1 consequently consequently ADV cana-1301 120 2 , , PUNCT cana-1301 120 3 𝐺 𝐺 PROPN cana-1301 120 4 is be AUX cana-1301 120 5 described describe VERB cana-1301 120 6 as as ADP cana-1301 120 7 an an DET cana-1301 120 8 m m NOUN cana-1301 120 9 - - NOUN cana-1301 120 10 bpfpg bpfpg NOUN cana-1301 120 11 with with ADP cana-1301 120 12 a a DET cana-1301 120 13 m m NOUN cana-1301 120 14 - - PUNCT cana-1301 120 15 bpf bpf NOUN cana-1301 120 16 planarity planarity NOUN cana-1301 120 17 value value NOUN cana-1301 120 18 Ξ³ Ξ³ X cana-1301 120 19 = = SYM cana-1301 120 20 〈 〈 PROPN cana-1301 120 21 [ [ X cana-1301 120 22 π‘ƒβ„ŽoΞ³+ π‘ƒβ„ŽoΞ³+ ADJ cana-1301 120 23 , , PUNCT cana-1301 120 24 π‘ƒβ„ŽoΞ³βˆ’]β„Ž=1 π‘ƒβ„ŽoΞ³βˆ’]β„Ž=1 PROPN cana-1301 120 25 π‘š π‘š PRON cana-1301 120 26 βŒͺ βŒͺ NOUN cana-1301 120 27 = = SYM cana-1301 120 28 〈 〈 PROPN cana-1301 120 29 [ [ PUNCT cana-1301 120 30 1 1 NUM cana-1301 120 31 1+{π‘ƒβ„ŽoΟ‡π‘Œ1 1+{π‘ƒβ„ŽoΟ‡π‘Œ1 NUM cana-1301 120 32 + + SYM cana-1301 120 33 + + ADJ cana-1301 120 34 π‘ƒβ„ŽoΟ‡π‘Œ2 π‘ƒβ„ŽoΟ‡π‘Œ2 NOUN cana-1301 120 35 + + X cana-1301 120 36 + + ADJ cana-1301 120 37 β‹―+π‘ƒβ„ŽoΟ‡π‘Œπ‘Ÿ β‹―+π‘ƒβ„ŽoΟ‡π‘Œπ‘Ÿ ADJ cana-1301 120 38 + + X cana-1301 120 39 } } PUNCT cana-1301 120 40 , , PUNCT cana-1301 120 41 βˆ’1 βˆ’1 NOUN cana-1301 120 42 1+{π‘ƒβ„ŽoΟ‡π‘Œ1 1+{π‘ƒβ„ŽoΟ‡π‘Œ1 PROPN cana-1301 120 43 βˆ’ βˆ’ PROPN cana-1301 121 1 + + NOUN cana-1301 121 2 π‘ƒβ„ŽoΟ‡π‘Œ2 π‘ƒβ„ŽoΟ‡π‘Œ2 NOUN cana-1301 121 3 βˆ’ βˆ’ PROPN cana-1301 122 1 + + ADJ cana-1301 122 2 β‹―+π‘ƒβ„ŽoΟ‡π‘Œπ‘Ÿ β‹―+π‘ƒβ„ŽoΟ‡π‘Œπ‘Ÿ NOUN cana-1301 122 3 βˆ’ βˆ’ PROPN cana-1301 122 4 } } PUNCT cana-1301 122 5 ] ] PUNCT cana-1301 122 6 β„Ž=1 β„Ž=1 PROPN cana-1301 122 7 π‘š π‘š X cana-1301 122 8 βŒͺ βŒͺ NOUN cana-1301 122 9 . . PUNCT cana-1301 123 1 it it PRON cana-1301 123 2 is be AUX cana-1301 123 3 obvious obvious ADJ cana-1301 123 4 that that SCONJ cana-1301 123 5 Ξ³ Ξ³ PROPN cana-1301 123 6 is be AUX cana-1301 123 7 bounded bound VERB cana-1301 123 8 , , PUNCT cana-1301 123 9 since since SCONJ cana-1301 123 10 0 0 NUM cana-1301 123 11 < < X cana-1301 123 12 π‘ƒβ„ŽoΞ³+ π‘ƒβ„ŽoΞ³+ PROPN cana-1301 123 13 ≀ ≀ ADV cana-1301 123 14 1 1 NUM cana-1301 123 15 and and CCONJ cana-1301 123 16 βˆ’1 βˆ’1 NOUN cana-1301 123 17 ≀ ≀ NOUN cana-1301 123 18 π‘ƒβ„ŽoΞ³βˆ’ π‘ƒβ„ŽoΞ³βˆ’ ADJ cana-1301 123 19 ≀ ≀ NUM cana-1301 123 20 0 0 PUNCT cana-1301 123 21 for for ADP cana-1301 123 22 all all PRON cana-1301 123 23 β„Ž β„Ž ADP cana-1301 123 24 = = SYM cana-1301 123 25 1 1 NUM cana-1301 123 26 , , PUNCT cana-1301 123 27 2 2 NUM cana-1301 123 28 , , PUNCT cana-1301 123 29 β‹― β‹― NOUN cana-1301 123 30 , , PUNCT cana-1301 123 31 π‘š. π‘š. VERB cana-1301 123 32 the the DET cana-1301 123 33 m m NOUN cana-1301 123 34 - - PUNCT cana-1301 123 35 bpf bpf NOUN cana-1301 123 36 planarity planarity NOUN cana-1301 123 37 value value NOUN cana-1301 123 38 of of ADP cana-1301 123 39 a a DET cana-1301 123 40 given give VERB cana-1301 123 41 geometrical geometrical ADJ cana-1301 123 42 representation representation NOUN cana-1301 123 43 of of ADP cana-1301 123 44 an an DET cana-1301 123 45 m m NOUN cana-1301 123 46 - - NOUN cana-1301 123 47 bpfpg bpfpg NOUN cana-1301 123 48 is be AUX cana-1301 123 49 〈 〈 PROPN cana-1301 123 50 [ [ X cana-1301 123 51 βˆ’1 βˆ’1 NOUN cana-1301 123 52 , , PUNCT cana-1301 123 53 1]β„Ž=1 1]β„Ž=1 NUM cana-1301 123 54 π‘š π‘š X cana-1301 123 55 βŒͺ βŒͺ NOUN cana-1301 123 56 if if SCONJ cana-1301 123 57 there there PRON cana-1301 123 58 is be VERB cana-1301 123 59 no no DET cana-1301 123 60 point point NOUN cana-1301 123 61 of of ADP cana-1301 123 62 junction junction NOUN cana-1301 123 63 for for ADP cana-1301 123 64 such such ADJ cana-1301 123 65 representation representation NOUN cana-1301 123 66 . . PUNCT cana-1301 124 1 in in ADP cana-1301 124 2 this this DET cana-1301 124 3 instance instance NOUN cana-1301 124 4 , , PUNCT cana-1301 124 5 the the DET cana-1301 124 6 crisp crisp ADJ cana-1301 124 7 planar planar ADJ cana-1301 124 8 graph graph NOUN cana-1301 124 9 serves serve VERB cana-1301 124 10 as as ADP cana-1301 124 11 the the DET cana-1301 124 12 underlying underlie VERB cana-1301 124 13 crisp crisp ADJ cana-1301 124 14 graph graph NOUN cana-1301 124 15 of of ADP cana-1301 124 16 this this DET cana-1301 124 17 m m NOUN cana-1301 124 18 - - PUNCT cana-1301 124 19 bpfg bpfg PROPN cana-1301 124 20 . . PUNCT cana-1301 125 1 example example NOUN cana-1301 125 2 3 3 NUM cana-1301 125 3 : : PUNCT cana-1301 125 4 take take VERB cana-1301 125 5 a a DET cana-1301 125 6 multigraph multigraph NOUN cana-1301 125 7 πΊβˆ— πΊβˆ— NOUN cana-1301 126 1 = = SYM cana-1301 126 2 ( ( PUNCT cana-1301 126 3 𝑉 𝑉 PROPN cana-1301 126 4 , , PUNCT cana-1301 126 5 𝐸 𝐸 PROPN cana-1301 126 6 ) ) PUNCT cana-1301 126 7 of of ADP cana-1301 126 8 𝐺 𝐺 PROPN cana-1301 126 9 = = SYM cana-1301 126 10 ( ( PUNCT cana-1301 126 11 𝑉 𝑉 PROPN cana-1301 126 12 , , PUNCT cana-1301 126 13 𝑄 𝑄 PROPN cana-1301 126 14 , , PUNCT cana-1301 126 15 𝑅 𝑅 PROPN cana-1301 126 16 ) ) PUNCT cana-1301 126 17 such such ADJ cana-1301 126 18 that that PRON cana-1301 126 19 𝑉 𝑉 PROPN cana-1301 126 20 = = PUNCT cana-1301 126 21 { { PUNCT cana-1301 126 22 πœ„ πœ„ PROPN cana-1301 126 23 , , PUNCT cana-1301 126 24 πœ… πœ… PROPN cana-1301 126 25 , , PUNCT cana-1301 126 26 𝜐 𝜐 NOUN cana-1301 126 27 , , PUNCT cana-1301 126 28 𝜏 𝜏 NOUN cana-1301 126 29 } } PUNCT cana-1301 126 30 , , PUNCT cana-1301 126 31 𝐸 𝐸 PROPN cana-1301 126 32 = = SYM cana-1301 126 33 { { PUNCT cana-1301 126 34 πœ„πœ… πœ„πœ… NOUN cana-1301 126 35 , , PUNCT cana-1301 126 36 πœ„πœ πœ„πœ NOUN cana-1301 126 37 , , PUNCT cana-1301 126 38 𝜐𝜏 𝜐𝜏 ADP cana-1301 126 39 , , PUNCT cana-1301 126 40 πœπœ… πœπœ… PRON cana-1301 126 41 } } PUNCT cana-1301 126 42 . . PUNCT cana-1301 127 1 fig fig NOUN cana-1301 127 2 . . PUNCT cana-1301 128 1 3 3 NUM cana-1301 128 2 . . NOUN cana-1301 128 3 2 2 NUM cana-1301 128 4 - - PUNCT cana-1301 128 5 bipolar bipolar ADJ cana-1301 128 6 fuzzy fuzzy ADJ cana-1301 128 7 planar planar ADJ cana-1301 128 8 graph graph NOUN cana-1301 128 9 communications communication NOUN cana-1301 128 10 on on ADP cana-1301 128 11 applied apply VERB cana-1301 128 12 nonlinear nonlinear ADJ cana-1301 128 13 analysis analysis NOUN cana-1301 128 14 issn issn NOUN cana-1301 128 15 : : PUNCT cana-1301 128 16 1074 1074 NUM cana-1301 128 17 - - PUNCT cana-1301 128 18 133x 133x NUM cana-1301 128 19 vol vol NOUN cana-1301 128 20 31 31 NUM cana-1301 128 21 no no NOUN cana-1301 128 22 . . PUNCT cana-1301 129 1 7s 7 NOUN cana-1301 129 2 ( ( PUNCT cana-1301 129 3 2024 2024 NUM cana-1301 129 4 ) ) PUNCT cana-1301 129 5 267 267 NUM cana-1301 129 6 https://internationalpubls.com https://internationalpubls.com X cana-1301 129 7 let let VERB cana-1301 129 8 𝑄 𝑄 PRON cana-1301 129 9 = = PUNCT cana-1301 129 10 { { PUNCT cana-1301 129 11 πœ„ πœ„ X cana-1301 129 12 〈 〈 PROPN cana-1301 129 13 [ [ X cana-1301 129 14 0.9,βˆ’0.5],[0.7,βˆ’0.9 0.9,βˆ’0.5],[0.7,βˆ’0.9 NUM cana-1301 129 15 ] ] X cana-1301 129 16 βŒͺ βŒͺ NOUN cana-1301 129 17 , , PUNCT cana-1301 129 18 πœ… πœ… NOUN cana-1301 129 19 〈 〈 PROPN cana-1301 129 20 [ [ X cana-1301 129 21 0.5,βˆ’0.3],[0.3,βˆ’0.7 0.5,βˆ’0.3],[0.3,βˆ’0.7 NUM cana-1301 129 22 ] ] VERB cana-1301 129 23 βŒͺ βŒͺ NOUN cana-1301 129 24 , , PUNCT cana-1301 129 25 𝜐 𝜐 PROPN cana-1301 129 26 〈 〈 PROPN cana-1301 129 27 [ [ X cana-1301 129 28 0.8,βˆ’0.4],[0.5,βˆ’0.7 0.8,βˆ’0.4],[0.5,βˆ’0.7 NOUN cana-1301 129 29 ] ] X cana-1301 129 30 βŒͺ βŒͺ NOUN cana-1301 129 31 , , PUNCT cana-1301 129 32 𝜏 𝜏 NOUN cana-1301 129 33 〈 〈 NOUN cana-1301 129 34 [ [ X cana-1301 129 35 0.9,βˆ’0.7],[0.5,βˆ’0.6 0.9,βˆ’0.7],[0.5,βˆ’0.6 NUM cana-1301 129 36 ] ] X cana-1301 129 37 βŒͺ βŒͺ NOUN cana-1301 129 38 } } PUNCT cana-1301 129 39 be be VERB cana-1301 129 40 an an DET cana-1301 129 41 m m NOUN cana-1301 129 42 - - PUNCT cana-1301 129 43 bpf bpf NOUN cana-1301 129 44 node node NOUN cana-1301 129 45 set set NOUN cana-1301 129 46 of of ADP cana-1301 129 47 𝑉 𝑉 PROPN cana-1301 129 48 and and CCONJ cana-1301 129 49 𝑅 𝑅 PROPN cana-1301 129 50 = = PROPN cana-1301 129 51 { { PUNCT cana-1301 129 52 πœ„πœ… πœ„πœ… NOUN cana-1301 129 53 〈 〈 PROPN cana-1301 129 54 [ [ X cana-1301 129 55 0.4,βˆ’0.2],[0.2,βˆ’0.6 0.4,βˆ’0.2],[0.2,βˆ’0.6 NUM cana-1301 129 56 ] ] SYM cana-1301 129 57 βŒͺ βŒͺ NOUN cana-1301 129 58 , , PUNCT cana-1301 129 59 πœ„πœ… πœ„πœ… NOUN cana-1301 129 60 〈 〈 PROPN cana-1301 129 61 [ [ X cana-1301 129 62 0.3,βˆ’0.1],[0.1,βˆ’0.5 0.3,βˆ’0.1],[0.1,βˆ’0.5 NUM cana-1301 129 63 ] ] X cana-1301 129 64 βŒͺ βŒͺ NOUN cana-1301 129 65 , , PUNCT cana-1301 129 66 𝜐𝜏 𝜐𝜏 ADP cana-1301 129 67 〈 〈 PROPN cana-1301 129 68 [ [ X cana-1301 129 69 0.7,βˆ’0.3],[0.3,βˆ’0.5 0.7,βˆ’0.3],[0.3,βˆ’0.5 NUM cana-1301 129 70 ] ] X cana-1301 129 71 βŒͺ βŒͺ NOUN cana-1301 129 72 , , PUNCT cana-1301 129 73 πœ„πœ πœ„πœ ADP cana-1301 129 74 〈 〈 PROPN cana-1301 129 75 [ [ X cana-1301 129 76 0.8,βˆ’0.4],[0.4,βˆ’0.5 0.8,βˆ’0.4],[0.4,βˆ’0.5 NUM cana-1301 129 77 ] ] X cana-1301 129 78 βŒͺ βŒͺ NOUN cana-1301 129 79 , , PUNCT cana-1301 129 80 πœπœ… πœπœ… NOUN cana-1301 129 81 〈 〈 PROPN cana-1301 129 82 [ [ X cana-1301 129 83 0.4,βˆ’0.2],[0.2,βˆ’0.5 0.4,βˆ’0.2],[0.2,βˆ’0.5 NUM cana-1301 129 84 ] ] X cana-1301 129 85 βŒͺ βŒͺ NOUN cana-1301 129 86 } } PUNCT cana-1301 129 87 . . PUNCT cana-1301 130 1 be be AUX cana-1301 130 2 a a DET cana-1301 130 3 m m NOUN cana-1301 130 4 - - PUNCT cana-1301 130 5 bpfml bpfml NOUN cana-1301 130 6 set set NOUN cana-1301 130 7 of of ADP cana-1301 130 8 𝑉 𝑉 PROPN cana-1301 130 9 Γ— Γ— PROPN cana-1301 130 10 𝑉 𝑉 PROPN cana-1301 130 11 is be AUX cana-1301 130 12 defined define VERB cana-1301 130 13 by by ADP cana-1301 130 14 an an DET cana-1301 130 15 m m NOUN cana-1301 130 16 - - NOUN cana-1301 130 17 bpfmg bpfmg NOUN cana-1301 130 18 as as SCONJ cana-1301 130 19 exposed expose VERB cana-1301 130 20 in in ADP cana-1301 130 21 fig fig NOUN cana-1301 130 22 . . PUNCT cana-1301 131 1 3 3 NUM cana-1301 131 2 has have VERB cana-1301 131 3 2 2 NUM cana-1301 131 4 point point NOUN cana-1301 131 5 of of ADP cana-1301 131 6 junctions junction NOUN cana-1301 131 7 π‘Œ1 π‘Œ1 NOUN cana-1301 131 8 and and CCONJ cana-1301 131 9 π‘Œ2 π‘Œ2 NOUN cana-1301 131 10 . . PUNCT cana-1301 132 1 π‘Œ1 π‘Œ1 PROPN cana-1301 132 2 is be AUX cana-1301 132 3 a a DET cana-1301 132 4 point point NOUN cana-1301 132 5 between between ADP cana-1301 132 6 the the DET cana-1301 132 7 lines line NOUN cana-1301 132 8 πœ„πœ… πœ„πœ… X cana-1301 132 9 〈 〈 PROPN cana-1301 132 10 [ [ X cana-1301 132 11 0.4,βˆ’0.2],[0.2,βˆ’0.6 0.4,βˆ’0.2],[0.2,βˆ’0.6 NUM cana-1301 132 12 ] ] SYM cana-1301 132 13 βŒͺ βŒͺ NOUN cana-1301 132 14 and and CCONJ cana-1301 132 15 𝜐𝜏 𝜐𝜏 ADP cana-1301 132 16 〈 〈 PROPN cana-1301 132 17 [ [ X cana-1301 132 18 0.7,βˆ’0.3],[0.3,βˆ’0.5 0.7,βˆ’0.3],[0.3,βˆ’0.5 NUM cana-1301 132 19 ] ] X cana-1301 132 20 βŒͺ βŒͺ NOUN cana-1301 132 21 and and CCONJ cana-1301 132 22 π‘Œ2 π‘Œ2 NOUN cana-1301 132 23 is be AUX cana-1301 132 24 between between ADP cana-1301 132 25 πœ„πœ… πœ„πœ… NOUN cana-1301 132 26 〈 〈 PROPN cana-1301 132 27 [ [ X cana-1301 132 28 0.3,βˆ’0.1],[0.1,βˆ’0.5 0.3,βˆ’0.1],[0.1,βˆ’0.5 NUM cana-1301 132 29 ] ] X cana-1301 132 30 βŒͺ βŒͺ NOUN cana-1301 132 31 and and CCONJ cana-1301 132 32 𝜐𝜏 𝜐𝜏 ADP cana-1301 132 33 〈 〈 PROPN cana-1301 132 34 [ [ X cana-1301 132 35 0.7,βˆ’0.3],[0.3,βˆ’0.5 0.7,βˆ’0.3],[0.3,βˆ’0.5 NUM cana-1301 132 36 ] ] X cana-1301 132 37 βŒͺ βŒͺ NOUN cana-1301 132 38 . . PUNCT cana-1301 133 1 for for ADP cana-1301 133 2 the the DET cana-1301 133 3 line line NOUN cana-1301 133 4 πœ„πœ… πœ„πœ… PRON cana-1301 133 5 〈 〈 PROPN cana-1301 133 6 [ [ X cana-1301 133 7 0.4,βˆ’0.2],[0.2,βˆ’0.6 0.4,βˆ’0.2],[0.2,βˆ’0.6 NUM cana-1301 133 8 ] ] SYM cana-1301 133 9 βŒͺ βŒͺ NOUN cana-1301 133 10 , , PUNCT cana-1301 133 11 πΈπœ„πœ… πΈπœ„πœ… PROPN cana-1301 133 12 = = SYM cana-1301 133 13 〈 〈 PROPN cana-1301 133 14 [ [ X cana-1301 133 15 0.8 0.8 NUM cana-1301 133 16 , , PUNCT cana-1301 133 17 0.67 0.67 NUM cana-1301 133 18 ] ] PUNCT cana-1301 133 19 , , PUNCT cana-1301 133 20 [ [ X cana-1301 133 21 0.67 0.67 NUM cana-1301 133 22 , , PUNCT cana-1301 133 23 0.86 0.86 NUM cana-1301 133 24 ] ] SYM cana-1301 133 25 βŒͺ βŒͺ NOUN cana-1301 133 26 . . PUNCT cana-1301 134 1 for for ADP cana-1301 134 2 the the DET cana-1301 134 3 line line NOUN cana-1301 134 4 πœ„πœ… πœ„πœ… PRON cana-1301 134 5 〈 〈 PROPN cana-1301 134 6 [ [ X cana-1301 134 7 0.3,βˆ’0.1],[0.1,βˆ’0.5 0.3,βˆ’0.1],[0.1,βˆ’0.5 NUM cana-1301 134 8 ] ] X cana-1301 134 9 βŒͺ βŒͺ NOUN cana-1301 134 10 , , PUNCT cana-1301 134 11 πΈπœ„πœ… πΈπœ„πœ… PROPN cana-1301 134 12 = = SYM cana-1301 134 13 〈 〈 PROPN cana-1301 134 14 [ [ X cana-1301 134 15 0.6 0.6 NUM cana-1301 134 16 , , PUNCT cana-1301 134 17 0.33 0.33 NUM cana-1301 134 18 ] ] PUNCT cana-1301 134 19 , , PUNCT cana-1301 134 20 [ [ X cana-1301 134 21 0.33 0.33 NUM cana-1301 134 22 , , PUNCT cana-1301 134 23 0.71 0.71 NUM cana-1301 134 24 ] ] SYM cana-1301 134 25 βŒͺ βŒͺ NOUN cana-1301 134 26 and and CCONJ cana-1301 134 27 for for ADP cana-1301 134 28 the the DET cana-1301 134 29 line line NOUN cana-1301 134 30 𝜐𝜏 𝜐𝜏 ADP cana-1301 134 31 〈 〈 PROPN cana-1301 134 32 [ [ X cana-1301 134 33 0.7,βˆ’0.3],[0.3,βˆ’0.5 0.7,βˆ’0.3],[0.3,βˆ’0.5 NUM cana-1301 134 34 ] ] X cana-1301 134 35 βŒͺ βŒͺ NOUN cana-1301 134 36 , , PUNCT cana-1301 134 37 e𝜐𝜏 e𝜐𝜏 ADV cana-1301 134 38 = = PUNCT cana-1301 134 39 〈 〈 PROPN cana-1301 134 40 [ [ X cana-1301 134 41 0.88 0.88 NUM cana-1301 134 42 , , PUNCT cana-1301 134 43 0.75 0.75 NUM cana-1301 134 44 ] ] PUNCT cana-1301 134 45 , , PUNCT cana-1301 134 46 [ [ X cana-1301 134 47 0.6 0.6 NUM cana-1301 134 48 , , PUNCT cana-1301 134 49 0.83 0.83 NUM cana-1301 134 50 ] ] SYM cana-1301 134 51 βŒͺ βŒͺ NOUN cana-1301 134 52 . . PUNCT cana-1301 135 1 for for ADP cana-1301 135 2 the the DET cana-1301 135 3 point point NOUN cana-1301 135 4 of of ADP cana-1301 135 5 junction junction NOUN cana-1301 135 6 π‘Œ1 π‘Œ1 NOUN cana-1301 135 7 , , PUNCT cana-1301 135 8 intersecting intersect VERB cana-1301 135 9 value value NOUN cana-1301 135 10 Ο‡π‘Œ1 Ο‡π‘Œ1 NOUN cana-1301 135 11 = = PUNCT cana-1301 135 12 〈 〈 PROPN cana-1301 135 13 [ [ X cana-1301 135 14 0.84 0.84 NUM cana-1301 135 15 , , PUNCT cana-1301 135 16 0.71 0.71 NUM cana-1301 135 17 ] ] PUNCT cana-1301 135 18 , , PUNCT cana-1301 135 19 [ [ X cana-1301 135 20 0.63 0.63 NUM cana-1301 135 21 , , PUNCT cana-1301 135 22 0.85 0.85 NUM cana-1301 135 23 ] ] SYM cana-1301 135 24 βŒͺ βŒͺ NOUN cana-1301 135 25 and and CCONJ cana-1301 135 26 that that SCONJ cana-1301 135 27 for for ADP cana-1301 135 28 the the DET cana-1301 135 29 next next ADJ cana-1301 135 30 point point NOUN cana-1301 135 31 of of ADP cana-1301 135 32 junction junction NOUN cana-1301 135 33 π‘Œ2 π‘Œ2 NOUN cana-1301 135 34 , , PUNCT cana-1301 135 35 Ο‡π‘Œ2 Ο‡π‘Œ2 X cana-1301 135 36 = = PUNCT cana-1301 136 1 〈 〈 PROPN cana-1301 137 1 [ [ X cana-1301 137 2 0.74 0.74 NUM cana-1301 137 3 , , PUNCT cana-1301 137 4 βˆ’0.54 βˆ’0.54 X cana-1301 137 5 ] ] X cana-1301 137 6 , , PUNCT cana-1301 137 7 [ [ X cana-1301 137 8 0.47 0.47 NUM cana-1301 137 9 , , PUNCT cana-1301 137 10 0.77 0.77 NUM cana-1301 137 11 ] ] SYM cana-1301 137 12 βŒͺ βŒͺ NOUN cana-1301 137 13 . . PUNCT cana-1301 138 1 therefore therefore ADV cana-1301 138 2 , , PUNCT cana-1301 138 3 the the DET cana-1301 138 4 m m NOUN cana-1301 138 5 - - PUNCT cana-1301 138 6 bpf bpf NOUN cana-1301 138 7 planarity planarity NOUN cana-1301 138 8 value value NOUN cana-1301 138 9 for for ADP cana-1301 138 10 the the DET cana-1301 138 11 m m PROPN cana-1301 138 12 - - NOUN cana-1301 138 13 bpfmg bpfmg NOUN cana-1301 138 14 show show VERB cana-1301 138 15 in in ADP cana-1301 138 16 fig fig NOUN cana-1301 138 17 . . PUNCT cana-1301 139 1 3 3 X cana-1301 139 2 . . X cana-1301 139 3 is be AUX cana-1301 139 4 〈 〈 PROPN cana-1301 139 5 [ [ X cana-1301 139 6 0.39 0.39 NUM cana-1301 139 7 , , PUNCT cana-1301 139 8 βˆ’0.44 βˆ’0.44 X cana-1301 139 9 ] ] X cana-1301 139 10 , , PUNCT cana-1301 139 11 [ [ X cana-1301 139 12 0.48 0.48 NUM cana-1301 139 13 , , PUNCT cana-1301 139 14 βˆ’0.38 βˆ’0.38 NOUN cana-1301 139 15 ] ] X cana-1301 139 16 βŒͺ βŒͺ NOUN cana-1301 139 17 . . PUNCT cana-1301 140 1 from from ADP cana-1301 140 2 the the DET cana-1301 140 3 next next ADJ cana-1301 140 4 theorem theorem NOUN cana-1301 140 5 , , PUNCT cana-1301 140 6 we we PRON cana-1301 140 7 can can AUX cana-1301 140 8 compute compute VERB cana-1301 140 9 an an DET cana-1301 140 10 m m NOUN cana-1301 140 11 - - PUNCT cana-1301 140 12 bpf bpf NOUN cana-1301 140 13 planarity planarity NOUN cana-1301 140 14 value value NOUN cana-1301 140 15 for for ADP cana-1301 140 16 an an DET cana-1301 140 17 m m NOUN cana-1301 140 18 - - PUNCT cana-1301 140 19 bpfcmg bpfcmg NOUN cana-1301 140 20 . . PUNCT cana-1301 141 1 theorem theorem VERB cana-1301 141 2 3.1 3.1 NUM cana-1301 141 3 : : PUNCT cana-1301 141 4 let let VERB cana-1301 141 5 𝐺 𝐺 PROPN cana-1301 141 6 be be AUX cana-1301 141 7 an an DET cana-1301 141 8 m m NOUN cana-1301 141 9 - - NOUN cana-1301 141 10 bpfcmg bpfcmg NOUN cana-1301 141 11 . . PUNCT cana-1301 142 1 an an DET cana-1301 142 2 m m NOUN cana-1301 142 3 - - PUNCT cana-1301 142 4 bpf bpf NOUN cana-1301 142 5 planarity planarity NOUN cana-1301 142 6 value value NOUN cana-1301 142 7 , , PUNCT cana-1301 142 8 Ξ³ Ξ³ NOUN cana-1301 142 9 = = SYM cana-1301 142 10 〈 〈 PROPN cana-1301 142 11 [ [ X cana-1301 142 12 π‘ƒβ„ŽoΞ³+ π‘ƒβ„ŽoΞ³+ ADJ cana-1301 142 13 , , PUNCT cana-1301 142 14 π‘ƒβ„ŽoΞ³βˆ’]β„Ž=1 π‘ƒβ„ŽoΞ³βˆ’]β„Ž=1 PROPN cana-1301 142 15 π‘š π‘š ADP cana-1301 142 16 βŒͺ βŒͺ NOUN cana-1301 142 17 of of ADP cana-1301 142 18 𝐺 𝐺 PROPN cana-1301 142 19 is be AUX cana-1301 142 20 given give VERB cana-1301 142 21 by by ADP cana-1301 142 22 π‘ƒβ„ŽoΞ³+ π‘ƒβ„ŽoΞ³+ PROPN cana-1301 142 23 = = SYM cana-1301 142 24 1 1 NUM cana-1301 142 25 1+ds 1+ds NUM cana-1301 142 26 and and CCONJ cana-1301 142 27 π‘ƒβ„ŽoΞ³βˆ’ π‘ƒβ„ŽoΞ³βˆ’ ADJ cana-1301 142 28 = = SYM cana-1301 142 29 βˆ’1 βˆ’1 NOUN cana-1301 142 30 1+ds 1+ds NUM cana-1301 142 31 , , PUNCT cana-1301 142 32 β„Ž β„Ž PROPN cana-1301 142 33 = = SYM cana-1301 142 34 1 1 NUM cana-1301 142 35 , , PUNCT cana-1301 142 36 2 2 NUM cana-1301 142 37 , , PUNCT cana-1301 142 38 β‹― β‹― PROPN cana-1301 142 39 , , PUNCT cana-1301 142 40 π‘š π‘š PROPN cana-1301 142 41 , , PUNCT cana-1301 142 42 where where SCONJ cana-1301 142 43 ds ds ADJ cana-1301 142 44 is be AUX cana-1301 142 45 the the DET cana-1301 142 46 number number NOUN cana-1301 142 47 of of ADP cana-1301 142 48 point point NOUN cana-1301 142 49 of of ADP cana-1301 142 50 junctions junction NOUN cana-1301 142 51 between between ADP cana-1301 142 52 the the DET cana-1301 142 53 lines line NOUN cana-1301 142 54 in in ADP cana-1301 142 55 𝐺. 𝐺. NOUN cana-1301 142 56 proof proof NOUN cana-1301 142 57 : : PUNCT cana-1301 142 58 as as SCONJ cana-1301 142 59 𝐺 𝐺 PROPN cana-1301 142 60 is be AUX cana-1301 142 61 complete complete ADJ cana-1301 142 62 , , PUNCT cana-1301 142 63 we we PRON cana-1301 142 64 getπ‘ƒβ„Žoψr getπ‘ƒβ„Žoψr VERB cana-1301 142 65 + + ADV cana-1301 142 66 ( ( PUNCT cana-1301 142 67 πœ„πœ…)𝑒 πœ„πœ…)𝑒 NOUN cana-1301 142 68 = = PUNCT cana-1301 142 69 min{π‘ƒβ„Žoψq min{π‘ƒβ„Žoψq NOUN cana-1301 143 1 + + ADJ cana-1301 143 2 ( ( PUNCT cana-1301 143 3 πœ„ πœ„ NOUN cana-1301 143 4 ) ) PUNCT cana-1301 143 5 , , PUNCT cana-1301 143 6 π‘ƒβ„Žoψq π‘ƒβ„Žoψq PROPN cana-1301 143 7 + + PROPN cana-1301 143 8 ( ( PUNCT cana-1301 143 9 πœ… πœ… NOUN cana-1301 143 10 ) ) PUNCT cana-1301 143 11 } } PUNCT cana-1301 143 12 , , PUNCT cana-1301 143 13 π‘ƒβ„Žoψr π‘ƒβ„Žoψr PROPN cana-1301 143 14 βˆ’(πœ„πœ…)𝑒 βˆ’(πœ„πœ…)𝑒 PROPN cana-1301 143 15 = = ADJ cana-1301 143 16 max{π‘ƒβ„Žoψq max{π‘ƒβ„Žoψq PROPN cana-1301 143 17 βˆ’(πœ„ βˆ’(πœ„ PROPN cana-1301 143 18 ) ) PUNCT cana-1301 143 19 , , PUNCT cana-1301 143 20 π‘ƒβ„Žoψq π‘ƒβ„Žoψq PROPN cana-1301 143 21 βˆ’(πœ… βˆ’(πœ… NOUN cana-1301 143 22 ) ) PUNCT cana-1301 143 23 } } PUNCT cana-1301 143 24 for for ADP cana-1301 143 25 all all DET cana-1301 143 26 πœ„ πœ„ PROPN cana-1301 143 27 , , PUNCT cana-1301 143 28 πœ… πœ… PROPN cana-1301 143 29 ∈ ∈ PROPN cana-1301 143 30 𝑉 𝑉 PROPN cana-1301 143 31 , , PUNCT cana-1301 143 32 β„Ž β„Ž X cana-1301 143 33 = = SYM cana-1301 143 34 1 1 NUM cana-1301 143 35 , , PUNCT cana-1301 143 36 2 2 NUM cana-1301 143 37 , , PUNCT cana-1301 143 38 β‹― β‹― PROPN cana-1301 143 39 , , PUNCT cana-1301 143 40 π‘š π‘š PROPN cana-1301 143 41 and and CCONJ cana-1301 143 42 𝑒 𝑒 X cana-1301 143 43 = = SYM cana-1301 143 44 1,2 1,2 NUM cana-1301 143 45 , , PUNCT cana-1301 143 46 β‹― β‹― PROPN cana-1301 143 47 , , PUNCT cana-1301 143 48 𝑑. 𝑑. AUX cana-1301 143 49 let let VERB cana-1301 143 50 π‘Œ1 π‘Œ1 NOUN cana-1301 143 51 , , PUNCT cana-1301 143 52 π‘Œ2 π‘Œ2 PROPN cana-1301 143 53 , , PUNCT cana-1301 143 54 β‹― β‹― PROPN cana-1301 143 55 , , PUNCT cana-1301 143 56 π‘Œπ‘Ÿ π‘Œπ‘Ÿ PROPN cana-1301 143 57 be be AUX cana-1301 143 58 the the DET cana-1301 143 59 points point NOUN cana-1301 143 60 of of ADP cana-1301 143 61 junction junction NOUN cana-1301 143 62 connecting connect VERB cana-1301 143 63 the the DET cana-1301 143 64 lines line NOUN cana-1301 143 65 in in ADP cana-1301 143 66 𝐺. 𝐺. NOUN cana-1301 143 67 for for ADP cana-1301 143 68 a a DET cana-1301 143 69 line line NOUN cana-1301 143 70 πœ„πœ… πœ„πœ… NOUN cana-1301 143 71 in in ADP cana-1301 143 72 𝐺 𝐺 PROPN cana-1301 143 73 , , PUNCT cana-1301 143 74 〈 〈 PROPN cana-1301 144 1 [ [ X cana-1301 144 2 π‘ƒβ„ŽoπΈπœ„πœ… π‘ƒβ„ŽoπΈπœ„πœ… ADJ cana-1301 144 3 + + SYM cana-1301 144 4 , , PUNCT cana-1301 144 5 π‘ƒβ„ŽoπΈπœ„πœ… π‘ƒβ„ŽoπΈπœ„πœ… PROPN cana-1301 144 6 βˆ’]β„Ž=1 βˆ’]β„Ž=1 NOUN cana-1301 144 7 π‘š π‘š ADP cana-1301 144 8 βŒͺ βŒͺ NOUN cana-1301 144 9 = = SYM cana-1301 144 10 〈 〈 PROPN cana-1301 144 11 [ [ PUNCT cana-1301 144 12 π‘ƒβ„ŽoΟˆπ‘… π‘ƒβ„ŽoΟˆπ‘… PROPN cana-1301 144 13 + + PROPN cana-1301 144 14 ( ( PUNCT cana-1301 144 15 πœ„πœ…)𝑒 πœ„πœ…)𝑒 NOUN cana-1301 144 16 min{π‘ƒβ„Žoψq min{π‘ƒβ„Žoψq NOUN cana-1301 145 1 + + NOUN cana-1301 145 2 ( ( PUNCT cana-1301 145 3 πœ„),π‘ƒβ„Žoψq πœ„),π‘ƒβ„Žoψq PROPN cana-1301 145 4 + + PROPN cana-1301 145 5 ( ( PUNCT cana-1301 145 6 πœ… πœ… NOUN cana-1301 145 7 ) ) PUNCT cana-1301 145 8 } } PUNCT cana-1301 145 9 , , PUNCT cana-1301 145 10 π‘ƒβ„ŽoΟˆπ‘… π‘ƒβ„ŽoΟˆπ‘… PROPN cana-1301 145 11 βˆ’(πœ„πœ…)𝑒 βˆ’(πœ„πœ…)𝑒 PROPN cana-1301 145 12 max{π‘ƒβ„Žoψq max{π‘ƒβ„Žoψq PROPN cana-1301 145 13 βˆ’(πœ„),π‘ƒβ„Žoψq βˆ’(πœ„),π‘ƒβ„Žoψq X cana-1301 145 14 βˆ’(πœ… βˆ’(πœ… PROPN cana-1301 145 15 ) ) PUNCT cana-1301 145 16 } } PUNCT cana-1301 145 17 ] ] PUNCT cana-1301 146 1 β„Ž=1 β„Ž=1 PROPN cana-1301 146 2 π‘š π‘š ADP cana-1301 146 3 βŒͺ βŒͺ NOUN cana-1301 146 4 = = SYM cana-1301 146 5 〈 〈 PROPN cana-1301 146 6 [ [ X cana-1301 146 7 1 1 NUM cana-1301 146 8 , , PUNCT cana-1301 146 9 1]β„Ž=1 1]β„Ž=1 NUM cana-1301 146 10 π‘š π‘š ADP cana-1301 146 11 βŒͺ βŒͺ NOUN cana-1301 146 12 . . PUNCT cana-1301 147 1 consequently consequently ADV cana-1301 147 2 , , PUNCT cana-1301 147 3 for for ADP cana-1301 147 4 the the DET cana-1301 147 5 point point NOUN cana-1301 147 6 π‘Œ1 π‘Œ1 NOUN cana-1301 147 7 which which PRON cana-1301 147 8 is be AUX cana-1301 147 9 the the DET cana-1301 147 10 point point NOUN cana-1301 147 11 of of ADP cana-1301 147 12 junctions junction NOUN cana-1301 147 13 connecting connect VERB cana-1301 147 14 the the DET cana-1301 147 15 lines line NOUN cana-1301 147 16 πœ„πœ… πœ„πœ… NOUN cana-1301 147 17 and and CCONJ cana-1301 147 18 𝜐𝜏 𝜐𝜏 ADP cana-1301 147 19 , , PUNCT cana-1301 147 20 the the DET cana-1301 147 21 junction junction NOUN cana-1301 147 22 value value NOUN cana-1301 147 23 is be AUX cana-1301 147 24 Ο‡π‘Œ1 Ο‡π‘Œ1 NOUN cana-1301 147 25 = = SYM cana-1301 147 26 〈 〈 PROPN cana-1301 147 27 [ [ X cana-1301 147 28 1 1 NUM cana-1301 147 29 , , PUNCT cana-1301 147 30 1]β„Ž=1 1]β„Ž=1 NUM cana-1301 147 31 π‘š π‘š ADP cana-1301 147 32 βŒͺ βŒͺ NOUN cana-1301 147 33 . . PUNCT cana-1301 148 1 hence hence ADV cana-1301 148 2 , , PUNCT cana-1301 148 3 Ο‡π‘Œβ„Ž Ο‡π‘Œβ„Ž PROPN cana-1301 148 4 = = PUNCT cana-1301 148 5 〈 〈 PROPN cana-1301 148 6 [ [ X cana-1301 148 7 1 1 NUM cana-1301 148 8 , , PUNCT cana-1301 148 9 1]β„Ž=1 1]β„Ž=1 NUM cana-1301 148 10 π‘š π‘š ADP cana-1301 148 11 βŒͺ βŒͺ NOUN cana-1301 148 12 for for ADP cana-1301 148 13 β„Ž β„Ž X cana-1301 148 14 = = SYM cana-1301 148 15 1 1 NUM cana-1301 148 16 , , PUNCT cana-1301 148 17 2 2 NUM cana-1301 148 18 , , PUNCT cana-1301 148 19 β‹― β‹― NOUN cana-1301 148 20 , , PUNCT cana-1301 148 21 π‘š. π‘š. VERB cana-1301 148 22 now now ADV cana-1301 148 23 for for ADP cana-1301 148 24 β„Ž β„Ž PROPN cana-1301 148 25 = = SYM cana-1301 148 26 1 1 NUM cana-1301 148 27 , , PUNCT cana-1301 148 28 2 2 NUM cana-1301 148 29 , , PUNCT cana-1301 148 30 β‹― β‹― PROPN cana-1301 148 31 , , PUNCT cana-1301 148 32 π‘š π‘š PROPN cana-1301 148 33 , , PUNCT cana-1301 148 34 〈 〈 PROPN cana-1301 148 35 [ [ X cana-1301 148 36 π‘ƒβ„ŽoΞ³+ π‘ƒβ„ŽoΞ³+ ADJ cana-1301 148 37 , , PUNCT cana-1301 148 38 π‘ƒβ„ŽoΞ³βˆ’ π‘ƒβ„ŽoΞ³βˆ’ ADJ cana-1301 148 39 ] ] NOUN cana-1301 148 40 βŒͺ βŒͺ NOUN cana-1301 148 41 = = SYM cana-1301 148 42 〈 〈 PROPN cana-1301 148 43 [ [ PUNCT cana-1301 148 44 1 1 NUM cana-1301 148 45 1 1 NUM cana-1301 148 46 + + CCONJ cana-1301 148 47 { { PUNCT cana-1301 148 48 π‘ƒβ„ŽoΟ‡π‘Œ1 π‘ƒβ„ŽoΟ‡π‘Œ1 NOUN cana-1301 148 49 + + CCONJ cana-1301 148 50 + + NUM cana-1301 148 51 π‘ƒβ„ŽoΟ‡π‘Œ2 π‘ƒβ„ŽoΟ‡π‘Œ2 NOUN cana-1301 148 52 + + X cana-1301 148 53 + + CCONJ cana-1301 148 54 β‹― β‹― VERB cana-1301 148 55 + + NUM cana-1301 148 56 π‘ƒβ„ŽoΟ‡π‘Œπ‘Ÿ π‘ƒβ„ŽoΟ‡π‘Œπ‘Ÿ NOUN cana-1301 148 57 + + CCONJ cana-1301 148 58 } } PUNCT cana-1301 148 59 , , PUNCT cana-1301 148 60 βˆ’1 βˆ’1 NOUN cana-1301 148 61 1 1 NUM cana-1301 148 62 + + CCONJ cana-1301 148 63 { { PUNCT cana-1301 148 64 π‘ƒβ„ŽoΟ‡π‘Œ1 π‘ƒβ„ŽoΟ‡π‘Œ1 NOUN cana-1301 148 65 βˆ’ βˆ’ NOUN cana-1301 148 66 + + NUM cana-1301 148 67 π‘ƒβ„ŽoΟ‡π‘Œ2 π‘ƒβ„ŽoΟ‡π‘Œ2 NOUN cana-1301 148 68 βˆ’ βˆ’ PROPN cana-1301 149 1 + + CCONJ cana-1301 149 2 β‹― β‹― PROPN cana-1301 149 3 + + CCONJ cana-1301 149 4 π‘ƒβ„ŽoΟ‡π‘Œπ‘Ÿ π‘ƒβ„ŽoΟ‡π‘Œπ‘Ÿ NOUN cana-1301 149 5 βˆ’ βˆ’ NOUN cana-1301 149 6 } } PUNCT cana-1301 149 7 ] ] PUNCT cana-1301 149 8 βŒͺ βŒͺ NOUN cana-1301 149 9 = = SYM cana-1301 149 10 〈 〈 PROPN cana-1301 149 11 [ [ PUNCT cana-1301 149 12 1 1 NUM cana-1301 149 13 1 1 NUM cana-1301 149 14 + + CCONJ cana-1301 149 15 { { PUNCT cana-1301 149 16 1 1 NUM cana-1301 149 17 + + SYM cana-1301 149 18 1 1 NUM cana-1301 150 1 + + CCONJ cana-1301 150 2 β‹― β‹― VERB cana-1301 150 3 + + CCONJ cana-1301 150 4 1 1 NUM cana-1301 150 5 } } PUNCT cana-1301 150 6 , , PUNCT cana-1301 150 7 βˆ’1 βˆ’1 NOUN cana-1301 150 8 1 1 NUM cana-1301 150 9 + + CCONJ cana-1301 150 10 { { PUNCT cana-1301 150 11 1 1 NUM cana-1301 150 12 + + SYM cana-1301 150 13 1 1 NUM cana-1301 151 1 + + CCONJ cana-1301 151 2 β‹― β‹― VERB cana-1301 151 3 + + CCONJ cana-1301 151 4 1 1 NUM cana-1301 151 5 } } PUNCT cana-1301 151 6 ] ] SYM cana-1301 151 7 βŒͺ βŒͺ NOUN cana-1301 151 8 = = SYM cana-1301 151 9 〈 〈 PROPN cana-1301 151 10 [ [ PUNCT cana-1301 151 11 1 1 NUM cana-1301 151 12 1 1 NUM cana-1301 151 13 + + CCONJ cana-1301 151 14 ds ds ADJ cana-1301 151 15 , , PUNCT cana-1301 151 16 βˆ’1 βˆ’1 NOUN cana-1301 151 17 1 1 NUM cana-1301 151 18 + + CCONJ cana-1301 151 19 ds ds ADJ cana-1301 151 20 ] ] PUNCT cana-1301 151 21 βŒͺ βŒͺ NOUN cana-1301 151 22 . . PUNCT cana-1301 152 1 therefore therefore ADV cana-1301 152 2 , , PUNCT cana-1301 152 3 m m NOUN cana-1301 152 4 - - PUNCT cana-1301 152 5 bpf bpf NOUN cana-1301 152 6 planarity planarity NOUN cana-1301 152 7 Ξ³ Ξ³ NOUN cana-1301 152 8 is be AUX cana-1301 152 9 given give VERB cana-1301 152 10 by〈[π‘ƒβ„ŽoΞ³+ by〈[π‘ƒβ„ŽoΞ³+ ADP cana-1301 152 11 , , PUNCT cana-1301 152 12 π‘ƒβ„ŽoΞ³βˆ’]β„Ž=1 π‘ƒβ„ŽoΞ³βˆ’]β„Ž=1 PROPN cana-1301 152 13 π‘š π‘š X cana-1301 152 14 βŒͺ βŒͺ PROPN cana-1301 152 15 where where SCONJ cana-1301 152 16 π‘ƒβ„ŽoΞ³+ π‘ƒβ„ŽoΞ³+ NOUN cana-1301 152 17 = = SYM cana-1301 152 18 1 1 NUM cana-1301 152 19 1+ds 1+ds NUM cana-1301 152 20 and and CCONJ cana-1301 152 21 π‘ƒβ„ŽoΞ³βˆ’ π‘ƒβ„ŽoΞ³βˆ’ ADJ cana-1301 152 22 = = SYM cana-1301 152 23 βˆ’1 βˆ’1 NOUN cana-1301 152 24 1+ds 1+ds NUM cana-1301 152 25 , , PUNCT cana-1301 152 26 β„Ž β„Ž PROPN cana-1301 152 27 = = SYM cana-1301 152 28 1 1 NUM cana-1301 152 29 , , PUNCT cana-1301 152 30 2 2 NUM cana-1301 152 31 , , PUNCT cana-1301 152 32 β‹― β‹― NOUN cana-1301 152 33 , , PUNCT cana-1301 152 34 π‘š. π‘š. ADJ cana-1301 152 35 communications communication NOUN cana-1301 152 36 on on ADP cana-1301 152 37 applied apply VERB cana-1301 152 38 nonlinear nonlinear ADJ cana-1301 152 39 analysis analysis NOUN cana-1301 152 40 issn issn NOUN cana-1301 152 41 : : PUNCT cana-1301 152 42 1074 1074 NUM cana-1301 152 43 - - PUNCT cana-1301 152 44 133x 133x NUM cana-1301 152 45 vol vol NOUN cana-1301 152 46 31 31 NUM cana-1301 152 47 no no NOUN cana-1301 152 48 . . PUNCT cana-1301 153 1 7s 7 NOUN cana-1301 153 2 ( ( PUNCT cana-1301 153 3 2024 2024 NUM cana-1301 153 4 ) ) PUNCT cana-1301 153 5 268 268 NUM cana-1301 153 6 https://internationalpubls.com https://internationalpubls.com X cana-1301 153 7 theorem theorem VERB cana-1301 153 8 3.2 3.2 NUM cana-1301 153 9 : : PUNCT cana-1301 153 10 let let VERB cana-1301 153 11 𝐺 𝐺 PRON cana-1301 153 12 be be AUX cana-1301 153 13 an an DET cana-1301 153 14 m m NOUN cana-1301 153 15 - - NOUN cana-1301 153 16 bpfpg bpfpg NOUN cana-1301 153 17 with with ADP cana-1301 153 18 m m PROPN cana-1301 153 19 - - PUNCT cana-1301 153 20 bpf bpf NOUN cana-1301 153 21 planarity planarity NOUN cana-1301 153 22 value value NOUN cana-1301 153 23 〈 〈 PROPN cana-1301 153 24 [ [ X cana-1301 153 25 π‘ƒβ„ŽoΞ³+ π‘ƒβ„ŽoΞ³+ ADJ cana-1301 153 26 , , PUNCT cana-1301 153 27 π‘ƒβ„ŽoΞ³βˆ’]β„Ž=1 π‘ƒβ„ŽoΞ³βˆ’]β„Ž=1 PROPN cana-1301 153 28 π‘š π‘š PRON cana-1301 153 29 βŒͺ βŒͺ NOUN cana-1301 153 30 is be AUX cana-1301 153 31 such such ADJ cana-1301 153 32 that that SCONJ cana-1301 153 33 π‘ƒβ„ŽoΞ³+ π‘ƒβ„ŽoΞ³+ PROPN cana-1301 153 34 β‰₯ β‰₯ X cana-1301 153 35 0.5 0.5 NUM cana-1301 153 36 , , PUNCT cana-1301 153 37 or or CCONJ cana-1301 153 38 π‘ƒβ„ŽoΞ³βˆ’ π‘ƒβ„ŽoΞ³βˆ’ ADJ cana-1301 153 39 ≀ ≀ NUM cana-1301 153 40 0.5 0.5 NUM cana-1301 153 41 for for ADP cana-1301 153 42 β„Ž β„Ž PROPN cana-1301 153 43 = = SYM cana-1301 153 44 1 1 NUM cana-1301 153 45 , , PUNCT cana-1301 153 46 2 2 NUM cana-1301 153 47 , , PUNCT cana-1301 153 48 β‹― β‹― NOUN cana-1301 153 49 , , PUNCT cana-1301 153 50 π‘š. π‘š. NOUN cana-1301 153 51 then then ADV cana-1301 153 52 the the DET cana-1301 153 53 number number NOUN cana-1301 153 54 of of ADP cana-1301 153 55 points point NOUN cana-1301 153 56 of of ADP cana-1301 153 57 junction junction NOUN cana-1301 153 58 between between ADP cana-1301 153 59 mbpf mbpf PROPN cana-1301 153 60 strong strong ADJ cana-1301 153 61 lines line NOUN cana-1301 153 62 in in ADP cana-1301 153 63 𝐺 𝐺 PROPN cana-1301 153 64 is be AUX cana-1301 153 65 at at ADP cana-1301 153 66 most most ADJ cana-1301 153 67 one one NUM cana-1301 153 68 . . PUNCT cana-1301 154 1 proof proof NOUN cana-1301 154 2 : : PUNCT cana-1301 154 3 let let VERB cana-1301 154 4 𝐺 𝐺 PRON cana-1301 154 5 be be AUX cana-1301 154 6 a a DET cana-1301 154 7 strong strong ADJ cana-1301 154 8 m m NOUN cana-1301 154 9 - - NOUN cana-1301 154 10 bpfpg bpfpg NOUN cana-1301 154 11 . . PUNCT cana-1301 155 1 assume assume VERB cana-1301 155 2 that that SCONJ cana-1301 155 3 𝐺 𝐺 PROPN cana-1301 155 4 has have VERB cana-1301 155 5 at at ADV cana-1301 155 6 least least ADV cana-1301 155 7 two two NUM cana-1301 155 8 point point NOUN cana-1301 155 9 of of ADP cana-1301 155 10 junctions junction NOUN cana-1301 155 11 π‘Œ1 π‘Œ1 NOUN cana-1301 155 12 and and CCONJ cana-1301 155 13 π‘Œ2 π‘Œ2 NOUN cana-1301 155 14 between between ADP cana-1301 155 15 the the DET cana-1301 155 16 two two NUM cana-1301 155 17 strong strong ADJ cana-1301 155 18 lines line NOUN cana-1301 155 19 in in ADP cana-1301 155 20 𝐺. 𝐺. NOUN cana-1301 155 21 for for ADP cana-1301 155 22 any any DET cana-1301 155 23 strong strong ADJ cana-1301 155 24 line line NOUN cana-1301 155 25 ( ( PUNCT cana-1301 155 26 πœ„πœ… πœ„πœ… PRON cana-1301 155 27 〈 〈 PROPN cana-1301 155 28 [ [ X cana-1301 155 29 π‘ƒβ„Žoψr π‘ƒβ„Žoψr PROPN cana-1301 155 30 + + PROPN cana-1301 155 31 ( ( PUNCT cana-1301 155 32 πœ„πœ…)𝑒 πœ„πœ…)𝑒 NOUN cana-1301 155 33 , , PUNCT cana-1301 155 34 π‘ƒβ„Žoψr π‘ƒβ„Žoψr PROPN cana-1301 155 35 βˆ’(πœ„πœ…)𝑒 βˆ’(πœ„πœ…)𝑒 PROPN cana-1301 155 36 ] ] PUNCT cana-1301 155 37 h=1 h=1 VERB cana-1301 155 38 m m VERB cana-1301 155 39 βŒͺ βŒͺ NOUN cana-1301 155 40 ) ) PUNCT cana-1301 156 1 π‘ƒβ„Žoψr π‘ƒβ„Žoψr PROPN cana-1301 156 2 + + PROPN cana-1301 156 3 ( ( PUNCT cana-1301 156 4 πœ„πœ…)𝑒 πœ„πœ…)𝑒 NOUN cana-1301 156 5 β‰₯ β‰₯ NOUN cana-1301 156 6 1 1 NUM cana-1301 156 7 2 2 NUM cana-1301 156 8 min{π‘ƒβ„Žoψq min{π‘ƒβ„Žoψq NOUN cana-1301 156 9 + + ADJ cana-1301 156 10 ( ( PUNCT cana-1301 156 11 πœ„ πœ„ NOUN cana-1301 156 12 ) ) PUNCT cana-1301 156 13 , , PUNCT cana-1301 156 14 π‘ƒβ„Žoψq π‘ƒβ„Žoψq PROPN cana-1301 156 15 + + PROPN cana-1301 156 16 ( ( PUNCT cana-1301 156 17 πœ… πœ… NOUN cana-1301 156 18 ) ) PUNCT cana-1301 156 19 } } PUNCT cana-1301 156 20 , , PUNCT cana-1301 156 21 π‘ƒβ„Žoψr π‘ƒβ„Žoψr PROPN cana-1301 156 22 βˆ’(πœ„πœ…)𝑒 βˆ’(πœ„πœ…)𝑒 PROPN cana-1301 156 23 ≀ ≀ NUM cana-1301 156 24 1 1 NUM cana-1301 156 25 2 2 NUM cana-1301 156 26 max{π‘ƒβ„Žoψq max{π‘ƒβ„Žoψq NOUN cana-1301 156 27 βˆ’(πœ„ βˆ’(πœ„ NOUN cana-1301 156 28 ) ) PUNCT cana-1301 156 29 , , PUNCT cana-1301 156 30 π‘ƒβ„Žoψq π‘ƒβ„Žoψq PROPN cana-1301 156 31 βˆ’(πœ… βˆ’(πœ… NOUN cana-1301 156 32 ) ) PUNCT cana-1301 156 33 } } PUNCT cana-1301 156 34 . . PUNCT cana-1301 157 1 this this PRON cana-1301 157 2 proves prove VERB cana-1301 157 3 that that SCONJ cana-1301 157 4 π‘ƒβ„ŽoΟ‡π‘Œ π‘ƒβ„ŽoΟ‡π‘Œ PROPN cana-1301 157 5 + + NUM cana-1301 157 6 β‰₯ β‰₯ NOUN cana-1301 157 7 0.5 0.5 NUM cana-1301 157 8 or or CCONJ cana-1301 157 9 π‘ƒβ„ŽoΟ‡π‘Œ π‘ƒβ„ŽoΟ‡π‘Œ PROPN cana-1301 157 10 βˆ’ βˆ’ PROPN cana-1301 157 11 ≀ ≀ ADV cana-1301 157 12 0.5 0.5 NUM cana-1301 157 13 for for ADP cana-1301 157 14 β„Ž β„Ž PROPN cana-1301 157 15 = = SYM cana-1301 157 16 1 1 NUM cana-1301 157 17 , , PUNCT cana-1301 157 18 2 2 NUM cana-1301 157 19 , , PUNCT cana-1301 157 20 β‹― β‹― NOUN cana-1301 157 21 , , PUNCT cana-1301 157 22 π‘š. π‘š. VERB cana-1301 157 23 thus thus ADV cana-1301 157 24 for for ADP cana-1301 157 25 2 2 NUM cana-1301 157 26 intersecting intersect VERB cana-1301 157 27 strong strong ADJ cana-1301 157 28 lines line NOUN cana-1301 157 29 ( ( PUNCT cana-1301 157 30 πœ„πœ… πœ„πœ… PRON cana-1301 157 31 〈 〈 PROPN cana-1301 157 32 [ [ X cana-1301 157 33 π‘ƒβ„Žoψr π‘ƒβ„Žoψr PROPN cana-1301 157 34 + + PROPN cana-1301 157 35 ( ( PUNCT cana-1301 157 36 πœ„πœ…)𝑒 πœ„πœ…)𝑒 NOUN cana-1301 157 37 , , PUNCT cana-1301 157 38 π‘ƒβ„Žoψr π‘ƒβ„Žoψr PROPN cana-1301 157 39 βˆ’(πœ„πœ…)𝑒 βˆ’(πœ„πœ…)𝑒 PROPN cana-1301 157 40 ] ] PUNCT cana-1301 157 41 h=1 h=1 NOUN cana-1301 157 42 m m VERB cana-1301 157 43 βŒͺ βŒͺ NOUN cana-1301 157 44 ) ) PUNCT cana-1301 157 45 and and CCONJ cana-1301 157 46 ( ( PUNCT cana-1301 157 47 𝜐𝜏 𝜐𝜏 ADP cana-1301 157 48 〈 〈 PROPN cana-1301 157 49 [ [ X cana-1301 157 50 π‘ƒβ„Žoψr π‘ƒβ„Žoψr PROPN cana-1301 157 51 + + PROPN cana-1301 157 52 ( ( PUNCT cana-1301 157 53 𝜐𝜏)𝑠 𝜐𝜏)𝑠 PROPN cana-1301 157 54 , , PUNCT cana-1301 157 55 π‘ƒβ„Žoψr π‘ƒβ„Žoψr PROPN cana-1301 157 56 βˆ’(𝜐𝜏)𝑠 βˆ’(𝜐𝜏)𝑠 PROPN cana-1301 157 57 ] ] PUNCT cana-1301 157 58 h=1 h=1 NOUN cana-1301 157 59 m m VERB cana-1301 157 60 βŒͺ βŒͺ NOUN cana-1301 157 61 ) ) PUNCT cana-1301 157 62 , , PUNCT cana-1301 157 63 π‘ƒβ„ŽoπΈπœ„πœ… π‘ƒβ„ŽoπΈπœ„πœ… PROPN cana-1301 157 64 + + CCONJ cana-1301 157 65 + + ADJ cana-1301 157 66 π‘ƒβ„Žo𝐸𝜐𝜏 π‘ƒβ„Žo𝐸𝜐𝜏 ADJ cana-1301 157 67 + + CCONJ cana-1301 157 68 2 2 NUM cana-1301 157 69 β‰₯ β‰₯ NOUN cana-1301 157 70 0.5 0.5 NUM cana-1301 157 71 , , PUNCT cana-1301 157 72 π‘ƒβ„ŽoπΈπœ„πœ… π‘ƒβ„ŽoπΈπœ„πœ… PROPN cana-1301 157 73 βˆ’ βˆ’ ADP cana-1301 158 1 + + ADJ cana-1301 158 2 π‘ƒβ„Žo𝐸𝜐𝜏 π‘ƒβ„Žo𝐸𝜐𝜏 ADJ cana-1301 158 3 βˆ’ βˆ’ PROPN cana-1301 158 4 2 2 NUM cana-1301 158 5 ≀ ≀ NUM cana-1301 158 6 0.5 0.5 NUM cana-1301 158 7 , , PUNCT cana-1301 158 8 for for ADP cana-1301 158 9 β„Ž β„Ž PROPN cana-1301 158 10 = = SYM cana-1301 158 11 1 1 NUM cana-1301 158 12 , , PUNCT cana-1301 158 13 2 2 NUM cana-1301 158 14 , , PUNCT cana-1301 158 15 β‹― β‹― NOUN cana-1301 158 16 , , PUNCT cana-1301 158 17 π‘š. π‘š. NOUN cana-1301 158 18 that that PRON cana-1301 158 19 is be AUX cana-1301 158 20 , , PUNCT cana-1301 158 21 π‘ƒβ„ŽoΟ‡π‘Œ1 π‘ƒβ„ŽoΟ‡π‘Œ1 NOUN cana-1301 158 22 + + CCONJ cana-1301 158 23 β‰₯ β‰₯ NOUN cana-1301 158 24 0.5 0.5 NUM cana-1301 158 25 , , PUNCT cana-1301 158 26 π‘ƒβ„ŽoΟ‡π‘Œ1 π‘ƒβ„ŽoΟ‡π‘Œ1 NOUN cana-1301 158 27 βˆ’ βˆ’ PROPN cana-1301 158 28 ≀ ≀ NUM cana-1301 158 29 0.5 0.5 NUM cana-1301 158 30 . . PUNCT cana-1301 159 1 similarlyπ‘ƒβ„ŽoΟ‡π‘Œ2 similarlyπ‘ƒβ„ŽoΟ‡π‘Œ2 NOUN cana-1301 159 2 + + NUM cana-1301 159 3 β‰₯ β‰₯ NOUN cana-1301 159 4 0.5 0.5 NUM cana-1301 159 5 , , PUNCT cana-1301 159 6 π‘ƒβ„ŽoΟ‡π‘Œ2 π‘ƒβ„ŽoΟ‡π‘Œ2 NOUN cana-1301 159 7 βˆ’ βˆ’ PROPN cana-1301 159 8 ≀ ≀ NUM cana-1301 159 9 0.5 0.5 NUM cana-1301 159 10 . . PUNCT cana-1301 160 1 this this PRON cana-1301 160 2 implies imply VERB cana-1301 160 3 that1 that1 NOUN cana-1301 161 1 + + CCONJ cana-1301 161 2 π‘ƒβ„ŽoΟ‡π‘Œ1 π‘ƒβ„ŽoΟ‡π‘Œ1 NOUN cana-1301 161 3 + + CCONJ cana-1301 161 4 + + NUM cana-1301 161 5 π‘ƒβ„ŽoΟ‡π‘Œ2 π‘ƒβ„ŽoΟ‡π‘Œ2 NOUN cana-1301 161 6 + + CCONJ cana-1301 161 7 β‰₯ β‰₯ NOUN cana-1301 161 8 2 2 NUM cana-1301 161 9 , , PUNCT cana-1301 161 10 1 1 NUM cana-1301 161 11 + + NUM cana-1301 161 12 π‘ƒβ„ŽoΟ‡π‘Œ1 π‘ƒβ„ŽoΟ‡π‘Œ1 NOUN cana-1301 161 13 βˆ’ βˆ’ NOUN cana-1301 161 14 + + NUM cana-1301 161 15 π‘ƒβ„ŽoΟ‡π‘Œ2 π‘ƒβ„ŽoΟ‡π‘Œ2 NOUN cana-1301 161 16 βˆ’ βˆ’ PROPN cana-1301 161 17 ≀ ≀ NUM cana-1301 161 18 2 2 NUM cana-1301 161 19 . . PUNCT cana-1301 162 1 therefore therefore ADV cana-1301 162 2 , , PUNCT cana-1301 162 3 π‘ƒβ„ŽoΞ³+ π‘ƒβ„ŽoΞ³+ NOUN cana-1301 162 4 = = SYM cana-1301 162 5 1 1 NUM cana-1301 162 6 1+π‘ƒβ„ŽoΟ‡π‘Œ1 1+π‘ƒβ„ŽoΟ‡π‘Œ1 NUM cana-1301 162 7 + + PUNCT cana-1301 163 1 + + ADJ cana-1301 163 2 π‘ƒβ„ŽoΟ‡π‘Œ2 π‘ƒβ„ŽoΟ‡π‘Œ2 NOUN cana-1301 163 3 + + CCONJ cana-1301 163 4 ≀ ≀ NUM cana-1301 163 5 0.5 0.5 NUM cana-1301 163 6 , , PUNCT cana-1301 163 7 π‘ƒβ„ŽoΞ³βˆ’ π‘ƒβ„ŽoΞ³βˆ’ NUM cana-1301 163 8 = = SYM cana-1301 163 9 βˆ’1 βˆ’1 NOUN cana-1301 163 10 1+π‘ƒβ„ŽoΟ‡π‘Œ1 1+π‘ƒβ„ŽoΟ‡π‘Œ1 NUM cana-1301 163 11 βˆ’ βˆ’ PROPN cana-1301 164 1 + + NOUN cana-1301 164 2 π‘ƒβ„ŽoΟ‡π‘Œ2 π‘ƒβ„ŽoΟ‡π‘Œ2 NOUN cana-1301 164 3 βˆ’ βˆ’ PROPN cana-1301 164 4 β‰₯ β‰₯ NOUN cana-1301 164 5 βˆ’0.5 βˆ’0.5 VERB cana-1301 164 6 for for ADP cana-1301 164 7 β„Ž β„Ž PROPN cana-1301 164 8 = = SYM cana-1301 164 9 1 1 NUM cana-1301 164 10 , , PUNCT cana-1301 164 11 2 2 NUM cana-1301 164 12 , , PUNCT cana-1301 164 13 β‹― β‹― NOUN cana-1301 164 14 , , PUNCT cana-1301 164 15 π‘š. π‘š. NOUN cana-1301 164 16 it it PRON cana-1301 164 17 contradicts contradict VERB cana-1301 164 18 the the DET cana-1301 164 19 assumption assumption NOUN cana-1301 164 20 that that SCONJ cana-1301 164 21 the the DET cana-1301 164 22 m m PROPN cana-1301 164 23 - - PUNCT cana-1301 164 24 bpfg bpfg PROPN cana-1301 164 25 is be AUX cana-1301 164 26 strong strong ADJ cana-1301 164 27 m m NOUN cana-1301 164 28 - - NOUN cana-1301 164 29 bpfpg bpfpg NOUN cana-1301 164 30 . . PUNCT cana-1301 165 1 therefore therefore ADV cana-1301 165 2 , , PUNCT cana-1301 165 3 there there PRON cana-1301 165 4 can can AUX cana-1301 165 5 not not PART cana-1301 165 6 be be AUX cana-1301 165 7 two two NUM cana-1301 165 8 points point NOUN cana-1301 165 9 where where SCONJ cana-1301 165 10 strong strong ADJ cana-1301 165 11 the the DET cana-1301 165 12 lines line NOUN cana-1301 165 13 are be AUX cana-1301 165 14 intersected intersect VERB cana-1301 165 15 . . PUNCT cana-1301 166 1 naturally naturally ADV cana-1301 166 2 , , PUNCT cana-1301 166 3 the the DET cana-1301 166 4 m m NOUN cana-1301 166 5 - - PUNCT cana-1301 166 6 bpf bpf NOUN cana-1301 166 7 planarity planarity NOUN cana-1301 166 8 value value NOUN cana-1301 166 9 drops drop VERB cana-1301 166 10 as as ADP cana-1301 166 11 the the DET cana-1301 166 12 number number NOUN cana-1301 166 13 of of ADP cana-1301 166 14 strong strong ADJ cana-1301 166 15 m m PROPN cana-1301 166 16 - - PUNCT cana-1301 166 17 bpf bpf NOUN cana-1301 166 18 line line NOUN cana-1301 166 19 junction junction NOUN cana-1301 166 20 points point NOUN cana-1301 166 21 rises rise NOUN cana-1301 166 22 . . PUNCT cana-1301 167 1 similar similar ADJ cana-1301 167 2 to to ADP cana-1301 167 3 this this PRON cana-1301 167 4 , , PUNCT cana-1301 167 5 if if SCONJ cana-1301 167 6 there there PRON cana-1301 167 7 is be VERB cana-1301 167 8 just just ADV cana-1301 167 9 one one NUM cana-1301 167 10 point point NOUN cana-1301 167 11 where where SCONJ cana-1301 167 12 2 2 NUM cana-1301 167 13 strong strong ADJ cana-1301 167 14 lines line NOUN cana-1301 167 15 cross cross VERB cana-1301 167 16 , , PUNCT cana-1301 167 17 an an DET cana-1301 167 18 m m NOUN cana-1301 167 19 - - PUNCT cana-1301 167 20 bpf bpf NOUN cana-1301 167 21 planarity planarity NOUN cana-1301 167 22 values value NOUN cana-1301 167 23 are be AUX cana-1301 167 24 π‘ƒβ„ŽoΞ³+ π‘ƒβ„ŽoΞ³+ ADJ cana-1301 167 25 < < X cana-1301 167 26 0.5 0.5 NUM cana-1301 167 27 , , PUNCT cana-1301 167 28 and and CCONJ cana-1301 167 29 π‘ƒβ„ŽoΞ³βˆ’ π‘ƒβ„ŽoΞ³βˆ’ VERB cana-1301 167 30 > > X cana-1301 167 31 βˆ’0.5 βˆ’0.5 PROPN cana-1301 167 32 for for ADP cana-1301 167 33 β„Ž β„Ž NOUN cana-1301 167 34 = = SYM cana-1301 167 35 1 1 NUM cana-1301 167 36 , , PUNCT cana-1301 167 37 2 2 NUM cana-1301 167 38 , , PUNCT cana-1301 167 39 β‹― β‹― NOUN cana-1301 167 40 , , PUNCT cana-1301 167 41 π‘š. π‘š. VERB cana-1301 167 42 a a DET cana-1301 167 43 strong strong ADJ cana-1301 167 44 m m NOUN cana-1301 167 45 - - NOUN cana-1301 167 46 bpfpg bpfpg NOUN cana-1301 167 47 is be AUX cana-1301 167 48 any any DET cana-1301 167 49 m m NOUN cana-1301 167 50 - - NOUN cana-1301 167 51 bpfpg bpfpg NOUN cana-1301 167 52 that that PRON cana-1301 167 53 has have VERB cana-1301 167 54 no no DET cana-1301 167 55 crossing crossing NOUN cana-1301 167 56 between between ADP cana-1301 167 57 the the DET cana-1301 167 58 lines line NOUN cana-1301 167 59 . . PUNCT cana-1301 168 1 thus thus ADV cana-1301 168 2 , , PUNCT cana-1301 168 3 we we PRON cana-1301 168 4 get get VERB cana-1301 168 5 the the DET cana-1301 168 6 conclusion conclusion NOUN cana-1301 168 7 that that SCONJ cana-1301 168 8 there there PRON cana-1301 168 9 is be VERB cana-1301 168 10 only only ADV cana-1301 168 11 one one NUM cana-1301 168 12 position position NOUN cana-1301 168 13 where where SCONJ cana-1301 168 14 the the DET cana-1301 168 15 strong strong ADJ cana-1301 168 16 lines line NOUN cana-1301 168 17 of of ADP cana-1301 168 18 𝐺 𝐺 PROPN cana-1301 168 19 can can AUX cana-1301 168 20 connect connect VERB cana-1301 168 21 . . PUNCT cana-1301 169 1 theorem theorem VERB cana-1301 169 2 3.3 3.3 NUM cana-1301 169 3 : : PUNCT cana-1301 169 4 let let VERB cana-1301 169 5 𝐺 𝐺 PRON cana-1301 169 6 be be AUX cana-1301 169 7 an an DET cana-1301 169 8 m m NOUN cana-1301 169 9 - - NOUN cana-1301 169 10 bpfpg bpfpg NOUN cana-1301 169 11 with with ADP cana-1301 169 12 an an DET cana-1301 169 13 m m NOUN cana-1301 169 14 - - PUNCT cana-1301 169 15 bpf bpf NOUN cana-1301 169 16 planarity planarity NOUN cana-1301 169 17 value value NOUN cana-1301 169 18 Ξ³ Ξ³ X cana-1301 169 19 = = SYM cana-1301 169 20 〈 〈 PROPN cana-1301 169 21 [ [ X cana-1301 169 22 π‘ƒβ„ŽoΞ³+ π‘ƒβ„ŽoΞ³+ ADJ cana-1301 169 23 , , PUNCT cana-1301 169 24 π‘ƒβ„ŽoΞ³βˆ’]β„Ž=1 π‘ƒβ„ŽoΞ³βˆ’]β„Ž=1 PROPN cana-1301 169 25 π‘š π‘š X cana-1301 169 26 βŒͺ βŒͺ NOUN cana-1301 169 27 . . PUNCT cana-1301 170 1 if if SCONJ cana-1301 170 2 π‘ƒβ„ŽoΞ³+ π‘ƒβ„ŽoΞ³+ PROPN cana-1301 170 3 β‰₯ β‰₯ X cana-1301 170 4 0.67 0.67 NUM cana-1301 170 5 , , PUNCT cana-1301 170 6 π‘ƒβ„ŽoΞ³βˆ’ π‘ƒβ„ŽoΞ³βˆ’ ADJ cana-1301 170 7 ≀ ≀ NUM cana-1301 170 8 βˆ’0.67 βˆ’0.67 ADP cana-1301 170 9 , , PUNCT cana-1301 170 10 then then ADV cana-1301 170 11 𝐺 𝐺 PROPN cana-1301 170 12 do do AUX cana-1301 170 13 not not PART cana-1301 170 14 have have VERB cana-1301 170 15 any any DET cana-1301 170 16 point point NOUN cana-1301 170 17 of of ADP cana-1301 170 18 junction junction NOUN cana-1301 170 19 between between ADP cana-1301 170 20 at at ADP cana-1301 170 21 two two NUM cana-1301 170 22 strong strong ADJ cana-1301 170 23 lines line NOUN cana-1301 170 24 . . PUNCT cana-1301 171 1 proof proof NOUN cana-1301 171 2 : : PUNCT cana-1301 171 3 if if SCONJ cana-1301 171 4 possible possible ADJ cana-1301 171 5 , , PUNCT cana-1301 171 6 let let VERB cana-1301 171 7 π‘Œ π‘Œ PRON cana-1301 171 8 be be AUX cana-1301 171 9 a a DET cana-1301 171 10 point point NOUN cana-1301 171 11 of of ADP cana-1301 171 12 junction junction NOUN cana-1301 171 13 between between ADP cana-1301 171 14 two two NUM cana-1301 171 15 m m NOUN cana-1301 171 16 - - PUNCT cana-1301 171 17 bpf bpf ADV cana-1301 171 18 strong strong ADJ cana-1301 171 19 lines line NOUN cana-1301 171 20 ( ( PUNCT cana-1301 171 21 πœ„πœ… πœ„πœ… PRON cana-1301 171 22 〈 〈 PROPN cana-1301 171 23 [ [ X cana-1301 171 24 π‘ƒβ„Žoψr π‘ƒβ„Žoψr PROPN cana-1301 171 25 + + PROPN cana-1301 171 26 ( ( PUNCT cana-1301 171 27 πœ„πœ…)𝑒 πœ„πœ…)𝑒 NOUN cana-1301 171 28 , , PUNCT cana-1301 171 29 π‘ƒβ„Žoψr π‘ƒβ„Žoψr PROPN cana-1301 171 30 βˆ’(πœ„πœ…)𝑒 βˆ’(πœ„πœ…)𝑒 PROPN cana-1301 171 31 ] ] PUNCT cana-1301 171 32 h=1 h=1 NOUN cana-1301 171 33 m m VERB cana-1301 171 34 βŒͺ βŒͺ NOUN cana-1301 171 35 ) ) PUNCT cana-1301 171 36 and and CCONJ cana-1301 171 37 ( ( PUNCT cana-1301 171 38 𝜐𝜏 𝜐𝜏 ADP cana-1301 171 39 〈 〈 PROPN cana-1301 171 40 [ [ X cana-1301 171 41 π‘ƒβ„Žoψr π‘ƒβ„Žoψr PROPN cana-1301 171 42 + + PROPN cana-1301 171 43 ( ( PUNCT cana-1301 171 44 𝜐𝜏)𝑠 𝜐𝜏)𝑠 PROPN cana-1301 171 45 , , PUNCT cana-1301 171 46 π‘ƒβ„Žoψr π‘ƒβ„Žoψr PROPN cana-1301 171 47 βˆ’(𝜐𝜏)𝑠 βˆ’(𝜐𝜏)𝑠 PROPN cana-1301 171 48 ] ] PUNCT cana-1301 171 49 h=1 h=1 NOUN cana-1301 171 50 m m VERB cana-1301 171 51 βŒͺ βŒͺ NOUN cana-1301 171 52 ) ) PUNCT cana-1301 171 53 . . PUNCT cana-1301 172 1 for for ADP cana-1301 172 2 any any DET cana-1301 172 3 m m NOUN cana-1301 172 4 - - PUNCT cana-1301 172 5 bpf bpf ADV cana-1301 172 6 strong strong ADJ cana-1301 172 7 line line NOUN cana-1301 172 8 ( ( PUNCT cana-1301 172 9 πœ„πœ… πœ„πœ… PRON cana-1301 172 10 〈 〈 PROPN cana-1301 172 11 [ [ X cana-1301 172 12 π‘ƒβ„Žoψr π‘ƒβ„Žoψr PROPN cana-1301 172 13 + + PROPN cana-1301 172 14 ( ( PUNCT cana-1301 172 15 πœ„πœ…)𝑒 πœ„πœ…)𝑒 NOUN cana-1301 172 16 , , PUNCT cana-1301 172 17 π‘ƒβ„Žoψr π‘ƒβ„Žoψr PROPN cana-1301 172 18 βˆ’(πœ„πœ…)𝑒 βˆ’(πœ„πœ…)𝑒 PROPN cana-1301 172 19 ] ] PUNCT cana-1301 172 20 h=1 h=1 NOUN cana-1301 172 21 m m VERB cana-1301 172 22 βŒͺ βŒͺ NOUN cana-1301 172 23 ) ) PUNCT cana-1301 172 24 , , PUNCT cana-1301 172 25 we we PRON cana-1301 172 26 have have VERB cana-1301 172 27 π‘ƒβ„ŽoπΈπœ„πœ… π‘ƒβ„ŽoπΈπœ„πœ… PROPN cana-1301 172 28 + + SYM cana-1301 172 29 β‰₯ β‰₯ NOUN cana-1301 172 30 0.5 0.5 NUM cana-1301 172 31 , , PUNCT cana-1301 172 32 π‘ƒβ„ŽoπΈπœ„πœ… π‘ƒβ„ŽoπΈπœ„πœ… ADJ cana-1301 172 33 βˆ’ βˆ’ NOUN cana-1301 172 34 ≀ ≀ NUM cana-1301 172 35 0.5 0.5 NUM cana-1301 172 36 for for ADP cana-1301 172 37 β„Ž β„Ž PROPN cana-1301 172 38 = = SYM cana-1301 172 39 1 1 NUM cana-1301 172 40 , , PUNCT cana-1301 172 41 2 2 NUM cana-1301 172 42 , , PUNCT cana-1301 172 43 β‹― β‹― PROPN cana-1301 172 44 , , PUNCT cana-1301 172 45 π‘š.for π‘š.for ADP cana-1301 172 46 the the DET cana-1301 172 47 minimum minimum ADJ cana-1301 172 48 value value NOUN cana-1301 172 49 of of ADP cana-1301 172 50 π‘ƒβ„ŽoπΈπœ„πœ… π‘ƒβ„ŽoπΈπœ„πœ… ADJ cana-1301 172 51 + + SYM cana-1301 172 52 and and CCONJ cana-1301 172 53 π‘ƒβ„Žo𝐸𝜐𝜏 π‘ƒβ„Žo𝐸𝜐𝜏 PROPN cana-1301 172 54 + + SYM cana-1301 172 55 , , PUNCT cana-1301 172 56 π‘ƒβ„ŽoΟ‡π‘Œ π‘ƒβ„ŽoΟ‡π‘Œ PROPN cana-1301 172 57 + + CCONJ cana-1301 172 58 = = SYM cana-1301 172 59 0.5 0.5 NUM cana-1301 172 60 for for ADP cana-1301 172 61 β„Ž β„Ž X cana-1301 172 62 = = SYM cana-1301 172 63 1 1 NUM cana-1301 172 64 , , PUNCT cana-1301 172 65 2 2 NUM cana-1301 172 66 , , PUNCT cana-1301 172 67 β‹― β‹― NOUN cana-1301 172 68 , , PUNCT cana-1301 172 69 π‘š. π‘š. ADV cana-1301 172 70 then then ADV cana-1301 172 71 , , PUNCT cana-1301 172 72 π‘ƒβ„ŽoΞ³+ π‘ƒβ„ŽoΞ³+ NOUN cana-1301 172 73 = = SYM cana-1301 172 74 1 1 NUM cana-1301 172 75 1 1 NUM cana-1301 172 76 + + NUM cana-1301 172 77 0.5 0.5 NUM cana-1301 172 78 < < X cana-1301 172 79 0.67 0.67 NUM cana-1301 172 80 . . PUNCT cana-1301 173 1 similarly similarly ADV cana-1301 173 2 , , PUNCT cana-1301 173 3 π‘ƒβ„ŽoΞ³βˆ’ π‘ƒβ„ŽoΞ³βˆ’ VERB cana-1301 173 4 > > X cana-1301 173 5 βˆ’0.67 βˆ’0.67 ADJ cana-1301 173 6 forβ„Ž forβ„Ž NOUN cana-1301 173 7 = = SYM cana-1301 173 8 1 1 NUM cana-1301 173 9 , , PUNCT cana-1301 173 10 2 2 NUM cana-1301 173 11 , , PUNCT cana-1301 173 12 β‹― β‹― PROPN cana-1301 173 13 , , PUNCT cana-1301 173 14 π‘š π‘š PROPN cana-1301 173 15 , , PUNCT cana-1301 173 16 a a DET cana-1301 173 17 contradiction contradiction NOUN cana-1301 173 18 . . PUNCT cana-1301 174 1 as as ADP cana-1301 174 2 a a DET cana-1301 174 3 result result NOUN cana-1301 174 4 , , PUNCT cana-1301 174 5 𝐺 𝐺 PROPN cana-1301 174 6 lacks lack VERB cana-1301 174 7 a a DET cana-1301 174 8 spot spot NOUN cana-1301 174 9 where where SCONJ cana-1301 174 10 two two NUM cana-1301 174 11 m m NOUN cana-1301 174 12 - - PUNCT cana-1301 174 13 bpf bpf ADV cana-1301 174 14 strong strong ADJ cana-1301 174 15 lines line NOUN cana-1301 174 16 connect connect VERB cana-1301 174 17 . . PUNCT cana-1301 175 1 next next ADJ cana-1301 175 2 , , PUNCT cana-1301 175 3 strong strong ADJ cana-1301 175 4 m m NOUN cana-1301 175 5 - - NOUN cana-1301 175 6 bpfpg bpfpg NOUN cana-1301 175 7 is be AUX cana-1301 175 8 defined define VERB cana-1301 175 9 as as ADP cana-1301 175 10 follows follow VERB cana-1301 175 11 . . PUNCT cana-1301 176 1 communications communication NOUN cana-1301 176 2 on on ADP cana-1301 176 3 applied apply VERB cana-1301 176 4 nonlinear nonlinear ADJ cana-1301 176 5 analysis analysis NOUN cana-1301 176 6 issn issn NOUN cana-1301 176 7 : : PUNCT cana-1301 176 8 1074 1074 NUM cana-1301 176 9 - - PUNCT cana-1301 176 10 133x 133x NUM cana-1301 176 11 vol vol NOUN cana-1301 176 12 31 31 NUM cana-1301 176 13 no no NOUN cana-1301 176 14 . . PUNCT cana-1301 177 1 7s 7 NOUN cana-1301 177 2 ( ( PUNCT cana-1301 177 3 2024 2024 NUM cana-1301 177 4 ) ) PUNCT cana-1301 177 5 269 269 NUM cana-1301 177 6 https://internationalpubls.com https://internationalpubls.com X cana-1301 177 7 definition definition NOUN cana-1301 177 8 3.8 3.8 NUM cana-1301 177 9 : : PUNCT cana-1301 177 10 an an DET cana-1301 177 11 m m PROPN cana-1301 177 12 - - ADJ cana-1301 177 13 bpfpg bpfpg ADJ cana-1301 177 14 𝐺 𝐺 PROPN cana-1301 177 15 is be AUX cana-1301 177 16 called call VERB cana-1301 177 17 strong strong ADJ cana-1301 177 18 m m NOUN cana-1301 177 19 - - NOUN cana-1301 177 20 bpfpg bpfpg NOUN cana-1301 177 21 if if SCONJ cana-1301 177 22 the the DET cana-1301 177 23 m m PROPN cana-1301 177 24 - - PUNCT cana-1301 177 25 bpf bpf NOUN cana-1301 177 26 planarity planarity NOUN cana-1301 177 27 value value NOUN cana-1301 177 28 〈 〈 PROPN cana-1301 177 29 [ [ X cana-1301 177 30 π‘ƒβ„ŽoΞ³+ π‘ƒβ„ŽoΞ³+ ADJ cana-1301 177 31 , , PUNCT cana-1301 177 32 π‘ƒβ„ŽoΞ³βˆ’]β„Ž=1 π‘ƒβ„ŽoΞ³βˆ’]β„Ž=1 PROPN cana-1301 177 33 π‘š π‘š ADP cana-1301 177 34 βŒͺ βŒͺ NOUN cana-1301 177 35 of of ADP cana-1301 177 36 the the DET cana-1301 177 37 graph graph NOUN cana-1301 177 38 is be AUX cana-1301 177 39 π‘ƒβ„ŽoΞ³+ π‘ƒβ„ŽoΞ³+ ADJ cana-1301 177 40 β‰₯ β‰₯ NOUN cana-1301 177 41 0.67 0.67 NUM cana-1301 177 42 , , PUNCT cana-1301 178 1 π‘ƒβ„ŽoΞ³βˆ’ π‘ƒβ„ŽoΞ³βˆ’ ADJ cana-1301 178 2 ≀ ≀ NUM cana-1301 178 3 βˆ’0.67 βˆ’0.67 PROPN cana-1301 178 4 . . PUNCT cana-1301 178 5 theorem theorem VERB cana-1301 178 6 3.4 3.4 NUM cana-1301 178 7 : : PUNCT cana-1301 178 8 any any DET cana-1301 178 9 complete complete ADJ cana-1301 178 10 m m NOUN cana-1301 178 11 - - PUNCT cana-1301 178 12 bpfg bpfg ADJ cana-1301 178 13 of of ADP cana-1301 178 14 5 5 NUM cana-1301 178 15 nodes node NOUN cana-1301 178 16 or or CCONJ cana-1301 178 17 complete complete ADJ cana-1301 178 18 bipartite bipartite PROPN cana-1301 178 19 m m PROPN cana-1301 178 20 - - PUNCT cana-1301 178 21 bpfg bpfg ADJ cana-1301 178 22 of of ADP cana-1301 178 23 6 6 NUM cana-1301 178 24 nodes node NOUN cana-1301 178 25 are be AUX cana-1301 178 26 not not PART cana-1301 178 27 strong strong ADJ cana-1301 178 28 m m NOUN cana-1301 178 29 - - NOUN cana-1301 178 30 bpfpg bpfpg NOUN cana-1301 178 31 . . PUNCT cana-1301 179 1 proof proof NOUN cana-1301 179 2 : : PUNCT cana-1301 179 3 let let VERB cana-1301 179 4 𝐺 𝐺 PROPN cana-1301 179 5 = = SYM cana-1301 179 6 ( ( PUNCT cana-1301 179 7 𝑉 𝑉 PROPN cana-1301 179 8 , , PUNCT cana-1301 179 9 𝑄 𝑄 PROPN cana-1301 179 10 , , PUNCT cana-1301 179 11 𝑅 𝑅 PROPN cana-1301 179 12 ) ) PUNCT cana-1301 179 13 be be AUX cana-1301 179 14 a a DET cana-1301 179 15 complete complete ADJ cana-1301 179 16 m m NOUN cana-1301 179 17 - - PUNCT cana-1301 179 18 bpfg bpfg ADJ cana-1301 179 19 of of ADP cana-1301 179 20 5 5 NUM cana-1301 179 21 nodes node NOUN cana-1301 179 22 , , PUNCT cana-1301 179 23 here here ADV cana-1301 179 24 𝑉 𝑉 PROPN cana-1301 179 25 = = PUNCT cana-1301 179 26 { { PUNCT cana-1301 179 27 πœ„ πœ„ PROPN cana-1301 179 28 , , PUNCT cana-1301 179 29 πœ… πœ… PROPN cana-1301 179 30 , , PUNCT cana-1301 179 31 𝛼 𝛼 PROPN cana-1301 179 32 , , PUNCT cana-1301 179 33 𝜐 𝜐 PROPN cana-1301 179 34 , , PUNCT cana-1301 179 35 𝜏 𝜏 NOUN cana-1301 179 36 } } PUNCT cana-1301 179 37 and and CCONJ cana-1301 179 38 𝑅 𝑅 PROPN cana-1301 179 39 = = SYM cana-1301 179 40 { { PUNCT cana-1301 179 41 ( ( PUNCT cana-1301 179 42 πœ„πœ… πœ„πœ… NOUN cana-1301 179 43 〈 〈 PROPN cana-1301 179 44 [ [ X cana-1301 179 45 π‘ƒβ„Žoψr π‘ƒβ„Žoψr PROPN cana-1301 179 46 + + PROPN cana-1301 179 47 ( ( PUNCT cana-1301 179 48 πœ„πœ… πœ„πœ… NOUN cana-1301 179 49 ) ) PUNCT cana-1301 179 50 , , PUNCT cana-1301 179 51 π‘ƒβ„Žoψr π‘ƒβ„Žoψr PROPN cana-1301 179 52 βˆ’(πœ„πœ…)]h=1 βˆ’(πœ„πœ…)]h=1 PROPN cana-1301 179 53 m m VERB cana-1301 179 54 βŒͺ βŒͺ NOUN cana-1301 179 55 ) ) PUNCT cana-1301 179 56 , , PUNCT cana-1301 179 57 |πœ„πœ… |πœ„πœ… PUNCT cana-1301 179 58 ∈ ∈ PROPN cana-1301 179 59 𝑉 𝑉 PROPN cana-1301 179 60 Γ— Γ— NOUN cana-1301 179 61 𝑉 𝑉 PROPN cana-1301 179 62 } } PUNCT cana-1301 179 63 . . PUNCT cana-1301 180 1 βˆ€ βˆ€ PUNCT cana-1301 180 2 , , PUNCT cana-1301 180 3 πœ„ πœ„ X cana-1301 180 4 , , PUNCT cana-1301 180 5 πœ… πœ… PROPN cana-1301 180 6 ∈ ∈ PROPN cana-1301 180 7 𝑉 𝑉 PROPN cana-1301 180 8 , , PUNCT cana-1301 180 9 we we PRON cana-1301 180 10 get get VERB cana-1301 180 11 , , PUNCT cana-1301 180 12 π‘ƒβ„Žoψr π‘ƒβ„Žoψr PROPN cana-1301 180 13 + + PROPN cana-1301 180 14 ( ( PUNCT cana-1301 180 15 πœ„πœ… πœ„πœ… NOUN cana-1301 180 16 ) ) PUNCT cana-1301 180 17 = = PUNCT cana-1301 180 18 min{π‘ƒβ„Žoψq min{π‘ƒβ„Žoψq X cana-1301 181 1 + + ADJ cana-1301 181 2 ( ( PUNCT cana-1301 181 3 πœ„ πœ„ NOUN cana-1301 181 4 ) ) PUNCT cana-1301 181 5 , , PUNCT cana-1301 181 6 π‘ƒβ„Žoψq π‘ƒβ„Žoψq PROPN cana-1301 181 7 + + PROPN cana-1301 181 8 ( ( PUNCT cana-1301 181 9 πœ… πœ… NOUN cana-1301 181 10 ) ) PUNCT cana-1301 181 11 } } PUNCT cana-1301 181 12 , , PUNCT cana-1301 181 13 π‘ƒβ„Žoψr π‘ƒβ„Žoψr PROPN cana-1301 181 14 βˆ’(πœ„πœ… βˆ’(πœ„πœ… PROPN cana-1301 181 15 ) ) PUNCT cana-1301 181 16 = = PRON cana-1301 181 17 max{π‘ƒβ„Žoψq max{π‘ƒβ„Žoψq PROPN cana-1301 181 18 βˆ’(πœ„ βˆ’(πœ„ PROPN cana-1301 181 19 ) ) PUNCT cana-1301 181 20 , , PUNCT cana-1301 181 21 π‘ƒβ„Žoψq π‘ƒβ„Žoψq PROPN cana-1301 181 22 βˆ’(πœ… βˆ’(πœ… NOUN cana-1301 181 23 ) ) PUNCT cana-1301 181 24 } } PUNCT cana-1301 181 25 for for ADP cana-1301 181 26 β„Ž β„Ž PROPN cana-1301 181 27 = = SYM cana-1301 181 28 1 1 NUM cana-1301 181 29 , , PUNCT cana-1301 181 30 2 2 NUM cana-1301 181 31 , , PUNCT cana-1301 181 32 β‹― β‹― NOUN cana-1301 181 33 , , PUNCT cana-1301 181 34 π‘š. π‘š. VERB cana-1301 181 35 a a DET cana-1301 181 36 complete complete ADJ cana-1301 181 37 m m NOUN cana-1301 181 38 - - PUNCT cana-1301 181 39 bpfg bpfg PROPN cana-1301 181 40 's 's PART cana-1301 181 41 m m PROPN cana-1301 181 42 - - PUNCT cana-1301 181 43 bpf bpf NOUN cana-1301 181 44 planarity planarity NOUN cana-1301 181 45 value value NOUN cana-1301 181 46 , , PUNCT cana-1301 181 47 according accord VERB cana-1301 181 48 to to ADP cana-1301 181 49 theorem theorem NOUN cana-1301 181 50 1 1 NUM cana-1301 181 51 , , PUNCT cana-1301 181 52 is be AUX cana-1301 181 53 Ξ³ Ξ³ X cana-1301 181 54 = = SYM cana-1301 181 55 〈 〈 PROPN cana-1301 181 56 [ [ X cana-1301 181 57 π‘ƒβ„ŽoΞ³+ π‘ƒβ„ŽoΞ³+ ADJ cana-1301 181 58 , , PUNCT cana-1301 181 59 π‘ƒβ„ŽoΞ³βˆ’]β„Ž=1 π‘ƒβ„ŽoΞ³βˆ’]β„Ž=1 PROPN cana-1301 181 60 π‘š π‘š X cana-1301 181 61 βŒͺ βŒͺ PROPN cana-1301 181 62 where where SCONJ cana-1301 181 63 π‘ƒβ„ŽoΞ³+ π‘ƒβ„ŽoΞ³+ NOUN cana-1301 181 64 = = SYM cana-1301 181 65 1 1 NUM cana-1301 181 66 1+ds 1+ds NUM cana-1301 181 67 and and CCONJ cana-1301 181 68 π‘ƒβ„ŽoΞ³βˆ’ π‘ƒβ„ŽoΞ³βˆ’ ADJ cana-1301 181 69 = = SYM cana-1301 181 70 βˆ’1 βˆ’1 NOUN cana-1301 181 71 1+ds 1+ds NUM cana-1301 181 72 , , PUNCT cana-1301 181 73 β„Ž β„Ž PROPN cana-1301 181 74 = = SYM cana-1301 181 75 1 1 NUM cana-1301 181 76 , , PUNCT cana-1301 181 77 2 2 NUM cana-1301 181 78 , , PUNCT cana-1301 181 79 β‹― β‹― PROPN cana-1301 181 80 , , PUNCT cana-1301 181 81 π‘š π‘š PROPN cana-1301 181 82 , , PUNCT cana-1301 181 83 ds ds X cana-1301 181 84 is be AUX cana-1301 181 85 the the DET cana-1301 181 86 number number NOUN cana-1301 181 87 of of ADP cana-1301 181 88 point point NOUN cana-1301 181 89 of of ADP cana-1301 181 90 junctions junction NOUN cana-1301 181 91 connecting connect VERB cana-1301 181 92 the the DET cana-1301 181 93 lines line NOUN cana-1301 181 94 in in ADP cana-1301 181 95 𝐺. 𝐺. NOUN cana-1301 181 96 we we PRON cana-1301 181 97 know know VERB cana-1301 181 98 that that SCONJ cana-1301 181 99 the the DET cana-1301 181 100 geometric geometric ADJ cana-1301 181 101 representation representation NOUN cana-1301 181 102 of of ADP cana-1301 181 103 the the DET cana-1301 181 104 underlying underlie VERB cana-1301 181 105 crisp crisp ADJ cana-1301 181 106 graph graph NOUN cana-1301 181 107 of of ADP cana-1301 181 108 an an DET cana-1301 181 109 m m NOUN cana-1301 181 110 - - PUNCT cana-1301 181 111 bpf bpf NOUN cana-1301 181 112 complete complete ADJ cana-1301 181 113 graph graph NOUN cana-1301 181 114 with with ADP cana-1301 181 115 5 5 NUM cana-1301 181 116 nodes node NOUN cana-1301 181 117 is be AUX cana-1301 181 118 non non ADJ cana-1301 181 119 - - ADJ cana-1301 181 120 planar planar ADJ cana-1301 181 121 , , PUNCT cana-1301 181 122 and and CCONJ cana-1301 181 123 that that SCONJ cana-1301 181 124 no no DET cana-1301 181 125 representation representation NOUN cana-1301 181 126 can can AUX cana-1301 181 127 avoid avoid VERB cana-1301 181 128 one one NUM cana-1301 181 129 point point NOUN cana-1301 181 130 of of ADP cana-1301 181 131 junction junction NOUN cana-1301 181 132 . . PUNCT cana-1301 182 1 so so ADV cana-1301 182 2 , , PUNCT cana-1301 182 3 〈 〈 PROPN cana-1301 182 4 [ [ X cana-1301 182 5 π‘ƒβ„ŽoΞ³+ π‘ƒβ„ŽoΞ³+ ADJ cana-1301 182 6 , , PUNCT cana-1301 182 7 π‘ƒβ„ŽoΞ³βˆ’]β„Ž=1 π‘ƒβ„ŽoΞ³βˆ’]β„Ž=1 PROPN cana-1301 182 8 π‘š π‘š X cana-1301 182 9 βŒͺ βŒͺ NOUN cana-1301 182 10 = = SYM cana-1301 182 11 〈 〈 PROPN cana-1301 183 1 [ [ X cana-1301 183 2 05 05 NUM cana-1301 183 3 , , PUNCT cana-1301 183 4 βˆ’0.5]β„Ž=1 βˆ’0.5]β„Ž=1 PROPN cana-1301 183 5 π‘š π‘š ADP cana-1301 183 6 βŒͺ βŒͺ NOUN cana-1301 183 7 . . PUNCT cana-1301 184 1 hence hence ADV cana-1301 184 2 , , PUNCT cana-1301 184 3 𝐺 𝐺 PROPN cana-1301 184 4 is be AUX cana-1301 184 5 not not PART cana-1301 184 6 astrong astrong NOUN cana-1301 184 7 m m PROPN cana-1301 184 8 - - NOUN cana-1301 184 9 bpfpg bpfpg NOUN cana-1301 184 10 . . PUNCT cana-1301 185 1 the the DET cana-1301 185 2 complete complete ADJ cana-1301 185 3 bipartite bipartite PROPN cana-1301 185 4 m m PROPN cana-1301 185 5 - - PUNCT cana-1301 185 6 bpfg bpfg ADJ cana-1301 185 7 of of ADP cana-1301 185 8 6 6 NUM cana-1301 185 9 nodes node NOUN cana-1301 185 10 can can AUX cana-1301 185 11 not not PART cana-1301 185 12 be be AUX cana-1301 185 13 shown show VERB cana-1301 185 14 to to PART cana-1301 185 15 constitute constitute VERB cana-1301 185 16 a a DET cana-1301 185 17 strong strong ADJ cana-1301 185 18 m m NOUN cana-1301 185 19 - - NOUN cana-1301 185 20 bpfpg bpfpg NOUN cana-1301 185 21 , , PUNCT cana-1301 185 22 however however ADV cana-1301 185 23 . . PUNCT cana-1301 186 1 4 4 X cana-1301 186 2 . . NUM cana-1301 186 3 faces face NOUN cana-1301 186 4 of of ADP cana-1301 186 5 an an DET cana-1301 186 6 m m ADJ cana-1301 186 7 - - ADJ cana-1301 186 8 bipolar bipolar ADJ cana-1301 186 9 fuzzy fuzzy ADJ cana-1301 186 10 planar planar ADJ cana-1301 186 11 graph graph NOUN cana-1301 186 12 the the DET cana-1301 186 13 parameter parameter NOUN cana-1301 186 14 of of ADP cana-1301 186 15 importance importance NOUN cana-1301 186 16 is be AUX cana-1301 186 17 the the DET cana-1301 186 18 face face NOUN cana-1301 186 19 of of ADP cana-1301 186 20 the the DET cana-1301 186 21 m m NOUN cana-1301 186 22 - - NOUN cana-1301 186 23 bpfpg bpfpg NOUN cana-1301 186 24 . . PUNCT cana-1301 187 1 a a DET cana-1301 187 2 region region NOUN cana-1301 187 3 enclosed enclose VERB cana-1301 187 4 by by ADP cana-1301 187 5 m m PROPN cana-1301 187 6 - - PUNCT cana-1301 187 7 bpf bpf NOUN cana-1301 187 8 lines line NOUN cana-1301 187 9 is be AUX cana-1301 187 10 known know VERB cana-1301 187 11 as as ADP cana-1301 187 12 the the DET cana-1301 187 13 face face NOUN cana-1301 187 14 of of ADP cana-1301 187 15 an an DET cana-1301 187 16 m m NOUN cana-1301 187 17 - - NOUN cana-1301 187 18 bpfpg bpfpg NOUN cana-1301 187 19 . . PUNCT cana-1301 188 1 each each DET cana-1301 188 2 m m PROPN cana-1301 188 3 - - PUNCT cana-1301 188 4 bpf bpf NOUN cana-1301 188 5 face face NOUN cana-1301 188 6 has have VERB cana-1301 188 7 m m NOUN cana-1301 188 8 - - PUNCT cana-1301 188 9 bpf bpf NOUN cana-1301 188 10 lines line NOUN cana-1301 188 11 that that PRON cana-1301 188 12 define define VERB cana-1301 188 13 its its PRON cana-1301 188 14 boundaries boundary NOUN cana-1301 188 15 . . PUNCT cana-1301 189 1 crisp crisp ADJ cana-1301 189 2 face face NOUN cana-1301 189 3 is be AUX cana-1301 189 4 created create VERB cana-1301 189 5 when when SCONJ cana-1301 189 6 all all PRON cana-1301 189 7 of of ADP cana-1301 189 8 the the DET cana-1301 189 9 lines line NOUN cana-1301 189 10 in in ADP cana-1301 189 11 the the DET cana-1301 189 12 boundary boundary NOUN cana-1301 189 13 of of ADP cana-1301 189 14 an an DET cana-1301 189 15 m m NOUN cana-1301 189 16 - - PUNCT cana-1301 189 17 bpf bpf NOUN cana-1301 189 18 face face NOUN cana-1301 189 19 have have VERB cana-1301 189 20 relationship relationship NOUN cana-1301 189 21 values value NOUN cana-1301 189 22 of of ADP cana-1301 189 23 [ [ X cana-1301 189 24 1 1 NUM cana-1301 189 25 , , PUNCT cana-1301 189 26 βˆ’1 βˆ’1 NOUN cana-1301 189 27 ] ] PUNCT cana-1301 189 28 . . PUNCT cana-1301 190 1 the the DET cana-1301 190 2 existence existence NOUN cana-1301 190 3 of of ADP cana-1301 190 4 the the DET cana-1301 190 5 m m PROPN cana-1301 190 6 - - PUNCT cana-1301 190 7 bpf bpf NOUN cana-1301 190 8 face face NOUN cana-1301 190 9 is be AUX cana-1301 190 10 negated negate VERB cana-1301 190 11 if if SCONJ cana-1301 190 12 one one NUM cana-1301 190 13 of of ADP cana-1301 190 14 these these DET cana-1301 190 15 lines line NOUN cana-1301 190 16 is be AUX cana-1301 190 17 eliminated eliminate VERB cana-1301 190 18 . . PUNCT cana-1301 191 1 therefore therefore ADV cana-1301 191 2 , , PUNCT cana-1301 191 3 the the DET cana-1301 191 4 minimum minimum ADJ cana-1301 191 5 energy energy NOUN cana-1301 191 6 of of ADP cana-1301 191 7 m m PROPN cana-1301 191 8 - - PUNCT cana-1301 191 9 bpf bpf NOUN cana-1301 191 10 lines line NOUN cana-1301 191 11 in in ADP cana-1301 191 12 a a DET cana-1301 191 13 border border NOUN cana-1301 191 14 determines determine VERB cana-1301 191 15 whether whether SCONJ cana-1301 191 16 an an DET cana-1301 191 17 m m NOUN cana-1301 191 18 - - PUNCT cana-1301 191 19 bpf bpf NOUN cana-1301 191 20 face face NOUN cana-1301 191 21 exists exist VERB cana-1301 191 22 . . PUNCT cana-1301 192 1 below below ADV cana-1301 192 2 is be AUX cana-1301 192 3 a a DET cana-1301 192 4 definition definition NOUN cana-1301 192 5 of of ADP cana-1301 192 6 an an DET cana-1301 192 7 m m NOUN cana-1301 192 8 - - PUNCT cana-1301 192 9 bpf bpf NOUN cana-1301 192 10 face face NOUN cana-1301 192 11 and and CCONJ cana-1301 192 12 its its PRON cana-1301 192 13 relationship relationship NOUN cana-1301 192 14 values value NOUN cana-1301 192 15 in in ADP cana-1301 192 16 an an DET cana-1301 192 17 m m NOUN cana-1301 192 18 - - PUNCT cana-1301 192 19 bpfg bpfg ADJ cana-1301 192 20 . . PUNCT cana-1301 193 1 definition definition NOUN cana-1301 193 2 4.1 4.1 NUM cana-1301 193 3 : : PUNCT cana-1301 193 4 let let VERB cana-1301 193 5 𝐺 𝐺 PROPN cana-1301 193 6 be be AUX cana-1301 193 7 an an DET cana-1301 193 8 m m NOUN cana-1301 193 9 - - NOUN cana-1301 193 10 bpfpg bpfpg NOUN cana-1301 193 11 and𝑅 and𝑅 NOUN cana-1301 193 12 = = PUNCT cana-1301 193 13 { { PUNCT cana-1301 193 14 ( ( PUNCT cana-1301 193 15 πœ„πœ… πœ„πœ… NOUN cana-1301 193 16 〈 〈 PROPN cana-1301 193 17 [ [ X cana-1301 193 18 π‘ƒβ„Žoψr π‘ƒβ„Žoψr PROPN cana-1301 193 19 + + PROPN cana-1301 193 20 ( ( PUNCT cana-1301 193 21 πœ„πœ…)𝑒 πœ„πœ…)𝑒 NOUN cana-1301 193 22 , , PUNCT cana-1301 193 23 π‘ƒβ„Žoψr π‘ƒβ„Žoψr PROPN cana-1301 193 24 βˆ’(πœ„πœ…)𝑒 βˆ’(πœ„πœ…)𝑒 PROPN cana-1301 193 25 ] ] PUNCT cana-1301 193 26 h=1 h=1 NOUN cana-1301 193 27 m m VERB cana-1301 193 28 βŒͺ βŒͺ NOUN cana-1301 193 29 ) ) PUNCT cana-1301 193 30 , , PUNCT cana-1301 193 31 𝑒 𝑒 X cana-1301 193 32 = = SYM cana-1301 193 33 1,2 1,2 NUM cana-1301 193 34 , , PUNCT cana-1301 193 35 β‹― β‹― VERB cana-1301 193 36 , , PUNCT cana-1301 193 37 𝑑|πœ„πœ… 𝑑|πœ„πœ… NOUN cana-1301 193 38 ∈ ∈ ADP cana-1301 193 39 𝑉 𝑉 PROPN cana-1301 193 40 Γ— Γ— NOUN cana-1301 193 41 𝑉 𝑉 PROPN cana-1301 193 42 } } PUNCT cana-1301 193 43 . . PUNCT cana-1301 194 1 an an DET cana-1301 194 2 m m NOUN cana-1301 194 3 - - PUNCT cana-1301 194 4 bpf bpf NOUN cana-1301 194 5 face face NOUN cana-1301 194 6 of of ADP cana-1301 194 7 𝐺 𝐺 PROPN cana-1301 194 8 is be AUX cana-1301 194 9 a a DET cana-1301 194 10 region region NOUN cana-1301 194 11 , , PUNCT cana-1301 194 12 bordered border VERB cana-1301 194 13 by by ADP cana-1301 194 14 the the DET cana-1301 194 15 set set NOUN cana-1301 194 16 of of ADP cana-1301 194 17 m m PROPN cana-1301 194 18 - - PUNCT cana-1301 194 19 bpf bpf NOUN cana-1301 194 20 lines line NOUN cana-1301 194 21 𝐸′ 𝐸′ PROPN cana-1301 194 22 βŠ‚ βŠ‚ PROPN cana-1301 194 23 𝑉 𝑉 PROPN cana-1301 194 24 Γ— Γ— PROPN cana-1301 194 25 𝑉 𝑉 PROPN cana-1301 194 26 , , PUNCT cana-1301 194 27 of of ADP cana-1301 194 28 a a DET cana-1301 194 29 geometric geometric ADJ cana-1301 194 30 sign sign NOUN cana-1301 194 31 of of ADP cana-1301 194 32 𝐺. 𝐺. NOUN cana-1301 194 33 the the DET cana-1301 194 34 positive positive ADJ cana-1301 194 35 and and CCONJ cana-1301 194 36 negative negative ADJ cana-1301 194 37 values value NOUN cana-1301 194 38 of of ADP cana-1301 194 39 the the DET cana-1301 194 40 m m NOUN cana-1301 194 41 - - PUNCT cana-1301 194 42 bpf bpf NOUN cana-1301 194 43 face face NOUN cana-1301 194 44 is be AUX cana-1301 194 45 〈 〈 PROPN cana-1301 194 46 [ [ X cana-1301 194 47 π‘ƒβ„Žo π‘ƒβ„Žo ADJ cana-1301 194 48 f+ f+ NOUN cana-1301 194 49 , , PUNCT cana-1301 194 50 π‘ƒβ„Žo π‘ƒβ„Žo PROPN cana-1301 194 51 fβˆ’]β„Ž=1 fβˆ’]β„Ž=1 PROPN cana-1301 194 52 π‘š π‘š PROPN cana-1301 194 53 βŒͺ βŒͺ NOUN cana-1301 194 54 , , PUNCT cana-1301 194 55 where where SCONJ cana-1301 194 56 π‘ƒβ„Žo π‘ƒβ„Žo PROPN cana-1301 194 57 f+ f+ NOUN cana-1301 194 58 = = SYM cana-1301 194 59 min min NOUN cana-1301 194 60 { { PUNCT cana-1301 194 61 ( ( PUNCT cana-1301 194 62 π‘ƒβ„ŽoΟˆπ‘… π‘ƒβ„ŽoΟˆπ‘… PROPN cana-1301 194 63 + + PROPN cana-1301 194 64 ( ( PUNCT cana-1301 194 65 πœ„πœ…)𝑒 πœ„πœ…)𝑒 NOUN cana-1301 194 66 min{π‘ƒβ„Žoψq min{π‘ƒβ„Žoψq NOUN cana-1301 195 1 + + NOUN cana-1301 195 2 ( ( PUNCT cana-1301 195 3 πœ„),π‘ƒβ„Žoψq πœ„),π‘ƒβ„Žoψq PROPN cana-1301 195 4 + + PROPN cana-1301 195 5 ( ( PUNCT cana-1301 195 6 πœ… πœ… NOUN cana-1301 195 7 ) ) PUNCT cana-1301 195 8 } } PUNCT cana-1301 195 9 ) ) PUNCT cana-1301 195 10 , , PUNCT cana-1301 195 11 𝑒 𝑒 X cana-1301 195 12 = = SYM cana-1301 195 13 1,2 1,2 NUM cana-1301 195 14 , , PUNCT cana-1301 195 15 β‹― β‹― VERB cana-1301 195 16 , , PUNCT cana-1301 195 17 𝑑|πœ„πœ… 𝑑|πœ„πœ… NOUN cana-1301 195 18 ∈ ∈ NOUN cana-1301 195 19 𝐸′ 𝐸′ NOUN cana-1301 195 20 } } PUNCT cana-1301 195 21 , , PUNCT cana-1301 195 22 π‘ƒβ„Žo π‘ƒβ„Žo PROPN cana-1301 195 23 fβˆ’ fβˆ’ NOUN cana-1301 195 24 = = SYM cana-1301 195 25 max max PROPN cana-1301 195 26 { { PUNCT cana-1301 195 27 ( ( PUNCT cana-1301 195 28 βˆ’ βˆ’ PROPN cana-1301 195 29 π‘ƒβ„ŽoΟˆπ‘… π‘ƒβ„ŽoΟˆπ‘… PROPN cana-1301 195 30 βˆ’(πœ„πœ…)𝑒 βˆ’(πœ„πœ…)𝑒 PROPN cana-1301 195 31 max{π‘ƒβ„Žoψq max{π‘ƒβ„Žoψq PROPN cana-1301 195 32 βˆ’(πœ„),π‘ƒβ„Žoψq βˆ’(πœ„),π‘ƒβ„Žoψq X cana-1301 195 33 βˆ’(πœ… βˆ’(πœ… PROPN cana-1301 195 34 ) ) PUNCT cana-1301 195 35 } } PUNCT cana-1301 195 36 ) ) PUNCT cana-1301 195 37 , , PUNCT cana-1301 195 38 𝑒 𝑒 X cana-1301 195 39 = = SYM cana-1301 195 40 1,2 1,2 NUM cana-1301 195 41 , , PUNCT cana-1301 195 42 β‹― β‹― VERB cana-1301 195 43 , , PUNCT cana-1301 195 44 𝑑|πœ„πœ… 𝑑|πœ„πœ… NOUN cana-1301 195 45 ∈ ∈ NOUN cana-1301 195 46 𝐸′ 𝐸′ NOUN cana-1301 195 47 } } PUNCT cana-1301 195 48 , , PUNCT cana-1301 195 49 β„Ž β„Ž X cana-1301 195 50 = = SYM cana-1301 195 51 1 1 NUM cana-1301 195 52 , , PUNCT cana-1301 195 53 2 2 NUM cana-1301 195 54 , , PUNCT cana-1301 195 55 β‹― β‹― NOUN cana-1301 195 56 , , PUNCT cana-1301 195 57 π‘š. π‘š. ADJ cana-1301 195 58 definition definition NOUN cana-1301 195 59 4.2 4.2 NUM cana-1301 195 60 : : PUNCT cana-1301 195 61 if if SCONJ cana-1301 195 62 an an DET cana-1301 195 63 m m NOUN cana-1301 195 64 - - PUNCT cana-1301 195 65 bpf bpf NOUN cana-1301 195 66 face face NOUN cana-1301 195 67 has have VERB cana-1301 195 68 a a DET cana-1301 195 69 positive positive ADJ cana-1301 195 70 relationship relationship NOUN cana-1301 195 71 value value NOUN cana-1301 195 72 π‘ƒβ„Žo π‘ƒβ„Žo PROPN cana-1301 195 73 f+ f+ X cana-1301 195 74 > > SYM cana-1301 195 75 0.5 0.5 NUM cana-1301 195 76 or or CCONJ cana-1301 195 77 a a DET cana-1301 195 78 negative negative ADJ cana-1301 195 79 relationship relationship NOUN cana-1301 195 80 value value NOUN cana-1301 195 81 π‘ƒβ„Žo π‘ƒβ„Žo PROPN cana-1301 195 82 f+ f+ X cana-1301 195 83 > > X cana-1301 195 84 -0.5 -0.5 PROPN cana-1301 195 85 , , PUNCT cana-1301 195 86 for for ADP cana-1301 195 87 β„Ž β„Ž PROPN cana-1301 195 88 = = SYM cana-1301 195 89 1 1 NUM cana-1301 195 90 , , PUNCT cana-1301 195 91 2 2 NUM cana-1301 195 92 , , PUNCT cana-1301 195 93 β‹― β‹― PROPN cana-1301 195 94 , , PUNCT cana-1301 195 95 π‘š π‘š PROPN cana-1301 195 96 , , PUNCT cana-1301 195 97 it it PRON cana-1301 195 98 is be AUX cana-1301 195 99 said say VERB cana-1301 195 100 to to PART cana-1301 195 101 be be AUX cana-1301 195 102 a a DET cana-1301 195 103 strong strong ADJ cana-1301 195 104 m m NOUN cana-1301 195 105 - - PUNCT cana-1301 195 106 bpf bpf NOUN cana-1301 195 107 face face NOUN cana-1301 195 108 ; ; PUNCT cana-1301 195 109 otherwise otherwise ADV cana-1301 195 110 , , PUNCT cana-1301 195 111 communications communication NOUN cana-1301 195 112 on on ADP cana-1301 195 113 applied apply VERB cana-1301 195 114 nonlinear nonlinear ADJ cana-1301 195 115 analysis analysis NOUN cana-1301 195 116 issn issn NOUN cana-1301 195 117 : : PUNCT cana-1301 195 118 1074 1074 NUM cana-1301 195 119 - - PUNCT cana-1301 195 120 133x 133x NUM cana-1301 195 121 vol vol NOUN cana-1301 195 122 31 31 NUM cana-1301 195 123 no no NOUN cana-1301 195 124 . . PUNCT cana-1301 196 1 7s 7 NOUN cana-1301 196 2 ( ( PUNCT cana-1301 196 3 2024 2024 NUM cana-1301 196 4 ) ) PUNCT cana-1301 196 5 270 270 NUM cana-1301 196 6 https://internationalpubls.com https://internationalpubls.com X cana-1301 197 1 it it PRON cana-1301 197 2 is be AUX cana-1301 197 3 called call VERB cana-1301 197 4 to to PART cana-1301 197 5 be be AUX cana-1301 197 6 a a DET cana-1301 197 7 weak weak ADJ cana-1301 197 8 face face NOUN cana-1301 197 9 . . PUNCT cana-1301 198 1 the the DET cana-1301 198 2 outer outer ADJ cana-1301 198 3 m m NOUN cana-1301 198 4 - - PUNCT cana-1301 198 5 bpf bpf NOUN cana-1301 198 6 face face NOUN cana-1301 198 7 is be AUX cana-1301 198 8 an an DET cana-1301 198 9 infinite infinite ADJ cana-1301 198 10 region region NOUN cana-1301 198 11 found find VERB cana-1301 198 12 in in ADP cana-1301 198 13 every every DET cana-1301 198 14 m m NOUN cana-1301 198 15 - - NOUN cana-1301 198 16 bpfpg bpfpg NOUN cana-1301 198 17 . . PUNCT cana-1301 199 1 remaining remain VERB cana-1301 199 2 faces face NOUN cana-1301 199 3 are be AUX cana-1301 199 4 call call VERB cana-1301 199 5 inner inner ADJ cana-1301 199 6 m m NOUN cana-1301 199 7 - - PUNCT cana-1301 199 8 bpf bpf NOUN cana-1301 199 9 faces face NOUN cana-1301 199 10 . . PUNCT cana-1301 200 1 example example NOUN cana-1301 200 2 4 4 NUM cana-1301 200 3 : : PUNCT cana-1301 200 4 take take VERB cana-1301 200 5 an an DET cana-1301 200 6 m m NOUN cana-1301 200 7 - - ADJ cana-1301 200 8 bpfpg bpfpg ADJ cana-1301 200 9 𝐺 𝐺 PROPN cana-1301 200 10 = = SYM cana-1301 200 11 ( ( PUNCT cana-1301 200 12 𝑉 𝑉 PROPN cana-1301 200 13 , , PUNCT cana-1301 200 14 𝑄 𝑄 PROPN cana-1301 200 15 , , PUNCT cana-1301 200 16 𝑅)as 𝑅)as PROPN cana-1301 200 17 shown show VERB cana-1301 200 18 in in ADP cana-1301 200 19 fig fig NOUN cana-1301 200 20 . . PUNCT cana-1301 201 1 4.where 4.where NUM cana-1301 201 2 𝑄 𝑄 NOUN cana-1301 201 3 = = SYM cana-1301 201 4 { { PUNCT cana-1301 201 5 π‘ž1 π‘ž1 PROPN cana-1301 201 6 〈 〈 PROPN cana-1301 201 7 [ [ X cana-1301 201 8 0.8,βˆ’0.7],[0.7,βˆ’0.65 0.8,βˆ’0.7],[0.7,βˆ’0.65 X cana-1301 201 9 ] ] X cana-1301 201 10 βŒͺ βŒͺ NOUN cana-1301 201 11 , , PUNCT cana-1301 201 12 π‘ž2 π‘ž2 PROPN cana-1301 201 13 〈 〈 PROPN cana-1301 201 14 [ [ X cana-1301 201 15 0.9,βˆ’0.52],[0.8,βˆ’0.7 0.9,βˆ’0.52],[0.8,βˆ’0.7 ADJ cana-1301 201 16 ] ] X cana-1301 201 17 βŒͺ βŒͺ NOUN cana-1301 201 18 , , PUNCT cana-1301 201 19 π‘ž3 π‘ž3 VERB cana-1301 201 20 〈 〈 PROPN cana-1301 201 21 [ [ X cana-1301 201 22 0.7,βˆ’0.9],[0.5,βˆ’0.7 0.7,βˆ’0.9],[0.5,βˆ’0.7 NUM cana-1301 201 23 ] ] X cana-1301 201 24 βŒͺ βŒͺ NOUN cana-1301 201 25 , , PUNCT cana-1301 201 26 π‘ž4 π‘ž4 PROPN cana-1301 201 27 〈 〈 PROPN cana-1301 201 28 [ [ X cana-1301 201 29 0.63,βˆ’0.51],[0.79,βˆ’0.54 0.63,βˆ’0.51],[0.79,βˆ’0.54 NUM cana-1301 201 30 ] ] X cana-1301 201 31 βŒͺ βŒͺ NOUN cana-1301 201 32 } } PUNCT cana-1301 201 33 , , PUNCT cana-1301 201 34 𝑅 𝑅 PROPN cana-1301 201 35 = = X cana-1301 201 36 { { PUNCT cana-1301 201 37 π‘ž1π‘ž2 π‘ž1π‘ž2 ADV cana-1301 201 38 〈 〈 PROPN cana-1301 201 39 [ [ X cana-1301 201 40 0.75,βˆ’0.51],[0.65,βˆ’0.54 0.75,βˆ’0.51],[0.65,βˆ’0.54 X cana-1301 201 41 ] ] X cana-1301 201 42 βŒͺ βŒͺ NOUN cana-1301 201 43 , , PUNCT cana-1301 201 44 π‘ž1π‘ž3 π‘ž1π‘ž3 PROPN cana-1301 201 45 〈 〈 PROPN cana-1301 201 46 [ [ X cana-1301 201 47 0.62,βˆ’0.63],[0.49,βˆ’0.53 0.62,βˆ’0.63],[0.49,βˆ’0.53 NOUN cana-1301 201 48 ] ] X cana-1301 201 49 βŒͺ βŒͺ NOUN cana-1301 201 50 , , PUNCT cana-1301 201 51 π‘ž2π‘ž3 π‘ž2π‘ž3 PROPN cana-1301 201 52 〈 〈 PROPN cana-1301 201 53 [ [ X cana-1301 201 54 0.69,βˆ’0.5],[0.49,βˆ’0.65 0.69,βˆ’0.5],[0.49,βˆ’0.65 PROPN cana-1301 201 55 ] ] SYM cana-1301 201 56 βŒͺ βŒͺ NOUN cana-1301 201 57 , , PUNCT cana-1301 201 58 π‘ž1π‘ž4 π‘ž1π‘ž4 X cana-1301 201 59 〈 〈 PROPN cana-1301 201 60 [ [ X cana-1301 201 61 0.23,βˆ’0.14],[0.6,βˆ’0.2 0.23,βˆ’0.14],[0.6,βˆ’0.2 PROPN cana-1301 201 62 ] ] X cana-1301 201 63 βŒͺ βŒͺ NOUN cana-1301 201 64 , , PUNCT cana-1301 201 65 π‘ž3π‘ž4 π‘ž3π‘ž4 X cana-1301 202 1 〈 〈 PROPN cana-1301 202 2 [ [ X cana-1301 202 3 0.59,βˆ’0.49],[0.47,βˆ’0.52 0.59,βˆ’0.49],[0.47,βˆ’0.52 ADJ cana-1301 202 4 ] ] X cana-1301 202 5 βŒͺ βŒͺ NOUN cana-1301 202 6 } } PUNCT cana-1301 202 7 . . PUNCT cana-1301 203 1 then then ADV cana-1301 203 2 m m PROPN cana-1301 203 3 - - NOUN cana-1301 203 4 bpfpg bpfpg NOUN cana-1301 203 5 has have VERB cana-1301 203 6 the the DET cana-1301 203 7 following follow VERB cana-1301 203 8 faces face NOUN cana-1301 203 9 : : PUNCT cana-1301 203 10 β€’ β€’ NUM cana-1301 203 11 m m NOUN cana-1301 203 12 - - PUNCT cana-1301 203 13 bpf bpf NOUN cana-1301 203 14 face face NOUN cana-1301 203 15 𝛽1 𝛽1 NOUN cana-1301 203 16 is be AUX cana-1301 203 17 bounded bound VERB cana-1301 203 18 by by ADP cana-1301 203 19 the the DET cana-1301 203 20 lines line NOUN cana-1301 203 21 π‘ž1π‘ž4 π‘ž1π‘ž4 VERB cana-1301 203 22 〈 〈 PROPN cana-1301 203 23 [ [ X cana-1301 203 24 0.23,βˆ’0.14],[0.6,βˆ’0.2 0.23,βˆ’0.14],[0.6,βˆ’0.2 PROPN cana-1301 203 25 ] ] X cana-1301 203 26 βŒͺ βŒͺ NOUN cana-1301 203 27 , , PUNCT cana-1301 203 28 π‘ž3π‘ž4 π‘ž3π‘ž4 X cana-1301 204 1 〈 〈 PROPN cana-1301 204 2 [ [ X cana-1301 204 3 0.59,βˆ’0.49],[0.47,βˆ’0.52 0.59,βˆ’0.49],[0.47,βˆ’0.52 NOUN cana-1301 204 4 ] ] ADJ cana-1301 204 5 βŒͺ βŒͺ NOUN cana-1301 204 6 , , PUNCT cana-1301 204 7 π‘ž1π‘ž3 π‘ž1π‘ž3 PROPN cana-1301 204 8 〈 〈 PROPN cana-1301 204 9 [ [ X cana-1301 204 10 0.62,βˆ’0.63],[0.49,βˆ’0.53 0.62,βˆ’0.63],[0.49,βˆ’0.53 NOUN cana-1301 204 11 ] ] X cana-1301 204 12 βŒͺ βŒͺ NOUN cana-1301 204 13 , , PUNCT cana-1301 204 14 β€’ β€’ NUM cana-1301 204 15 m m NOUN cana-1301 204 16 - - PUNCT cana-1301 204 17 bpf bpf NOUN cana-1301 204 18 face face NOUN cana-1301 204 19 𝛽2 𝛽2 NOUN cana-1301 204 20 is be AUX cana-1301 204 21 surrounded surround VERB cana-1301 204 22 by by ADP cana-1301 204 23 the the DET cana-1301 204 24 lines line NOUN cana-1301 204 25 π‘ž1π‘ž2 π‘ž1π‘ž2 VERB cana-1301 204 26 〈 〈 PROPN cana-1301 204 27 [ [ X cana-1301 204 28 0.75,βˆ’0.51],[0.65,βˆ’0.54 0.75,βˆ’0.51],[0.65,βˆ’0.54 X cana-1301 204 29 ] ] X cana-1301 204 30 βŒͺ βŒͺ NOUN cana-1301 204 31 , , PUNCT cana-1301 204 32 π‘ž1π‘ž3 π‘ž1π‘ž3 PROPN cana-1301 204 33 〈 〈 PROPN cana-1301 204 34 [ [ X cana-1301 204 35 0.62,βˆ’0.63],[0.49,βˆ’0.53 0.62,βˆ’0.63],[0.49,βˆ’0.53 NOUN cana-1301 204 36 ] ] X cana-1301 204 37 βŒͺ βŒͺ NOUN cana-1301 204 38 , , PUNCT cana-1301 204 39 π‘ž2π‘ž3 π‘ž2π‘ž3 PROPN cana-1301 204 40 〈 〈 PROPN cana-1301 204 41 [ [ X cana-1301 204 42 0.69,βˆ’0.5],[0.49,βˆ’0.65 0.69,βˆ’0.5],[0.49,βˆ’0.65 PROPN cana-1301 204 43 ] ] SYM cana-1301 204 44 βŒͺ βŒͺ NOUN cana-1301 204 45 , , PUNCT cana-1301 204 46 β€’ β€’ NUM cana-1301 204 47 outer outer ADJ cana-1301 204 48 m m NOUN cana-1301 204 49 - - PUNCT cana-1301 204 50 bpf bpf NOUN cana-1301 204 51 face face NOUN cana-1301 204 52 𝛽3 𝛽3 NOUN cana-1301 204 53 surrounded surround VERB cana-1301 204 54 by by ADP cana-1301 204 55 the the DET cana-1301 204 56 lines line NOUN cana-1301 204 57 π‘ž1π‘ž2 π‘ž1π‘ž2 VERB cana-1301 204 58 〈 〈 PROPN cana-1301 204 59 [ [ X cana-1301 204 60 0.75,βˆ’0.51],[0.65,βˆ’0.54 0.75,βˆ’0.51],[0.65,βˆ’0.54 X cana-1301 204 61 ] ] X cana-1301 204 62 βŒͺ βŒͺ NOUN cana-1301 204 63 , , PUNCT cana-1301 204 64 π‘ž2π‘ž3 π‘ž2π‘ž3 PROPN cana-1301 204 65 〈 〈 PROPN cana-1301 204 66 [ [ X cana-1301 204 67 0.69,βˆ’0.5],[0.49,βˆ’0.65 0.69,βˆ’0.5],[0.49,βˆ’0.65 PROPN cana-1301 204 68 ] ] SYM cana-1301 204 69 βŒͺ βŒͺ NOUN cana-1301 204 70 , , PUNCT cana-1301 204 71 π‘ž1π‘ž4 π‘ž1π‘ž4 X cana-1301 204 72 〈 〈 PROPN cana-1301 204 73 [ [ X cana-1301 204 74 0.23,βˆ’0.14],[0.6,βˆ’0.2 0.23,βˆ’0.14],[0.6,βˆ’0.2 PROPN cana-1301 204 75 ] ] X cana-1301 204 76 βŒͺ βŒͺ NOUN cana-1301 204 77 , , PUNCT cana-1301 204 78 π‘ž3π‘ž4 π‘ž3π‘ž4 X cana-1301 205 1 〈 〈 PROPN cana-1301 205 2 [ [ X cana-1301 205 3 0.59,βˆ’0.49],[0.47,βˆ’0.52 0.59,βˆ’0.49],[0.47,βˆ’0.52 ADJ cana-1301 205 4 ] ] ADJ cana-1301 205 5 βŒͺ βŒͺ NOUN cana-1301 205 6 . . PUNCT cana-1301 206 1 fig fig NOUN cana-1301 206 2 . . PUNCT cana-1301 207 1 4 4 NUM cana-1301 207 2 faces face NOUN cana-1301 207 3 in in ADP cana-1301 207 4 m m ADJ cana-1301 207 5 - - ADJ cana-1301 207 6 bipolar bipolar ADJ cana-1301 207 7 fuzzy fuzzy ADJ cana-1301 207 8 planar planar ADJ cana-1301 207 9 graph graph NOUN cana-1301 207 10 clearly clearly ADV cana-1301 207 11 the the DET cana-1301 207 12 relationship relationship NOUN cana-1301 207 13 value value NOUN cana-1301 207 14 of of ADP cana-1301 207 15 m m PROPN cana-1301 207 16 - - PUNCT cana-1301 207 17 bpf bpf NOUN cana-1301 207 18 face face NOUN cana-1301 207 19 𝛽1 𝛽1 PROPN cana-1301 207 20 is〈[0.37 is〈[0.37 PROPN cana-1301 207 21 , , PUNCT cana-1301 207 22 βˆ’0.27 βˆ’0.27 NOUN cana-1301 207 23 ] ] X cana-1301 207 24 , , PUNCT cana-1301 207 25 [ [ X cana-1301 207 26 0.86 0.86 NUM cana-1301 207 27 , , PUNCT cana-1301 207 28 βˆ’0.37 βˆ’0.37 NOUN cana-1301 207 29 ] ] X cana-1301 207 30 βŒͺ βŒͺ NOUN cana-1301 207 31 . . PUNCT cana-1301 208 1 the the DET cana-1301 208 2 relationship relationship NOUN cana-1301 208 3 value value NOUN cana-1301 208 4 of of ADP cana-1301 208 5 m m PROPN cana-1301 208 6 - - PUNCT cana-1301 208 7 bpf bpf NOUN cana-1301 208 8 face face NOUN cana-1301 208 9 𝛽3is〈[0.37 𝛽3is〈[0.37 VERB cana-1301 208 10 , , PUNCT cana-1301 208 11 βˆ’0.27 βˆ’0.27 NOUN cana-1301 208 12 ] ] X cana-1301 208 13 , , PUNCT cana-1301 208 14 [ [ X cana-1301 208 15 0.86 0.86 NUM cana-1301 208 16 , , PUNCT cana-1301 208 17 βˆ’0.37 βˆ’0.37 NOUN cana-1301 208 18 ] ] X cana-1301 208 19 βŒͺ βŒͺ NOUN cana-1301 208 20 . . PUNCT cana-1301 209 1 thus thus ADV cana-1301 209 2 𝛽1and 𝛽1and ADP cana-1301 209 3 𝛽3 𝛽3 NOUN cana-1301 209 4 are be AUX cana-1301 209 5 weak weak ADJ cana-1301 209 6 m m NOUN cana-1301 209 7 - - PUNCT cana-1301 209 8 bpf bpf NOUN cana-1301 209 9 faces face NOUN cana-1301 209 10 and and CCONJ cana-1301 209 11 𝛽2 𝛽2 PROPN cana-1301 209 12 is be AUX cana-1301 209 13 a a DET cana-1301 209 14 strong strong ADJ cana-1301 209 15 m m NOUN cana-1301 209 16 - - PUNCT cana-1301 209 17 bpf bpf NOUN cana-1301 209 18 face face NOUN cana-1301 209 19 with with ADP cana-1301 209 20 the the DET cana-1301 209 21 relationship relationship NOUN cana-1301 209 22 value〈[0.89 value〈[0.89 PROPN cana-1301 209 23 , , PUNCT cana-1301 209 24 βˆ’0.90 βˆ’0.90 X cana-1301 209 25 ] ] PUNCT cana-1301 209 26 , , PUNCT cana-1301 210 1 [ [ X cana-1301 210 2 0.93 0.93 NUM cana-1301 210 3 , , PUNCT cana-1301 210 4 βˆ’0.82 βˆ’0.82 NOUN cana-1301 210 5 ] ] X cana-1301 210 6 βŒͺ βŒͺ NOUN cana-1301 210 7 . . PUNCT cana-1301 211 1 we we PRON cana-1301 211 2 are be AUX cana-1301 211 3 now now ADV cana-1301 211 4 introducing introduce VERB cana-1301 211 5 dual dual ADJ cana-1301 211 6 of of ADP cana-1301 211 7 m m NOUN cana-1301 211 8 - - NOUN cana-1301 211 9 bpfpg bpfpg NOUN cana-1301 211 10 . . PUNCT cana-1301 212 1 each each DET cana-1301 212 2 m m PROPN cana-1301 212 3 - - PUNCT cana-1301 212 4 bpf bpf NOUN cana-1301 212 5 line line NOUN cana-1301 212 6 between between ADP cana-1301 212 7 two two NUM cana-1301 212 8 nodes node NOUN cana-1301 212 9 corresponds correspond VERB cana-1301 212 10 to to ADP cana-1301 212 11 each each DET cana-1301 212 12 line line NOUN cana-1301 212 13 in in ADP cana-1301 212 14 the the DET cana-1301 212 15 border border NOUN cana-1301 212 16 between between ADP cana-1301 212 17 two two NUM cana-1301 212 18 faces face NOUN cana-1301 212 19 of of ADP cana-1301 212 20 the the DET cana-1301 212 21 m m NOUN cana-1301 212 22 - - NOUN cana-1301 212 23 bpfpg bpfpg NOUN cana-1301 212 24 , , PUNCT cana-1301 212 25 and and CCONJ cana-1301 212 26 in in ADP cana-1301 212 27 the the DET cana-1301 212 28 m m PROPN cana-1301 212 29 - - PUNCT cana-1301 212 30 bpfdg bpfdg NOUN cana-1301 212 31 , , PUNCT cana-1301 212 32 nodes node NOUN cana-1301 212 33 correspond correspond VERB cana-1301 212 34 to to ADP cana-1301 212 35 the the DET cana-1301 212 36 strong strong ADJ cana-1301 212 37 m m NOUN cana-1301 212 38 - - PUNCT cana-1301 212 39 bpf bpf NOUN cana-1301 212 40 faces face NOUN cana-1301 212 41 of of ADP cana-1301 212 42 the the DET cana-1301 212 43 m m NOUN cana-1301 212 44 - - NOUN cana-1301 212 45 bpfpg bpfpg NOUN cana-1301 212 46 . . PUNCT cana-1301 213 1 below below ADV cana-1301 213 2 is be AUX cana-1301 213 3 the the DET cana-1301 213 4 formal formal ADJ cana-1301 213 5 definition definition NOUN cana-1301 213 6 . . PUNCT cana-1301 214 1 communications communication NOUN cana-1301 214 2 on on ADP cana-1301 214 3 applied apply VERB cana-1301 214 4 nonlinear nonlinear ADJ cana-1301 214 5 analysis analysis NOUN cana-1301 214 6 issn issn NOUN cana-1301 214 7 : : PUNCT cana-1301 214 8 1074 1074 NUM cana-1301 214 9 - - PUNCT cana-1301 214 10 133x 133x NUM cana-1301 214 11 vol vol NOUN cana-1301 214 12 31 31 NUM cana-1301 214 13 no no NOUN cana-1301 214 14 . . PUNCT cana-1301 215 1 7s 7 NOUN cana-1301 215 2 ( ( PUNCT cana-1301 215 3 2024 2024 NUM cana-1301 215 4 ) ) PUNCT cana-1301 215 5 271 271 NUM cana-1301 215 6 https://internationalpubls.com https://internationalpubls.com X cana-1301 215 7 definition definition NOUN cana-1301 215 8 4.3 4.3 NUM cana-1301 215 9 : : PUNCT cana-1301 215 10 let let VERB cana-1301 215 11 𝐺 𝐺 PRON cana-1301 215 12 be be AUX cana-1301 215 13 an an DET cana-1301 215 14 m m NOUN cana-1301 215 15 - - NOUN cana-1301 215 16 bpfpg bpfpg NOUN cana-1301 215 17 and and CCONJ cana-1301 215 18 let let VERB cana-1301 215 19 𝑅 𝑅 PROPN cana-1301 215 20 = = PRON cana-1301 215 21 { { PUNCT cana-1301 215 22 ( ( PUNCT cana-1301 215 23 πœ„πœ… πœ„πœ… NOUN cana-1301 215 24 〈 〈 PROPN cana-1301 215 25 [ [ X cana-1301 215 26 π‘ƒβ„Žoψr π‘ƒβ„Žoψr PROPN cana-1301 215 27 + + PROPN cana-1301 215 28 ( ( PUNCT cana-1301 215 29 πœ„πœ…)𝑒 πœ„πœ…)𝑒 NOUN cana-1301 215 30 , , PUNCT cana-1301 215 31 π‘ƒβ„Žoψr π‘ƒβ„Žoψr PROPN cana-1301 215 32 βˆ’(πœ„πœ…)𝑒 βˆ’(πœ„πœ…)𝑒 PROPN cana-1301 215 33 ] ] PUNCT cana-1301 215 34 h=1 h=1 NOUN cana-1301 215 35 m m VERB cana-1301 215 36 βŒͺ βŒͺ NOUN cana-1301 215 37 ) ) PUNCT cana-1301 215 38 , , PUNCT cana-1301 215 39 𝑒 𝑒 X cana-1301 215 40 = = SYM cana-1301 215 41 1,2 1,2 NUM cana-1301 215 42 , , PUNCT cana-1301 215 43 β‹― β‹― VERB cana-1301 215 44 , , PUNCT cana-1301 215 45 𝑑|πœ„πœ… 𝑑|πœ„πœ… NOUN cana-1301 215 46 ∈ ∈ ADP cana-1301 215 47 𝑉 𝑉 PROPN cana-1301 215 48 Γ— Γ— NOUN cana-1301 215 49 𝑉 𝑉 PROPN cana-1301 215 50 } } PUNCT cana-1301 215 51 . . PUNCT cana-1301 216 1 let let VERB cana-1301 216 2 𝛽1 𝛽1 NOUN cana-1301 216 3 , , PUNCT cana-1301 216 4 𝛽2 𝛽2 NOUN cana-1301 216 5 , , PUNCT cana-1301 216 6 β‹― β‹― PROPN cana-1301 216 7 , , PUNCT cana-1301 216 8 π›½π‘Ÿbe π›½π‘Ÿbe VERB cana-1301 216 9 the the DET cana-1301 216 10 strong strong ADJ cana-1301 216 11 m m NOUN cana-1301 216 12 - - PUNCT cana-1301 216 13 bpf bpf NOUN cana-1301 216 14 faces face NOUN cana-1301 216 15 of of ADP cana-1301 216 16 𝐺. 𝐺. NOUN cana-1301 216 17 an an DET cana-1301 216 18 m m NOUN cana-1301 216 19 - - PUNCT cana-1301 216 20 bpfdg bpfdg NOUN cana-1301 216 21 of of ADP cana-1301 216 22 𝐺 𝐺 PROPN cana-1301 216 23 is be AUX cana-1301 216 24 an an DET cana-1301 216 25 m m NOUN cana-1301 216 26 - - NOUN cana-1301 216 27 bpfpg bpfpg NOUN cana-1301 217 1 𝐺𝑑 𝐺𝑑 NOUN cana-1301 217 2 = = SYM cana-1301 217 3 ( ( PUNCT cana-1301 217 4 𝑉𝑑 𝑉𝑑 PROPN cana-1301 217 5 , , PUNCT cana-1301 217 6 𝑄𝑑 𝑄𝑑 PROPN cana-1301 217 7 , , PUNCT cana-1301 217 8 𝑅𝑑 𝑅𝑑 PROPN cana-1301 217 9 ) ) PUNCT cana-1301 217 10 , , PUNCT cana-1301 217 11 where where SCONJ cana-1301 217 12 𝑉𝑑 𝑉𝑑 PROPN cana-1301 217 13 = = SYM cana-1301 217 14 { { PUNCT cana-1301 217 15 π‘žπ‘– π‘žπ‘– PROPN cana-1301 217 16 , , PUNCT cana-1301 217 17 𝑖 𝑖 NOUN cana-1301 217 18 = = SYM cana-1301 217 19 1 1 NUM cana-1301 217 20 , , PUNCT cana-1301 217 21 2 2 NUM cana-1301 217 22 , , PUNCT cana-1301 217 23 β‹― β‹― NOUN cana-1301 217 24 , , PUNCT cana-1301 217 25 π‘Ÿ π‘Ÿ NOUN cana-1301 217 26 } } PUNCT cana-1301 217 27 , , PUNCT cana-1301 217 28 and and CCONJ cana-1301 217 29 the the DET cana-1301 217 30 node node NOUN cana-1301 217 31 π‘žπ‘–of π‘žπ‘–of PROPN cana-1301 218 1 𝐺𝑑 𝐺𝑑 X cana-1301 218 2 is be AUX cana-1301 218 3 measured measure VERB cana-1301 218 4 for for ADP cana-1301 218 5 the the DET cana-1301 218 6 face face NOUN cana-1301 218 7 𝛽𝑖of 𝛽𝑖of NOUN cana-1301 218 8 𝐺. 𝐺. PROPN cana-1301 218 9 the the DET cana-1301 218 10 positive positive ADJ cana-1301 218 11 and and CCONJ cana-1301 218 12 negative negative ADJ cana-1301 218 13 relationship relationship NOUN cana-1301 218 14 values value NOUN cana-1301 218 15 of of ADP cana-1301 218 16 nodes node NOUN cana-1301 218 17 are be AUX cana-1301 218 18 taken take VERB cana-1301 218 19 by by ADP cana-1301 218 20 the the DET cana-1301 218 21 mapping mapping NOUN cana-1301 218 22 𝑄𝑑 𝑄𝑑 PROPN cana-1301 218 23 = = SYM cana-1301 218 24 〈 〈 PROPN cana-1301 218 25 [ [ X cana-1301 218 26 π‘ƒβ„Žoψ π‘ƒβ„Žoψ NOUN cana-1301 218 27 𝑄𝑑 𝑄𝑑 PROPN cana-1301 218 28 + + X cana-1301 218 29 , , PUNCT cana-1301 218 30 π‘ƒβ„Žoψ π‘ƒβ„Žoψ PROPN cana-1301 218 31 𝑄𝑑 𝑄𝑑 PROPN cana-1301 218 32 βˆ’ βˆ’ PROPN cana-1301 218 33 ] ] PUNCT cana-1301 219 1 h=1 h=1 VERB cana-1301 219 2 m m VERB cana-1301 219 3 βŒͺ βŒͺ NOUN cana-1301 219 4 , , PUNCT cana-1301 219 5 π‘ƒβ„Žoψ π‘ƒβ„Žoψ PROPN cana-1301 219 6 𝑄𝑑 𝑄𝑑 PROPN cana-1301 219 7 + + CCONJ cana-1301 219 8 : : PUNCT cana-1301 219 9 𝑉𝑑 𝑉𝑑 PROPN cana-1301 219 10 β†’ β†’ SYM cana-1301 219 11 [ [ X cana-1301 219 12 0 0 NUM cana-1301 219 13 , , PUNCT cana-1301 219 14 1 1 NUM cana-1301 219 15 ] ] PUNCT cana-1301 219 16 and and CCONJ cana-1301 219 17 π‘ƒβ„Žoψ π‘ƒβ„Žoψ PROPN cana-1301 219 18 𝑄𝑑 𝑄𝑑 PROPN cana-1301 219 19 βˆ’ βˆ’ NOUN cana-1301 219 20 : : PUNCT cana-1301 219 21 𝑉𝑑 𝑉𝑑 PROPN cana-1301 219 22 β†’ β†’ SYM cana-1301 219 23 [ [ X cana-1301 219 24 βˆ’1 βˆ’1 NOUN cana-1301 219 25 , , PUNCT cana-1301 219 26 0 0 NUM cana-1301 219 27 ] ] PUNCT cana-1301 219 28 such such ADJ cana-1301 219 29 that that SCONJ cana-1301 219 30 π‘ƒβ„Žoψ π‘ƒβ„Žoψ PROPN cana-1301 219 31 𝑄𝑑 𝑄𝑑 PROPN cana-1301 219 32 + + CCONJ cana-1301 219 33 ( ( PUNCT cana-1301 219 34 π‘žπ‘– π‘žπ‘– ADJ cana-1301 219 35 ) ) PUNCT cana-1301 219 36 = = VERB cana-1301 219 37 max{π‘ƒβ„Žoψ max{π‘ƒβ„Žoψ NOUN cana-1301 220 1 𝑅𝑑 𝑅𝑑 INTJ cana-1301 220 2 + + CCONJ cana-1301 220 3 ( ( PUNCT cana-1301 220 4 πœ„πœ…)𝑒 πœ„πœ…)𝑒 NOUN cana-1301 220 5 , , PUNCT cana-1301 220 6 𝑒 𝑒 X cana-1301 220 7 = = SYM cana-1301 220 8 1,2 1,2 NUM cana-1301 220 9 , , PUNCT cana-1301 220 10 β‹― β‹― PROPN cana-1301 220 11 , , PUNCT cana-1301 220 12 𝑑 𝑑 PROPN cana-1301 220 13 | | ADV cana-1301 220 14 πœ„πœ… πœ„πœ… INTJ cana-1301 220 15 𝑖𝑠 𝑖𝑠 NOUN cana-1301 220 16 π‘Ž π‘Ž DET cana-1301 220 17 𝑙𝑖𝑛𝑒 𝑙𝑖𝑛𝑒 NOUN cana-1301 220 18 π‘œπ‘“ π‘œπ‘“ ADP cana-1301 220 19 π‘‘β„Žπ‘’ π‘‘β„Žπ‘’ INTJ cana-1301 220 20 π‘π‘œπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ π‘π‘œπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ NOUN cana-1301 220 21 π‘œπ‘“ π‘œπ‘“ ADP cana-1301 220 22 π‘‘β„Žπ‘’ π‘‘β„Žπ‘’ X cana-1301 220 23 π‘ π‘‘π‘Ÿπ‘œπ‘›π‘” π‘ π‘‘π‘Ÿπ‘œπ‘›π‘” ADJ cana-1301 220 24 π‘š π‘š ADP cana-1301 220 25 βˆ’ βˆ’ PROPN cana-1301 220 26 𝐡𝑃𝐹 𝐡𝑃𝐹 PROPN cana-1301 220 27 π‘“π‘Žπ‘π‘’ π‘“π‘Žπ‘π‘’ ADV cana-1301 220 28 𝛽𝑖 𝛽𝑖 NOUN cana-1301 220 29 } } PUNCT cana-1301 220 30 , , PUNCT cana-1301 220 31 π‘ƒβ„Žoψ π‘ƒβ„Žoψ PROPN cana-1301 220 32 𝑄𝑑 𝑄𝑑 PROPN cana-1301 220 33 βˆ’ βˆ’ PROPN cana-1301 220 34 ( ( PUNCT cana-1301 220 35 π‘žπ‘– π‘žπ‘– ADJ cana-1301 220 36 ) ) PUNCT cana-1301 220 37 = = PUNCT cana-1301 221 1 min{π‘ƒβ„Žoψ min{π‘ƒβ„Žoψ NOUN cana-1301 221 2 𝑅𝑑 𝑅𝑑 INTJ cana-1301 221 3 βˆ’ βˆ’ PROPN cana-1301 221 4 ( ( PUNCT cana-1301 221 5 πœ„πœ…)𝑒 πœ„πœ…)𝑒 NOUN cana-1301 221 6 , , PUNCT cana-1301 221 7 𝑒 𝑒 X cana-1301 221 8 = = SYM cana-1301 221 9 1,2 1,2 NUM cana-1301 221 10 , , PUNCT cana-1301 221 11 β‹― β‹― PROPN cana-1301 221 12 , , PUNCT cana-1301 221 13 𝑑| 𝑑| VERB cana-1301 221 14 πœ„πœ… πœ„πœ… NOUN cana-1301 221 15 𝑖𝑠 𝑖𝑠 NOUN cana-1301 221 16 π‘Ž π‘Ž DET cana-1301 221 17 𝑙𝑖𝑛𝑒 𝑙𝑖𝑛𝑒 NOUN cana-1301 221 18 π‘œπ‘“ π‘œπ‘“ ADP cana-1301 221 19 π‘‘β„Žπ‘’ π‘‘β„Žπ‘’ INTJ cana-1301 221 20 π‘π‘œπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ π‘π‘œπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ NOUN cana-1301 221 21 π‘œπ‘“ π‘œπ‘“ ADP cana-1301 221 22 π‘‘β„Žπ‘’ π‘‘β„Žπ‘’ X cana-1301 221 23 π‘ π‘‘π‘Ÿπ‘œπ‘›π‘” π‘ π‘‘π‘Ÿπ‘œπ‘›π‘” ADJ cana-1301 221 24 π‘š π‘š ADP cana-1301 221 25 βˆ’ βˆ’ PROPN cana-1301 221 26 𝐡𝑃𝐹 𝐡𝑃𝐹 PROPN cana-1301 221 27 π‘“π‘Žπ‘π‘’ π‘“π‘Žπ‘π‘’ ADV cana-1301 221 28 𝛽𝑖 𝛽𝑖 VERB cana-1301 221 29 } } PUNCT cana-1301 221 30 . . PUNCT cana-1301 222 1 there there PRON cana-1301 222 2 may may AUX cana-1301 222 3 exist exist VERB cana-1301 222 4 excess excess NOUN cana-1301 222 5 of of ADP cana-1301 222 6 one one NUM cana-1301 222 7 common common ADJ cana-1301 222 8 line line NOUN cana-1301 222 9 corresponding correspond VERB cana-1301 222 10 to to ADP cana-1301 222 11 two two NUM cana-1301 222 12 faces face NOUN cana-1301 222 13 𝛽𝑖 𝛽𝑖 VERB cana-1301 222 14 and and CCONJ cana-1301 222 15 𝛽𝑗 𝛽𝑗 NOUN cana-1301 222 16 of of ADP cana-1301 222 17 𝐺. 𝐺. NOUN cana-1301 222 18 therefore therefore ADV cana-1301 222 19 , , PUNCT cana-1301 222 20 there there PRON cana-1301 222 21 may may AUX cana-1301 222 22 be be AUX cana-1301 222 23 in in ADP cana-1301 222 24 excess excess NOUN cana-1301 222 25 of of ADP cana-1301 222 26 one one NUM cana-1301 222 27 line line NOUN cana-1301 222 28 corresponding correspond VERB cana-1301 222 29 to to ADP cana-1301 222 30 two two NUM cana-1301 222 31 nodesπ‘žπ‘– nodesπ‘žπ‘– NOUN cana-1301 222 32 and and CCONJ cana-1301 222 33 π‘žπ‘— π‘žπ‘— VERB cana-1301 222 34 in in ADP cana-1301 222 35 m m NOUN cana-1301 222 36 - - PUNCT cana-1301 222 37 bpfdg bpfdg VERB cana-1301 223 1 𝐺𝑑 𝐺𝑑 NOUN cana-1301 223 2 .let .let PUNCT cana-1301 224 1 〈 〈 PROPN cana-1301 224 2 [ [ X cana-1301 224 3 π‘ƒβ„Žoψr π‘ƒβ„Žoψr PROPN cana-1301 224 4 + + PROPN cana-1301 224 5 r r PROPN cana-1301 224 6 ( ( PUNCT cana-1301 224 7 π‘žπ‘–π‘žπ‘— π‘žπ‘–π‘žπ‘— PROPN cana-1301 224 8 ) ) PUNCT cana-1301 224 9 , , PUNCT cana-1301 225 1 π‘ƒβ„Žoψr π‘ƒβ„Žoψr PROPN cana-1301 225 2 βˆ’r βˆ’r PROPN cana-1301 225 3 ( ( PUNCT cana-1301 225 4 π‘žπ‘–π‘žπ‘— π‘žπ‘–π‘žπ‘— PROPN cana-1301 225 5 ) ) PUNCT cana-1301 225 6 ] ] PUNCT cana-1301 226 1 β„Ž=1 β„Ž=1 PROPN cana-1301 226 2 π‘š π‘š ADP cana-1301 226 3 βŒͺ βŒͺ NOUN cana-1301 226 4 denotes denote VERB cana-1301 226 5 the the DET cana-1301 226 6 positive positive ADJ cana-1301 226 7 and and CCONJ cana-1301 226 8 negative negative ADJ cana-1301 226 9 relationship relationship NOUN cana-1301 226 10 values value NOUN cana-1301 226 11 of of ADP cana-1301 226 12 the the DET cana-1301 226 13 rthline rthline NOUN cana-1301 226 14 between between ADP cana-1301 226 15 π‘žπ‘–and π‘žπ‘–and NOUN cana-1301 226 16 π‘žπ‘— π‘žπ‘— VERB cana-1301 226 17 , , PUNCT cana-1301 226 18 the the DET cana-1301 226 19 positive positive ADJ cana-1301 226 20 and and CCONJ cana-1301 226 21 negative negative ADJ cana-1301 226 22 relationship relationship NOUN cana-1301 226 23 values value NOUN cana-1301 226 24 of of ADP cana-1301 226 25 the the DET cana-1301 226 26 m m PROPN cana-1301 226 27 - - PUNCT cana-1301 226 28 bpf bpf NOUN cana-1301 226 29 lines line NOUN cana-1301 226 30 of of ADP cana-1301 226 31 the the DET cana-1301 226 32 mbpfdg mbpfdg NOUN cana-1301 226 33 are be AUX cana-1301 226 34 given give VERB cana-1301 226 35 byπ‘ƒβ„Žoψ byπ‘ƒβ„Žoψ NOUN cana-1301 226 36 𝑅𝑑 𝑅𝑑 ADP cana-1301 226 37 + + NOUN cana-1301 226 38 r r NOUN cana-1301 226 39 ( ( PUNCT cana-1301 226 40 π‘žπ‘–π‘žπ‘—)π‘Ÿ π‘žπ‘–π‘žπ‘—)π‘Ÿ PUNCT cana-1301 226 41 = = PUNCT cana-1301 227 1 π‘ƒβ„ŽoΟˆπ‘… π‘ƒβ„ŽoΟˆπ‘… PROPN cana-1301 227 2 + + NUM cana-1301 227 3 r r NOUN cana-1301 227 4 ( ( PUNCT cana-1301 227 5 πœ„πœ…)𝑒,π‘ƒβ„Žoψ πœ„πœ…)𝑒,π‘ƒβ„Žoψ NOUN cana-1301 227 6 𝑅𝑑 𝑅𝑑 ADP cana-1301 227 7 βˆ’r βˆ’r PROPN cana-1301 227 8 ( ( PUNCT cana-1301 227 9 π‘žπ‘–π‘žπ‘—)π‘Ÿ π‘žπ‘–π‘žπ‘—)π‘Ÿ PUNCT cana-1301 227 10 = = SYM cana-1301 227 11 π‘ƒβ„ŽoΟˆπ‘… π‘ƒβ„ŽoΟˆπ‘… PROPN cana-1301 227 12 βˆ’r βˆ’r ADJ cana-1301 227 13 ( ( PUNCT cana-1301 227 14 πœ„πœ…)𝑒 πœ„πœ…)𝑒 PROPN cana-1301 227 15 , , PUNCT cana-1301 227 16 where(πœ„πœ…)𝑒is where(πœ„πœ…)𝑒is PROPN cana-1301 227 17 an an DET cana-1301 227 18 line line NOUN cana-1301 227 19 in in ADP cana-1301 227 20 the the DET cana-1301 227 21 boundary boundary ADJ cana-1301 227 22 corresponding correspond VERB cana-1301 227 23 2 2 NUM cana-1301 227 24 strong strong ADJ cana-1301 227 25 m m NOUN cana-1301 227 26 - - PUNCT cana-1301 227 27 bpf bpf NOUN cana-1301 227 28 faces face NOUN cana-1301 227 29 𝛽𝑖 𝛽𝑖 ADJ cana-1301 227 30 and and CCONJ cana-1301 227 31 𝛽𝑗and 𝛽𝑗and VERB cana-1301 227 32 π‘Ÿ π‘Ÿ NOUN cana-1301 227 33 = = SYM cana-1301 227 34 1,2 1,2 NUM cana-1301 227 35 , , PUNCT cana-1301 227 36 β‹― β‹― VERB cana-1301 227 37 , , PUNCT cana-1301 227 38 𝑠,where 𝑠,where ADV cana-1301 227 39 𝑠is 𝑠is VERB cana-1301 227 40 the the DET cana-1301 227 41 number number NOUN cana-1301 227 42 of of ADP cana-1301 227 43 common common ADJ cana-1301 227 44 lines line NOUN cana-1301 227 45 in in ADP cana-1301 227 46 the the DET cana-1301 227 47 boundary boundary NOUN cana-1301 227 48 between between ADP cana-1301 227 49 𝛽𝑖 𝛽𝑖 NOUN cana-1301 227 50 and and CCONJ cana-1301 227 51 𝛽𝑗 𝛽𝑗 NOUN cana-1301 227 52 or or CCONJ cana-1301 227 53 the the DET cana-1301 227 54 number number NOUN cana-1301 227 55 of of ADP cana-1301 227 56 lines line NOUN cana-1301 227 57 between between ADP cana-1301 227 58 π‘žπ‘– π‘žπ‘– PROPN cana-1301 227 59 and and CCONJ cana-1301 227 60 π‘žπ‘— π‘žπ‘— INTJ cana-1301 227 61 . . PUNCT cana-1301 228 1 if if SCONJ cana-1301 228 2 there there PRON cana-1301 228 3 be be VERB cana-1301 228 4 any any DET cana-1301 228 5 strong strong ADJ cana-1301 228 6 pendant pendant ADJ cana-1301 228 7 line line NOUN cana-1301 228 8 in in ADP cana-1301 228 9 the the DET cana-1301 228 10 m m NOUN cana-1301 228 11 - - NOUN cana-1301 228 12 bpfpg bpfpg NOUN cana-1301 228 13 , , PUNCT cana-1301 228 14 then then ADV cana-1301 228 15 there there PRON cana-1301 228 16 will will AUX cana-1301 228 17 be be AUX cana-1301 228 18 a a DET cana-1301 228 19 self self NOUN cana-1301 228 20 loop loop NOUN cana-1301 228 21 in in ADP cana-1301 228 22 𝐺𝑑 𝐺𝑑 PROPN cana-1301 228 23 corresponding correspond VERB cana-1301 228 24 to to ADP cana-1301 228 25 this this DET cana-1301 228 26 pendant pendant ADJ cana-1301 228 27 line line NOUN cana-1301 228 28 . . PUNCT cana-1301 229 1 the the DET cana-1301 229 2 line line NOUN cana-1301 229 3 positive positive ADJ cana-1301 229 4 and and CCONJ cana-1301 229 5 negative negative ADJ cana-1301 229 6 relationship relationship NOUN cana-1301 229 7 value value NOUN cana-1301 229 8 of of ADP cana-1301 229 9 the the DET cana-1301 229 10 self self NOUN cana-1301 229 11 loop loop NOUN cana-1301 229 12 is be AUX cana-1301 229 13 equal equal ADJ cana-1301 229 14 to to ADP cana-1301 229 15 the the DET cana-1301 229 16 positive positive ADJ cana-1301 229 17 and and CCONJ cana-1301 229 18 negative negative ADJ cana-1301 229 19 relationship relationship NOUN cana-1301 229 20 values value NOUN cana-1301 229 21 of of ADP cana-1301 229 22 the the DET cana-1301 229 23 pendant pendant ADJ cana-1301 229 24 line line NOUN cana-1301 229 25 . . PUNCT cana-1301 230 1 m m PROPN cana-1301 230 2 - - PUNCT cana-1301 230 3 bpfdg bpfdg NOUN cana-1301 230 4 of of ADP cana-1301 230 5 m m PROPN cana-1301 230 6 - - NOUN cana-1301 230 7 bpfpg bpfpg NOUN cana-1301 230 8 does do AUX cana-1301 230 9 not not PART cana-1301 230 10 contain contain VERB cana-1301 230 11 point point NOUN cana-1301 230 12 of of ADP cana-1301 230 13 junction junction NOUN cana-1301 230 14 of of ADP cana-1301 230 15 the the DET cana-1301 230 16 lines line NOUN cana-1301 230 17 for for ADP cana-1301 230 18 a a DET cana-1301 230 19 certain certain ADJ cana-1301 230 20 representation representation NOUN cana-1301 230 21 , , PUNCT cana-1301 230 22 so so CCONJ cana-1301 230 23 it it PRON cana-1301 230 24 is be AUX cana-1301 230 25 m m NOUN cana-1301 230 26 - - NOUN cana-1301 230 27 bpfpg bpfpg NOUN cana-1301 230 28 with with ADP cana-1301 230 29 planarity planarity NOUN cana-1301 230 30 value value NOUN cana-1301 230 31 〈 〈 PROPN cana-1301 230 32 [ [ X cana-1301 230 33 1 1 NUM cana-1301 230 34 , , PUNCT cana-1301 230 35 βˆ’1]β„Ž=1 βˆ’1]β„Ž=1 PROPN cana-1301 230 36 π‘š π‘š ADP cana-1301 230 37 βŒͺ βŒͺ NOUN cana-1301 230 38 . . PUNCT cana-1301 231 1 hence hence ADV cana-1301 231 2 the the DET cana-1301 231 3 m m NOUN cana-1301 231 4 - - PUNCT cana-1301 231 5 bpf bpf NOUN cana-1301 231 6 face face NOUN cana-1301 231 7 of of ADP cana-1301 231 8 m m PROPN cana-1301 231 9 - - PUNCT cana-1301 231 10 bpfdg bpfdg PROPN cana-1301 231 11 can can AUX cana-1301 231 12 be be AUX cana-1301 231 13 similarly similarly ADV cana-1301 231 14 described describe VERB cana-1301 231 15 as as ADP cana-1301 231 16 in in ADP cana-1301 231 17 m m NOUN cana-1301 231 18 - - NOUN cana-1301 231 19 bpfpgs bpfpgs NOUN cana-1301 231 20 . . PUNCT cana-1301 232 1 example example NOUN cana-1301 232 2 5 5 NUM cana-1301 232 3 : : PUNCT cana-1301 232 4 consider consider VERB cana-1301 232 5 an an DET cana-1301 232 6 m m NOUN cana-1301 232 7 - - ADJ cana-1301 232 8 bpfpg bpfpg ADJ cana-1301 232 9 𝐺 𝐺 PROPN cana-1301 232 10 = = SYM cana-1301 232 11 ( ( PUNCT cana-1301 232 12 𝑉 𝑉 PROPN cana-1301 232 13 , , PUNCT cana-1301 232 14 𝑄 𝑄 PROPN cana-1301 232 15 , , PUNCT cana-1301 232 16 𝑅)of 𝑅)of VERB cana-1301 232 17 πΊβˆ— πΊβˆ— NOUN cana-1301 233 1 = = PUNCT cana-1301 233 2 ( ( PUNCT cana-1301 233 3 𝑉 𝑉 PROPN cana-1301 233 4 , , PUNCT cana-1301 233 5 𝐸)as 𝐸)as NOUN cana-1301 233 6 given give VERB cana-1301 233 7 away away ADP cana-1301 233 8 in in ADP cana-1301 233 9 fig fig NOUN cana-1301 233 10 . . PUNCT cana-1301 234 1 5 5 NUM cana-1301 234 2 such such ADJ cana-1301 234 3 that that DET cana-1301 234 4 𝑉 𝑉 PROPN cana-1301 234 5 = = SYM cana-1301 234 6 { { PUNCT cana-1301 234 7 𝑝 𝑝 PROPN cana-1301 234 8 , , PUNCT cana-1301 234 9 π‘ž π‘ž PROPN cana-1301 234 10 , , PUNCT cana-1301 234 11 π‘Ÿ π‘Ÿ X cana-1301 234 12 , , PUNCT cana-1301 234 13 𝑠 𝑠 PROPN cana-1301 234 14 } } PUNCT cana-1301 234 15 , , PUNCT cana-1301 234 16 𝐸 𝐸 PROPN cana-1301 234 17 = = SYM cana-1301 234 18 { { PUNCT cana-1301 234 19 π‘π‘ž π‘π‘ž PROPN cana-1301 234 20 , , PUNCT cana-1301 234 21 π‘žπ‘Ÿ π‘žπ‘Ÿ PROPN cana-1301 234 22 , , PUNCT cana-1301 234 23 π‘Ÿπ‘  π‘Ÿπ‘  PROPN cana-1301 234 24 , , PUNCT cana-1301 234 25 𝑠𝑝 𝑠𝑝 PROPN cana-1301 234 26 , , PUNCT cana-1301 234 27 π‘π‘Ÿ π‘π‘Ÿ NOUN cana-1301 234 28 } } PUNCT cana-1301 234 29 fig fig NOUN cana-1301 234 30 . . PUNCT cana-1301 235 1 5.2 5.2 NUM cana-1301 235 2 - - PUNCT cana-1301 235 3 bpfg bpfg NOUN cana-1301 236 1 and and CCONJ cana-1301 236 2 it it PRON cana-1301 236 3 ’s ’ VERB cana-1301 236 4 2 2 NUM cana-1301 236 5 - - PUNCT cana-1301 236 6 bpfdg bpfdg NOUN cana-1301 236 7 communications communication NOUN cana-1301 236 8 on on ADP cana-1301 236 9 applied apply VERB cana-1301 236 10 nonlinear nonlinear ADJ cana-1301 236 11 analysis analysis NOUN cana-1301 236 12 issn issn NOUN cana-1301 236 13 : : PUNCT cana-1301 236 14 1074 1074 NUM cana-1301 236 15 - - PUNCT cana-1301 236 16 133x 133x NUM cana-1301 236 17 vol vol NOUN cana-1301 236 18 31 31 NUM cana-1301 236 19 no no NOUN cana-1301 236 20 . . PUNCT cana-1301 237 1 7s 7 NOUN cana-1301 237 2 ( ( PUNCT cana-1301 237 3 2024 2024 NUM cana-1301 237 4 ) ) PUNCT cana-1301 237 5 272 272 NUM cana-1301 237 6 https://internationalpubls.com https://internationalpubls.com X cana-1301 237 7 𝑄 𝑄 PROPN cana-1301 237 8 = = PUNCT cana-1301 237 9 { { PUNCT cana-1301 237 10 𝑝 𝑝 PROPN cana-1301 237 11 〈 〈 PROPN cana-1301 237 12 [ [ X cana-1301 237 13 0.9,βˆ’0.7],[0.8,βˆ’0.5 0.9,βˆ’0.7],[0.8,βˆ’0.5 NUM cana-1301 237 14 ] ] X cana-1301 237 15 βŒͺ βŒͺ NOUN cana-1301 237 16 , , PUNCT cana-1301 237 17 π‘ž π‘ž X cana-1301 237 18 〈 〈 PROPN cana-1301 237 19 [ [ X cana-1301 237 20 0.7,βˆ’0.8],[0.8,βˆ’0.9 0.7,βˆ’0.8],[0.8,βˆ’0.9 NUM cana-1301 237 21 ] ] X cana-1301 237 22 βŒͺ βŒͺ NOUN cana-1301 237 23 , , PUNCT cana-1301 237 24 π‘Ÿ π‘Ÿ X cana-1301 237 25 〈 〈 NOUN cana-1301 237 26 [ [ X cana-1301 237 27 0.5,βˆ’0.23],[0.7,βˆ’0.9 0.5,βˆ’0.23],[0.7,βˆ’0.9 NOUN cana-1301 237 28 ] ] X cana-1301 237 29 βŒͺ βŒͺ NOUN cana-1301 237 30 , , PUNCT cana-1301 237 31 𝑠 𝑠 PROPN cana-1301 237 32 〈 〈 PROPN cana-1301 237 33 [ [ X cana-1301 237 34 0.6,βˆ’0.8],[0.7,βˆ’0.35 0.6,βˆ’0.8],[0.7,βˆ’0.35 X cana-1301 237 35 ] ] X cana-1301 237 36 βŒͺ βŒͺ NOUN cana-1301 237 37 } } PUNCT cana-1301 237 38 , , PUNCT cana-1301 237 39 𝑅 𝑅 PROPN cana-1301 237 40 = = PROPN cana-1301 237 41 { { PUNCT cana-1301 237 42 π‘π‘ž π‘π‘ž NOUN cana-1301 237 43 〈 〈 PROPN cana-1301 237 44 [ [ X cana-1301 237 45 0.65,βˆ’0.69],[0.75,βˆ’0.45 0.65,βˆ’0.69],[0.75,βˆ’0.45 NOUN cana-1301 237 46 ] ] NOUN cana-1301 237 47 βŒͺ βŒͺ NOUN cana-1301 237 48 , , PUNCT cana-1301 237 49 π‘žπ‘Ÿ π‘žπ‘Ÿ PROPN cana-1301 237 50 〈 〈 PROPN cana-1301 237 51 [ [ X cana-1301 237 52 0.49,βˆ’0.21],[0.65,βˆ’0.75 0.49,βˆ’0.21],[0.65,βˆ’0.75 NOUN cana-1301 237 53 ] ] X cana-1301 237 54 βŒͺ βŒͺ NOUN cana-1301 237 55 , , PUNCT cana-1301 237 56 π‘Ÿπ‘  π‘Ÿπ‘  PROPN cana-1301 237 57 〈 〈 PROPN cana-1301 237 58 [ [ X cana-1301 237 59 0.49,βˆ’0.2],[0.61,βˆ’0.32 0.49,βˆ’0.2],[0.61,βˆ’0.32 PROPN cana-1301 237 60 ] ] SYM cana-1301 237 61 βŒͺ βŒͺ NOUN cana-1301 237 62 , , PUNCT cana-1301 237 63 𝑠𝑝 𝑠𝑝 ADP cana-1301 237 64 〈 〈 PROPN cana-1301 237 65 [ [ X cana-1301 237 66 0.55,βˆ’0.45],[0.69,βˆ’0.23 0.55,βˆ’0.45],[0.69,βˆ’0.23 X cana-1301 237 67 ] ] X cana-1301 237 68 βŒͺ βŒͺ NOUN cana-1301 237 69 , , PUNCT cana-1301 237 70 π‘π‘Ÿ π‘π‘Ÿ PROPN cana-1301 237 71 〈 〈 PROPN cana-1301 237 72 [ [ X cana-1301 237 73 0.41,βˆ’0.2],[0.66,βˆ’0.49 0.41,βˆ’0.2],[0.66,βˆ’0.49 PROPN cana-1301 237 74 ] ] X cana-1301 237 75 βŒͺ βŒͺ NOUN cana-1301 237 76 , , PUNCT cana-1301 237 77 π‘žπ‘Ÿ π‘žπ‘Ÿ PROPN cana-1301 237 78 〈 〈 PROPN cana-1301 237 79 [ [ PROPN cana-1301 237 80 0.46,βˆ’0.2],[0.65,βˆ’0.81 0.46,βˆ’0.2],[0.65,βˆ’0.81 PROPN cana-1301 237 81 ] ] X cana-1301 237 82 βŒͺ βŒͺ NOUN cana-1301 237 83 } } PUNCT cana-1301 237 84 . . PUNCT cana-1301 238 1 the the DET cana-1301 238 2 m m PROPN cana-1301 238 3 - - NOUN cana-1301 238 4 bpfpg bpfpg NOUN cana-1301 238 5 has have VERB cana-1301 238 6 the the DET cana-1301 238 7 following follow VERB cana-1301 238 8 faces face NOUN cana-1301 238 9 : : PUNCT cana-1301 238 10 β€’ β€’ NUM cana-1301 238 11 m m NOUN cana-1301 238 12 - - PUNCT cana-1301 238 13 bpf bpf NOUN cana-1301 238 14 face face NOUN cana-1301 238 15 𝛽1 𝛽1 NOUN cana-1301 238 16 is be AUX cana-1301 238 17 bounded bound VERB cana-1301 238 18 by by ADP cana-1301 238 19 π‘π‘ž π‘π‘ž NOUN cana-1301 238 20 〈 〈 PROPN cana-1301 238 21 [ [ X cana-1301 238 22 0.65,βˆ’0.69],[0.75,βˆ’0.45 0.65,βˆ’0.69],[0.75,βˆ’0.45 NOUN cana-1301 238 23 ] ] NOUN cana-1301 238 24 βŒͺ βŒͺ NOUN cana-1301 238 25 , , PUNCT cana-1301 238 26 π‘π‘Ÿ π‘π‘Ÿ PROPN cana-1301 238 27 〈 〈 PROPN cana-1301 238 28 [ [ X cana-1301 238 29 0.41,βˆ’0.2],[0.66,βˆ’0.49 0.41,βˆ’0.2],[0.66,βˆ’0.49 PROPN cana-1301 238 30 ] ] X cana-1301 238 31 βŒͺ βŒͺ NOUN cana-1301 238 32 , , PUNCT cana-1301 238 33 π‘žπ‘Ÿ π‘žπ‘Ÿ PROPN cana-1301 238 34 〈 〈 PROPN cana-1301 238 35 [ [ X cana-1301 238 36 0.49,βˆ’0.21],[0.65,βˆ’0.75 0.49,βˆ’0.21],[0.65,βˆ’0.75 NOUN cana-1301 238 37 ] ] X cana-1301 238 38 βŒͺ βŒͺ X cana-1301 238 39 β€’ β€’ NUM cana-1301 238 40 m m PROPN cana-1301 238 41 - - PUNCT cana-1301 238 42 bpf bpf NOUN cana-1301 238 43 face face NOUN cana-1301 238 44 𝛽2 𝛽2 NOUN cana-1301 238 45 bounded bound VERB cana-1301 238 46 by by ADP cana-1301 238 47 the the DET cana-1301 238 48 lines line NOUN cana-1301 238 49 π‘π‘Ÿ π‘π‘Ÿ PROPN cana-1301 238 50 〈 〈 PROPN cana-1301 238 51 [ [ X cana-1301 238 52 0.41,βˆ’0.2],[0.66,βˆ’0.49 0.41,βˆ’0.2],[0.66,βˆ’0.49 PROPN cana-1301 238 53 ] ] X cana-1301 238 54 βŒͺ βŒͺ NOUN cana-1301 238 55 , , PUNCT cana-1301 238 56 𝑠𝑝 𝑠𝑝 ADP cana-1301 238 57 〈 〈 PROPN cana-1301 238 58 [ [ X cana-1301 238 59 0.55,βˆ’0.45],[0.69,βˆ’0.23 0.55,βˆ’0.45],[0.69,βˆ’0.23 X cana-1301 238 60 ] ] X cana-1301 238 61 βŒͺ βŒͺ NOUN cana-1301 238 62 , , PUNCT cana-1301 238 63 π‘Ÿπ‘  π‘Ÿπ‘  PROPN cana-1301 238 64 〈 〈 PROPN cana-1301 238 65 [ [ X cana-1301 238 66 0.49,βˆ’0.2],[0.61,βˆ’0.32 0.49,βˆ’0.2],[0.61,βˆ’0.32 PROPN cana-1301 238 67 ] ] SYM cana-1301 238 68 βŒͺ βŒͺ NOUN cana-1301 238 69 β€’ β€’ NUM cana-1301 238 70 m m PROPN cana-1301 238 71 - - PUNCT cana-1301 238 72 bpf bpf NOUN cana-1301 238 73 face face NOUN cana-1301 238 74 𝛽3 𝛽3 PROPN cana-1301 238 75 is be AUX cana-1301 238 76 bounded bound VERB cana-1301 238 77 by by ADP cana-1301 238 78 the the DET cana-1301 238 79 lines line NOUN cana-1301 238 80 π‘žπ‘Ÿ π‘žπ‘Ÿ PROPN cana-1301 238 81 〈 〈 PROPN cana-1301 238 82 [ [ X cana-1301 238 83 0.49,βˆ’0.21],[0.65,βˆ’0.75 0.49,βˆ’0.21],[0.65,βˆ’0.75 NOUN cana-1301 238 84 ] ] X cana-1301 238 85 βŒͺ βŒͺ NOUN cana-1301 238 86 , , PUNCT cana-1301 238 87 π‘žπ‘Ÿ π‘žπ‘Ÿ PROPN cana-1301 238 88 〈 〈 PROPN cana-1301 238 89 [ [ PROPN cana-1301 238 90 0.46,βˆ’0.2],[0.65,βˆ’0.81 0.46,βˆ’0.2],[0.65,βˆ’0.81 PROPN cana-1301 238 91 ] ] SYM cana-1301 238 92 βŒͺ βŒͺ NOUN cana-1301 238 93 and and CCONJ cana-1301 238 94 β€’ β€’ ADJ cana-1301 238 95 outer outer ADJ cana-1301 238 96 m m NOUN cana-1301 238 97 - - PUNCT cana-1301 238 98 bpf bpf NOUN cana-1301 238 99 𝛽4 𝛽4 PROPN cana-1301 238 100 is be AUX cana-1301 238 101 surrounded surround VERB cana-1301 238 102 by by ADP cana-1301 238 103 π‘π‘ž π‘π‘ž NOUN cana-1301 238 104 〈 〈 PROPN cana-1301 238 105 [ [ X cana-1301 238 106 0.65,βˆ’0.69],[0.75,βˆ’0.45 0.65,βˆ’0.69],[0.75,βˆ’0.45 NOUN cana-1301 238 107 ] ] NOUN cana-1301 238 108 βŒͺ βŒͺ NOUN cana-1301 238 109 , , PUNCT cana-1301 238 110 π‘žπ‘Ÿ π‘žπ‘Ÿ PROPN cana-1301 238 111 〈 〈 PROPN cana-1301 238 112 [ [ PROPN cana-1301 238 113 0.46,βˆ’0.2],[0.65,βˆ’0.81 0.46,βˆ’0.2],[0.65,βˆ’0.81 PROPN cana-1301 238 114 ] ] X cana-1301 238 115 βŒͺ βŒͺ NOUN cana-1301 238 116 , , PUNCT cana-1301 238 117 π‘Ÿπ‘  π‘Ÿπ‘  PROPN cana-1301 238 118 〈 〈 PROPN cana-1301 238 119 [ [ X cana-1301 238 120 0.49,βˆ’0.2],[0.61,βˆ’0.32 0.49,βˆ’0.2],[0.61,βˆ’0.32 PROPN cana-1301 238 121 ] ] SYM cana-1301 238 122 βŒͺ βŒͺ NOUN cana-1301 238 123 , , PUNCT cana-1301 238 124 𝑠𝑝 𝑠𝑝 ADP cana-1301 238 125 〈 〈 PROPN cana-1301 238 126 [ [ X cana-1301 238 127 0.55,βˆ’0.45],[0.69,βˆ’0.23 0.55,βˆ’0.45],[0.69,βˆ’0.23 X cana-1301 238 128 ] ] X cana-1301 238 129 βŒͺ βŒͺ NOUN cana-1301 238 130 . . PUNCT cana-1301 239 1 according accord VERB cana-1301 239 2 to to ADP cana-1301 239 3 routine routine ADJ cana-1301 239 4 calculations calculation NOUN cana-1301 239 5 , , PUNCT cana-1301 239 6 all all DET cana-1301 239 7 faces face NOUN cana-1301 239 8 are be AUX cana-1301 239 9 strong strong ADJ cana-1301 239 10 m m NOUN cana-1301 239 11 - - PUNCT cana-1301 239 12 bpf bpf NOUN cana-1301 239 13 faces face NOUN cana-1301 239 14 . . PUNCT cana-1301 240 1 we we PRON cana-1301 240 2 consider consider VERB cana-1301 240 3 into into ADP cana-1301 240 4 an an DET cana-1301 240 5 account account NOUN cana-1301 240 6 a a DET cana-1301 240 7 node node NOUN cana-1301 240 8 for for ADP cana-1301 240 9 the the DET cana-1301 240 10 m m NOUN cana-1301 240 11 - - PUNCT cana-1301 240 12 bpfdg bpfdg NOUN cana-1301 240 13 for for SCONJ cana-1301 240 14 each each DET cana-1301 240 15 strong strong ADJ cana-1301 240 16 m m NOUN cana-1301 240 17 - - PUNCT cana-1301 240 18 bpf bpf NOUN cana-1301 240 19 face.so face.so NUM cana-1301 240 20 the the DET cana-1301 240 21 node node ADJ cana-1301 240 22 set𝑉𝑑 set𝑉𝑑 NOUN cana-1301 240 23 = = SYM cana-1301 240 24 { { PUNCT cana-1301 240 25 π‘ž1 π‘ž1 ADJ cana-1301 240 26 , , PUNCT cana-1301 240 27 π‘ž2 π‘ž2 NOUN cana-1301 240 28 , , PUNCT cana-1301 240 29 π‘ž3 π‘ž3 NOUN cana-1301 240 30 , , PUNCT cana-1301 240 31 π‘ž4 π‘ž4 PROPN cana-1301 240 32 } } PUNCT cana-1301 240 33 , , PUNCT cana-1301 240 34 where where SCONJ cana-1301 240 35 the the DET cana-1301 240 36 node node NOUN cana-1301 240 37 π‘žπ‘— π‘žπ‘— X cana-1301 240 38 is be AUX cana-1301 240 39 given give VERB cana-1301 240 40 consequent consequent NOUN cana-1301 240 41 to to ADP cana-1301 240 42 the the DET cana-1301 240 43 strong strong ADJ cana-1301 240 44 m m NOUN cana-1301 240 45 - - PUNCT cana-1301 240 46 bpf bpf NOUN cana-1301 240 47 face face NOUN cana-1301 240 48 𝛽𝑗 𝛽𝑗 PRON cana-1301 240 49 , , PUNCT cana-1301 240 50 𝑗 𝑗 NOUN cana-1301 240 51 = = SYM cana-1301 240 52 1 1 NUM cana-1301 240 53 , , PUNCT cana-1301 240 54 2,3,4 2,3,4 NUM cana-1301 240 55 . . PUNCT cana-1301 241 1 thus thus ADV cana-1301 241 2 , , PUNCT cana-1301 241 3 𝑄𝑑 𝑄𝑑 PROPN cana-1301 241 4 = = PUNCT cana-1301 241 5 { { PUNCT cana-1301 241 6 π‘ž1 π‘ž1 PROPN cana-1301 241 7 〈 〈 PROPN cana-1301 241 8 [ [ X cana-1301 241 9 0.65,βˆ’0.69],[0.75,βˆ’0.75 0.65,βˆ’0.69],[0.75,βˆ’0.75 ADJ cana-1301 241 10 ] ] ADJ cana-1301 241 11 βŒͺ βŒͺ NOUN cana-1301 241 12 , , PUNCT cana-1301 241 13 π‘ž2 π‘ž2 PROPN cana-1301 241 14 〈 〈 PROPN cana-1301 241 15 [ [ X cana-1301 241 16 0.55,βˆ’0.45],[0.69,βˆ’0.49 0.55,βˆ’0.45],[0.69,βˆ’0.49 NOUN cana-1301 241 17 ] ] X cana-1301 241 18 βŒͺ βŒͺ NOUN cana-1301 241 19 , , PUNCT cana-1301 241 20 π‘ž3 π‘ž3 PROPN cana-1301 241 21 〈 〈 PROPN cana-1301 241 22 [ [ X cana-1301 241 23 0.49,βˆ’0.21],[0.65,βˆ’0.81 0.49,βˆ’0.21],[0.65,βˆ’0.81 NOUN cana-1301 241 24 ] ] X cana-1301 241 25 βŒͺ βŒͺ NOUN cana-1301 241 26 , , PUNCT cana-1301 241 27 π‘ž4 π‘ž4 PROPN cana-1301 241 28 〈 〈 PROPN cana-1301 241 29 [ [ X cana-1301 241 30 0.65,βˆ’0.69],[0.75,βˆ’0.81 0.65,βˆ’0.69],[0.75,βˆ’0.81 PROPN cana-1301 241 31 ] ] X cana-1301 241 32 βŒͺ βŒͺ NOUN cana-1301 241 33 } } PUNCT cana-1301 241 34 , , PUNCT cana-1301 241 35 the the DET cana-1301 241 36 faces face NOUN cana-1301 241 37 𝛽2and 𝛽2and PROPN cana-1301 241 38 𝛽4 𝛽4 PROPN cana-1301 241 39 in in ADP cana-1301 241 40 𝐺share 𝐺share PROPN cana-1301 241 41 two two NUM cana-1301 241 42 common common ADJ cana-1301 241 43 lines line NOUN cana-1301 241 44 , , PUNCT cana-1301 241 45 𝑝𝑠 𝑝𝑠 CCONJ cana-1301 241 46 and and CCONJ cana-1301 241 47 π‘Ÿπ‘ . π‘Ÿπ‘ . NOUN cana-1301 241 48 consequently consequently ADV cana-1301 241 49 , , PUNCT cana-1301 241 50 there there PRON cana-1301 241 51 are be VERB cana-1301 241 52 two two NUM cana-1301 241 53 lines line NOUN cana-1301 241 54 in in ADP cana-1301 241 55 the the DET cana-1301 241 56 m m NOUN cana-1301 241 57 - - PUNCT cana-1301 241 58 bpfdg bpfdg NOUN cana-1301 241 59 of of ADP cana-1301 241 60 g g NOUN cana-1301 241 61 between between ADP cana-1301 241 62 the the DET cana-1301 241 63 nodes node NOUN cana-1301 241 64 π‘ž2 π‘ž2 NOUN cana-1301 241 65 and and CCONJ cana-1301 241 66 π‘ž4 π‘ž4 NOUN cana-1301 241 67 . . PUNCT cana-1301 242 1 that that DET cana-1301 242 2 line line NOUN cana-1301 242 3 ’s ’ VERB cana-1301 242 4 positive positive ADJ cana-1301 242 5 and and CCONJ cana-1301 242 6 negative negative ADJ cana-1301 242 7 relationship relationship NOUN cana-1301 242 8 values value NOUN cana-1301 242 9 are be AUX cana-1301 242 10 given give VERB cana-1301 242 11 by by ADP cana-1301 242 12 { { PUNCT cana-1301 242 13 π‘ž2π‘ž4 π‘ž2π‘ž4 INTJ cana-1301 242 14 〈 〈 NOUN cana-1301 242 15 [ [ X cana-1301 242 16 0.49,βˆ’0.2],[0.61,βˆ’0.32 0.49,βˆ’0.2],[0.61,βˆ’0.32 PROPN cana-1301 242 17 ] ] SYM cana-1301 242 18 βŒͺ βŒͺ NOUN cana-1301 242 19 , , PUNCT cana-1301 242 20 π‘ž2π‘ž4 π‘ž2π‘ž4 PROPN cana-1301 242 21 〈 〈 ADP cana-1301 242 22 [ [ X cana-1301 242 23 0.55,βˆ’0.45],[0.69,βˆ’0.23 0.55,βˆ’0.45],[0.69,βˆ’0.23 X cana-1301 242 24 ] ] X cana-1301 242 25 βŒͺ βŒͺ NOUN cana-1301 242 26 } } PUNCT cana-1301 242 27 . . PUNCT cana-1301 243 1 the the DET cana-1301 243 2 other other ADJ cana-1301 243 3 lines line NOUN cana-1301 243 4 of of ADP cana-1301 243 5 the the DET cana-1301 243 6 mbpfdg mbpfdg NOUN cana-1301 243 7 's 's PART cana-1301 243 8 of of ADP cana-1301 243 9 positive positive ADJ cana-1301 243 10 and and CCONJ cana-1301 243 11 negative negative ADJ cana-1301 243 12 relationship relationship NOUN cana-1301 243 13 values value NOUN cana-1301 243 14 are be AUX cana-1301 243 15 computed compute VERB cana-1301 243 16 as as SCONJ cana-1301 243 17 follows follow VERB cana-1301 243 18 . . PUNCT cana-1301 244 1 { { PUNCT cana-1301 245 1 π‘ž1π‘ž3 π‘ž1π‘ž3 PROPN cana-1301 245 2 〈 〈 PROPN cana-1301 245 3 [ [ X cana-1301 245 4 0.49,βˆ’0.21],[0.65,βˆ’0.75 0.49,βˆ’0.21],[0.65,βˆ’0.75 NOUN cana-1301 245 5 ] ] X cana-1301 245 6 βŒͺ βŒͺ NOUN cana-1301 245 7 , , PUNCT cana-1301 245 8 π‘ž1π‘ž2 π‘ž1π‘ž2 PROPN cana-1301 245 9 〈 〈 PROPN cana-1301 245 10 [ [ X cana-1301 245 11 0.41,βˆ’0.2],[0.66,βˆ’0.49 0.41,βˆ’0.2],[0.66,βˆ’0.49 PROPN cana-1301 245 12 ] ] X cana-1301 245 13 βŒͺ βŒͺ NOUN cana-1301 245 14 , , PUNCT cana-1301 245 15 π‘ž1π‘ž4 π‘ž1π‘ž4 X cana-1301 245 16 〈 〈 NOUN cana-1301 245 17 [ [ X cana-1301 245 18 0.65,βˆ’0.69],[0.75,βˆ’0.45 0.65,βˆ’0.69],[0.75,βˆ’0.45 NOUN cana-1301 245 19 ] ] NOUN cana-1301 245 20 βŒͺ βŒͺ NOUN cana-1301 245 21 , , PUNCT cana-1301 245 22 π‘ž3π‘ž4 π‘ž3π‘ž4 X cana-1301 246 1 〈 〈 PROPN cana-1301 246 2 [ [ SYM cana-1301 246 3 0.46,βˆ’0.2],[0.65,βˆ’0.81 0.46,βˆ’0.2],[0.65,βˆ’0.81 PROPN cana-1301 246 4 ] ] X cana-1301 246 5 βŒͺ βŒͺ NOUN cana-1301 246 6 } } PUNCT cana-1301 246 7 . . PUNCT cana-1301 247 1 thus thus ADV cana-1301 247 2 , , PUNCT cana-1301 247 3 the the DET cana-1301 247 4 line line NOUN cana-1301 247 5 set set VERB cana-1301 247 6 of of ADP cana-1301 247 7 m m PROPN cana-1301 247 8 - - PUNCT cana-1301 247 9 bpfdg bpfdg PROPN cana-1301 247 10 is be AUX cana-1301 247 11 𝑅𝑑 𝑅𝑑 PROPN cana-1301 247 12 = = PUNCT cana-1301 247 13 { { PUNCT cana-1301 247 14 π‘ž2π‘ž4 π‘ž2π‘ž4 INTJ cana-1301 247 15 〈 〈 NOUN cana-1301 247 16 [ [ X cana-1301 247 17 0.49,βˆ’0.2],[0.61,βˆ’0.32 0.49,βˆ’0.2],[0.61,βˆ’0.32 PROPN cana-1301 247 18 ] ] SYM cana-1301 247 19 βŒͺ βŒͺ NOUN cana-1301 247 20 , , PUNCT cana-1301 247 21 π‘ž2π‘ž4 π‘ž2π‘ž4 PROPN cana-1301 247 22 〈 〈 ADP cana-1301 247 23 [ [ X cana-1301 247 24 0.55,βˆ’0.45],[0.69,βˆ’0.23 0.55,βˆ’0.45],[0.69,βˆ’0.23 X cana-1301 247 25 ] ] X cana-1301 247 26 βŒͺ βŒͺ NOUN cana-1301 247 27 , , PUNCT cana-1301 247 28 π‘ž1π‘ž3 π‘ž1π‘ž3 PROPN cana-1301 247 29 〈 〈 PROPN cana-1301 247 30 [ [ X cana-1301 247 31 0.49,βˆ’0.21],[0.65,βˆ’0.75 0.49,βˆ’0.21],[0.65,βˆ’0.75 NOUN cana-1301 247 32 ] ] X cana-1301 247 33 βŒͺ βŒͺ NOUN cana-1301 247 34 , , PUNCT cana-1301 247 35 π‘ž1π‘ž2 π‘ž1π‘ž2 PROPN cana-1301 247 36 〈 〈 PROPN cana-1301 247 37 [ [ X cana-1301 247 38 0.41,βˆ’0.2],[0.66,βˆ’0.49 0.41,βˆ’0.2],[0.66,βˆ’0.49 PROPN cana-1301 247 39 ] ] X cana-1301 247 40 βŒͺ βŒͺ NOUN cana-1301 247 41 , , PUNCT cana-1301 247 42 π‘ž1π‘ž4 π‘ž1π‘ž4 X cana-1301 247 43 〈 〈 NOUN cana-1301 247 44 [ [ X cana-1301 247 45 0.65,βˆ’0.69],[0.75,βˆ’0.45 0.65,βˆ’0.69],[0.75,βˆ’0.45 NOUN cana-1301 247 46 ] ] NOUN cana-1301 247 47 βŒͺ βŒͺ NOUN cana-1301 247 48 , , PUNCT cana-1301 247 49 π‘ž3π‘ž4 π‘ž3π‘ž4 X cana-1301 248 1 〈 〈 PROPN cana-1301 248 2 [ [ SYM cana-1301 248 3 0.46,βˆ’0.2],[0.65,βˆ’0.81 0.46,βˆ’0.2],[0.65,βˆ’0.81 PROPN cana-1301 248 4 ] ] X cana-1301 248 5 βŒͺ βŒͺ NOUN cana-1301 248 6 } } PUNCT cana-1301 248 7 . . PUNCT cana-1301 249 1 in in ADP cana-1301 249 2 fig fig NOUN cana-1301 249 3 . . PUNCT cana-1301 250 1 5 5 NUM cana-1301 250 2 , , PUNCT cana-1301 250 3 the the DET cana-1301 250 4 m m NOUN cana-1301 250 5 - - PUNCT cana-1301 250 6 bpfdg bpfdg PROPN cana-1301 250 7 𝐺𝑑of 𝐺𝑑of PROPN cana-1301 250 8 𝐺 𝐺 PROPN cana-1301 250 9 is be AUX cana-1301 250 10 drawn draw VERB cana-1301 250 11 by by ADP cana-1301 250 12 dotted dotted ADJ cana-1301 250 13 line line NOUN cana-1301 250 14 . . PUNCT cana-1301 251 1 in in ADP cana-1301 251 2 m m NOUN cana-1301 251 3 - - NOUN cana-1301 251 4 bpfdgs bpfdgs ADJ cana-1301 251 5 , , PUNCT cana-1301 251 6 weak weak ADJ cana-1301 251 7 lines line NOUN cana-1301 251 8 in in ADP cana-1301 251 9 planar planar ADJ cana-1301 251 10 graphs graph NOUN cana-1301 251 11 are be AUX cana-1301 251 12 not not PART cana-1301 251 13 taken take VERB cana-1301 251 14 into into ADP cana-1301 251 15 an an DET cana-1301 251 16 account account NOUN cana-1301 251 17 for for ADP cana-1301 251 18 any any DET cana-1301 251 19 computations computation NOUN cana-1301 251 20 . . PUNCT cana-1301 252 1 communications communication NOUN cana-1301 252 2 on on ADP cana-1301 252 3 applied apply VERB cana-1301 252 4 nonlinear nonlinear ADJ cana-1301 252 5 analysis analysis NOUN cana-1301 252 6 issn issn NOUN cana-1301 252 7 : : PUNCT cana-1301 252 8 1074 1074 NUM cana-1301 252 9 - - PUNCT cana-1301 252 10 133x 133x NUM cana-1301 252 11 vol vol NOUN cana-1301 252 12 31 31 NUM cana-1301 252 13 no no NOUN cana-1301 252 14 . . PUNCT cana-1301 253 1 7s 7 NOUN cana-1301 253 2 ( ( PUNCT cana-1301 253 3 2024 2024 NUM cana-1301 253 4 ) ) PUNCT cana-1301 253 5 273 273 NUM cana-1301 253 6 https://internationalpubls.com https://internationalpubls.com X cana-1301 253 7 theorem theorem VERB cana-1301 253 8 4.1 4.1 NUM cana-1301 253 9 : : PUNCT cana-1301 253 10 let let VERB cana-1301 253 11 𝐺 𝐺 PROPN cana-1301 253 12 be be AUX cana-1301 253 13 an an DET cana-1301 253 14 m m NOUN cana-1301 253 15 - - NOUN cana-1301 253 16 bpfpg bpfpg NOUN cana-1301 253 17 whose whose DET cana-1301 253 18 number number NOUN cana-1301 253 19 of of ADP cana-1301 253 20 nodes node NOUN cana-1301 253 21 , , PUNCT cana-1301 253 22 the the DET cana-1301 253 23 number number NOUN cana-1301 253 24 of of ADP cana-1301 253 25 m m NOUN cana-1301 253 26 - - PUNCT cana-1301 253 27 bpf bpf NOUN cana-1301 253 28 lines line NOUN cana-1301 253 29 and and CCONJ cana-1301 253 30 number number NOUN cana-1301 253 31 of of ADP cana-1301 253 32 strong strong ADJ cana-1301 253 33 faces face NOUN cana-1301 253 34 are be AUX cana-1301 253 35 denoted denote VERB cana-1301 253 36 by by ADP cana-1301 253 37 π‘₯ π‘₯ PROPN cana-1301 253 38 , , PUNCT cana-1301 253 39 𝑦 𝑦 NOUN cana-1301 253 40 , , PUNCT cana-1301 253 41 𝑧 𝑧 X cana-1301 253 42 respectively respectively ADV cana-1301 253 43 . . PUNCT cana-1301 254 1 let let VERB cana-1301 254 2 𝐺𝑑be 𝐺𝑑be PROPN cana-1301 254 3 the the DET cana-1301 254 4 m m NOUN cana-1301 254 5 - - PUNCT cana-1301 254 6 bpfdg bpfdg NOUN cana-1301 254 7 of of ADP cana-1301 254 8 𝐺. 𝐺. NOUN cana-1301 254 9 then then ADV cana-1301 254 10 : : PUNCT cana-1301 255 1 i. i. PROPN cana-1301 255 2 no no PROPN cana-1301 255 3 . . PUNCT cana-1301 256 1 of of ADP cana-1301 256 2 nodes node NOUN cana-1301 256 3 of of ADP cana-1301 256 4 𝐺𝑑= 𝐺𝑑= PROPN cana-1301 256 5 π‘₯ π‘₯ PROPN cana-1301 256 6 , , PUNCT cana-1301 256 7 ii.no ii.no ADJ cana-1301 256 8 . . PUNCT cana-1301 256 9 of of ADP cana-1301 256 10 lines line NOUN cana-1301 256 11 of of ADP cana-1301 256 12 𝐺𝑑= 𝐺𝑑= PROPN cana-1301 256 13 𝑦 𝑦 PROPN cana-1301 256 14 , , PUNCT cana-1301 256 15 iii.no iii.no NUM cana-1301 256 16 . . NOUN cana-1301 256 17 of of ADP cana-1301 256 18 m m PROPN cana-1301 256 19 - - PUNCT cana-1301 256 20 bpf bpf NOUN cana-1301 256 21 faces face NOUN cana-1301 256 22 of of ADP cana-1301 256 23 𝐺𝑑= 𝐺𝑑= PROPN cana-1301 256 24 𝑧. 𝑧. NOUN cana-1301 256 25 proof proof NOUN cana-1301 256 26 : : PUNCT cana-1301 256 27 the the DET cana-1301 256 28 definition definition NOUN cana-1301 256 29 of of ADP cana-1301 256 30 m m PROPN cana-1301 256 31 - - PUNCT cana-1301 256 32 bpfdg bpfdg NOUN cana-1301 256 33 provides provide VERB cana-1301 256 34 evidence evidence NOUN cana-1301 256 35 for for ADP cana-1301 256 36 ( ( PUNCT cana-1301 256 37 i i NOUN cana-1301 256 38 ) ) PUNCT cana-1301 256 39 ( ( PUNCT cana-1301 256 40 ii ii NOUN cana-1301 256 41 ) ) PUNCT cana-1301 256 42 , , PUNCT cana-1301 256 43 and and CCONJ cana-1301 256 44 ( ( PUNCT cana-1301 256 45 iii iii NOUN cana-1301 256 46 ) ) PUNCT cana-1301 256 47 . . PUNCT cana-1301 257 1 theorem theorem VERB cana-1301 257 2 4.2 4.2 NUM cana-1301 257 3 : : PUNCT cana-1301 257 4 let let VERB cana-1301 257 5 𝐺 𝐺 PRON cana-1301 257 6 be be AUX cana-1301 257 7 a a DET cana-1301 257 8 strong strong ADJ cana-1301 257 9 m m NOUN cana-1301 257 10 - - NOUN cana-1301 257 11 bpfpg bpfpg NOUN cana-1301 257 12 without without ADP cana-1301 257 13 weak weak ADJ cana-1301 257 14 lines line NOUN cana-1301 257 15 and and CCONJ cana-1301 257 16 the the DET cana-1301 257 17 m m NOUN cana-1301 257 18 - - PUNCT cana-1301 257 19 bpfdg bpfdg NOUN cana-1301 257 20 of of ADP cana-1301 257 21 𝐺 𝐺 PROPN cana-1301 257 22 be be AUX cana-1301 257 23 𝐺𝑑. 𝐺𝑑. PROPN cana-1301 257 24 then then ADV cana-1301 257 25 the the DET cana-1301 257 26 positive positive ADJ cana-1301 257 27 relationship relationship NOUN cana-1301 257 28 and and CCONJ cana-1301 257 29 negative negative ADJ cana-1301 257 30 relationship relationship NOUN cana-1301 257 31 values value NOUN cana-1301 257 32 of of ADP cana-1301 257 33 m m NOUN cana-1301 257 34 - - PUNCT cana-1301 257 35 bpf bpf NOUN cana-1301 257 36 lines line NOUN cana-1301 257 37 of of ADP cana-1301 257 38 𝐺𝑑are 𝐺𝑑are NOUN cana-1301 257 39 equal equal ADJ cana-1301 257 40 to to ADP cana-1301 257 41 positive positive ADJ cana-1301 257 42 relationship relationship NOUN cana-1301 257 43 and and CCONJ cana-1301 257 44 negative negative ADJ cana-1301 257 45 relationship relationship NOUN cana-1301 257 46 values value NOUN cana-1301 257 47 of of ADP cana-1301 257 48 m m NOUN cana-1301 257 49 - - PUNCT cana-1301 257 50 bpf bpf NOUN cana-1301 257 51 lines line NOUN cana-1301 257 52 of of ADP cana-1301 257 53 𝐺. 𝐺. NOUN cana-1301 257 54 proof proof NOUN cana-1301 257 55 : : PUNCT cana-1301 257 56 the the DET cana-1301 257 57 dual dual ADJ cana-1301 257 58 graph graph NOUN cana-1301 258 1 𝐺𝑑 𝐺𝑑 ADV cana-1301 258 2 of of ADP cana-1301 258 3 𝐺 𝐺 PROPN cana-1301 258 4 is be AUX cana-1301 258 5 a a DET cana-1301 258 6 strong strong ADJ cana-1301 258 7 m m NOUN cana-1301 258 8 - - NOUN cana-1301 258 9 bpfpg bpfpg NOUN cana-1301 258 10 as as SCONJ cana-1301 258 11 there there PRON cana-1301 258 12 is be VERB cana-1301 258 13 no no DET cana-1301 258 14 point point NOUN cana-1301 258 15 of of ADP cana-1301 258 16 junction junction NOUN cana-1301 258 17 between between ADP cana-1301 258 18 any any DET cana-1301 258 19 lines line NOUN cana-1301 258 20 . . PUNCT cana-1301 259 1 let let VERB cana-1301 259 2 { { PUNCT cana-1301 259 3 𝛽1 𝛽1 NOUN cana-1301 259 4 , , PUNCT cana-1301 259 5 𝛽2 𝛽2 NOUN cana-1301 259 6 , , PUNCT cana-1301 259 7 β‹― β‹― PROPN cana-1301 259 8 , , PUNCT cana-1301 259 9 π›½π‘Ÿ}be π›½π‘Ÿ}be PROPN cana-1301 259 10 the the DET cana-1301 259 11 set set NOUN cana-1301 259 12 of of ADP cana-1301 259 13 strong strong ADJ cana-1301 259 14 faces face NOUN cana-1301 259 15 of of ADP cana-1301 259 16 𝐺. 𝐺. NOUN cana-1301 259 17 by by ADP cana-1301 259 18 the the DET cana-1301 259 19 definition definition NOUN cana-1301 259 20 of of ADP cana-1301 259 21 m m PROPN cana-1301 259 22 - - PUNCT cana-1301 259 23 bpfdg bpfdg NOUN cana-1301 259 24 we we PRON cana-1301 259 25 know know VERB cana-1301 259 26 that that SCONJ cana-1301 259 27 π‘ƒβ„Žoψ π‘ƒβ„Žoψ NOUN cana-1301 259 28 𝑅𝑑 𝑅𝑑 ADP cana-1301 259 29 + + NOUN cana-1301 259 30 r r NOUN cana-1301 259 31 ( ( PUNCT cana-1301 259 32 π‘žπ‘–π‘žπ‘— π‘žπ‘–π‘žπ‘— PROPN cana-1301 259 33 ) ) PUNCT cana-1301 259 34 = = PUNCT cana-1301 260 1 π‘ƒβ„ŽoΟˆπ‘… π‘ƒβ„ŽoΟˆπ‘… PROPN cana-1301 261 1 + + NOUN cana-1301 261 2 r r NOUN cana-1301 261 3 ( ( PUNCT cana-1301 261 4 πœ„πœ… πœ„πœ… NOUN cana-1301 261 5 ) ) PUNCT cana-1301 261 6 , , PUNCT cana-1301 261 7 π‘ƒβ„Žoψ π‘ƒβ„Žoψ PROPN cana-1301 261 8 𝑅𝑑 𝑅𝑑 PROPN cana-1301 261 9 βˆ’r βˆ’r PROPN cana-1301 261 10 ( ( PUNCT cana-1301 261 11 π‘žπ‘–π‘žπ‘— π‘žπ‘–π‘žπ‘— PROPN cana-1301 261 12 ) ) PUNCT cana-1301 261 13 = = PUNCT cana-1301 262 1 π‘ƒβ„ŽoΟˆπ‘… π‘ƒβ„ŽoΟˆπ‘… PROPN cana-1301 262 2 βˆ’r βˆ’r PROPN cana-1301 262 3 ( ( PUNCT cana-1301 262 4 πœ„πœ…)where πœ„πœ…)where NOUN cana-1301 262 5 πœ„πœ…is πœ„πœ…is NOUN cana-1301 262 6 a a DET cana-1301 262 7 common common ADJ cana-1301 262 8 line line NOUN cana-1301 262 9 between between ADP cana-1301 262 10 2 2 NUM cana-1301 262 11 strong strong ADJ cana-1301 262 12 m m NOUN cana-1301 262 13 - - PUNCT cana-1301 262 14 bpf bpf NOUN cana-1301 262 15 faces face NOUN cana-1301 262 16 𝛽𝑖and 𝛽𝑖and VERB cana-1301 262 17 𝛽𝑗 𝛽𝑗 NOUN cana-1301 262 18 and and CCONJ cana-1301 262 19 π‘Ÿ π‘Ÿ NOUN cana-1301 262 20 = = SYM cana-1301 262 21 1,2 1,2 NUM cana-1301 262 22 , , PUNCT cana-1301 262 23 β‹― β‹― PROPN cana-1301 262 24 , , PUNCT cana-1301 262 25 𝑠 𝑠 PROPN cana-1301 262 26 , , PUNCT cana-1301 262 27 where where SCONJ cana-1301 262 28 𝑠 𝑠 ADP cana-1301 262 29 being be AUX cana-1301 262 30 the the DET cana-1301 262 31 no no NOUN cana-1301 262 32 . . PUNCT cana-1301 262 33 of of ADP cana-1301 262 34 common common ADJ cana-1301 262 35 lines line NOUN cana-1301 262 36 in in ADP cana-1301 262 37 the the DET cana-1301 262 38 boundary boundary NOUN cana-1301 262 39 between between ADP cana-1301 262 40 𝛽𝑖and 𝛽𝑖and NOUN cana-1301 262 41 𝛽𝑗 𝛽𝑗 NOUN cana-1301 262 42 . . PUNCT cana-1301 263 1 the the DET cana-1301 263 2 no no NOUN cana-1301 263 3 . . PROPN cana-1301 263 4 of of ADP cana-1301 263 5 m m PROPN cana-1301 263 6 - - PUNCT cana-1301 263 7 bpf bpf NOUN cana-1301 263 8 lines line NOUN cana-1301 263 9 of of ADP cana-1301 263 10 two two NUM cana-1301 263 11 graphs graph NOUN cana-1301 263 12 𝐺 𝐺 PROPN cana-1301 263 13 and and CCONJ cana-1301 263 14 𝐺𝑑are 𝐺𝑑are PROPN cana-1301 263 15 same same ADJ cana-1301 263 16 as as SCONJ cana-1301 263 17 𝐺 𝐺 PROPN cana-1301 263 18 has have VERB cana-1301 263 19 no no DET cana-1301 263 20 weak weak ADJ cana-1301 263 21 lines line NOUN cana-1301 263 22 . . PUNCT cana-1301 264 1 so so ADV cana-1301 264 2 , , PUNCT cana-1301 264 3 for for ADP cana-1301 264 4 each each DET cana-1301 264 5 m m NOUN cana-1301 264 6 - - PUNCT cana-1301 264 7 bpf bpf NOUN cana-1301 264 8 line line NOUN cana-1301 264 9 of of ADP cana-1301 264 10 𝐺 𝐺 PROPN cana-1301 264 11 there there PRON cana-1301 264 12 is be VERB cana-1301 264 13 an an DET cana-1301 264 14 m m NOUN cana-1301 264 15 - - PUNCT cana-1301 264 16 bpf bpf NOUN cana-1301 264 17 line line NOUN cana-1301 264 18 in in ADP cana-1301 264 19 𝐺𝑑 𝐺𝑑 PROPN cana-1301 264 20 with with ADP cana-1301 264 21 the the DET cana-1301 264 22 same same ADJ cana-1301 264 23 relationship relationship NOUN cana-1301 264 24 value value NOUN cana-1301 264 25 . . PUNCT cana-1301 265 1 we we PRON cana-1301 265 2 're be AUX cana-1301 265 3 now now ADV cana-1301 265 4 studying study VERB cana-1301 265 5 at at ADP cana-1301 265 6 isomorphism isomorphism NOUN cana-1301 265 7 between between ADP cana-1301 265 8 m m NOUN cana-1301 265 9 - - NOUN cana-1301 265 10 bpfpgs bpfpgs NOUN cana-1301 265 11 . . PUNCT cana-1301 266 1 definition definition NOUN cana-1301 266 2 4.4 4.4 NUM cana-1301 266 3 : : PUNCT cana-1301 266 4 let let VERB cana-1301 266 5 πΊπ‘Ž πΊπ‘Ž VERB cana-1301 266 6 = = SYM cana-1301 266 7 ( ( PUNCT cana-1301 266 8 π‘‰π‘Ž π‘‰π‘Ž PROPN cana-1301 266 9 , , PUNCT cana-1301 266 10 π‘„π‘Ž π‘„π‘Ž PROPN cana-1301 266 11 , , PUNCT cana-1301 266 12 π‘…π‘Ž)and π‘…π‘Ž)and CCONJ cana-1301 267 1 𝐺𝑏 𝐺𝑏 PROPN cana-1301 267 2 = = SYM cana-1301 267 3 ( ( PUNCT cana-1301 267 4 𝑉𝑏 𝑉𝑏 PROPN cana-1301 267 5 , , PUNCT cana-1301 267 6 𝑄𝑏 𝑄𝑏 PROPN cana-1301 267 7 , , PUNCT cana-1301 267 8 𝑅𝑏)be 𝑅𝑏)be NOUN cana-1301 267 9 two two NUM cana-1301 267 10 m m NOUN cana-1301 267 11 - - NOUN cana-1301 267 12 bpfpgs bpfpgs NOUN cana-1301 267 13 of of ADP cana-1301 267 14 the the DET cana-1301 267 15 graphs graph NOUN cana-1301 267 16 πΊπ‘Ž πΊπ‘Ž NOUN cana-1301 267 17 βˆ— βˆ— NOUN cana-1301 267 18 = = SYM cana-1301 267 19 ( ( PUNCT cana-1301 267 20 π‘‰π‘Ž π‘‰π‘Ž PROPN cana-1301 267 21 , , PUNCT cana-1301 267 22 πΈπ‘Ž πΈπ‘Ž PROPN cana-1301 267 23 ) ) PUNCT cana-1301 267 24 and and CCONJ cana-1301 267 25 𝐺𝑏 𝐺𝑏 PROPN cana-1301 267 26 βˆ— βˆ— NOUN cana-1301 267 27 = = PUNCT cana-1301 267 28 ( ( PUNCT cana-1301 267 29 𝑉𝑏 𝑉𝑏 PROPN cana-1301 267 30 , , PUNCT cana-1301 267 31 𝐸𝑏 𝐸𝑏 PROPN cana-1301 267 32 ) ) PUNCT cana-1301 267 33 respectively respectively ADV cana-1301 267 34 . . PUNCT cana-1301 268 1 ( ( PUNCT cana-1301 268 2 i i NOUN cana-1301 268 3 ) ) PUNCT cana-1301 268 4 an an DET cana-1301 268 5 isomorphism isomorphism NOUN cana-1301 268 6 between between ADP cana-1301 268 7 πΊπ‘Ž πΊπ‘Ž PROPN cana-1301 268 8 and and CCONJ cana-1301 268 9 𝐺𝑏 𝐺𝑏 PROPN cana-1301 268 10 is be AUX cana-1301 268 11 a a DET cana-1301 268 12 bijective bijective ADJ cana-1301 268 13 transformation transformation NOUN cana-1301 268 14 ΞΆ ΞΆ NOUN cana-1301 268 15 : : PUNCT cana-1301 268 16 π‘‰π‘Ž π‘‰π‘Ž PROPN cana-1301 268 17 β†’ β†’ SYM cana-1301 268 18 𝑉𝑏such 𝑉𝑏such PROPN cana-1301 268 19 that that SCONJ cana-1301 268 20 for for ADP cana-1301 268 21 each each DET cana-1301 268 22 β„Ž β„Ž PROPN cana-1301 268 23 = = SYM cana-1301 268 24 1 1 NUM cana-1301 268 25 , , PUNCT cana-1301 268 26 2 2 NUM cana-1301 268 27 , , PUNCT cana-1301 268 28 β‹― β‹― PROPN cana-1301 268 29 , , PUNCT cana-1301 268 30 π‘š π‘š X cana-1301 268 31 ( ( PUNCT cana-1301 268 32 a a X cana-1301 268 33 ) ) PUNCT cana-1301 268 34 π‘ƒβ„Žoψqa π‘ƒβ„Žoψqa PROPN cana-1301 268 35 + + CCONJ cana-1301 268 36 ( ( PUNCT cana-1301 268 37 πœ„π‘Ž πœ„π‘Ž NOUN cana-1301 268 38 ) ) PUNCT cana-1301 268 39 = = SYM cana-1301 269 1 π‘ƒβ„Žoψqb π‘ƒβ„Žoψqb PROPN cana-1301 269 2 + + CCONJ cana-1301 269 3 ( ( PUNCT cana-1301 269 4 ΞΆ(πœ„π‘Ž ΞΆ(πœ„π‘Ž NOUN cana-1301 269 5 ) ) PUNCT cana-1301 269 6 ) ) PUNCT cana-1301 269 7 , , PUNCT cana-1301 269 8 π‘ƒβ„Žoψqa π‘ƒβ„Žoψqa PROPN cana-1301 269 9 βˆ’ βˆ’ PROPN cana-1301 269 10 ( ( PUNCT cana-1301 269 11 πœ„π‘Ž πœ„π‘Ž NOUN cana-1301 269 12 ) ) PUNCT cana-1301 269 13 = = SYM cana-1301 269 14 π‘ƒβ„Žoψqb π‘ƒβ„Žoψqb PROPN cana-1301 269 15 βˆ’ βˆ’ PROPN cana-1301 269 16 ( ( PUNCT cana-1301 269 17 ΞΆ(πœ„π‘Ž))for ΞΆ(πœ„π‘Ž))for ADP cana-1301 269 18 all all PRON cana-1301 269 19 πœ„π‘Ž πœ„π‘Ž NOUN cana-1301 269 20 ∈ ∈ NOUN cana-1301 270 1 π‘‰π‘Ž π‘‰π‘Ž PROPN cana-1301 270 2 ( ( PUNCT cana-1301 270 3 b b NOUN cana-1301 270 4 ) ) PUNCT cana-1301 270 5 π‘ƒβ„Žoψra π‘ƒβ„Žoψra PROPN cana-1301 270 6 + + CCONJ cana-1301 270 7 ( ( PUNCT cana-1301 270 8 πœ„π‘Žπœ…π‘Ž πœ„π‘Žπœ…π‘Ž PROPN cana-1301 270 9 ) ) PUNCT cana-1301 270 10 = = PUNCT cana-1301 271 1 π‘ƒβ„Žoψrb π‘ƒβ„Žoψrb NOUN cana-1301 271 2 + + CCONJ cana-1301 271 3 ( ( PUNCT cana-1301 271 4 ΞΆ(πœ„π‘Ž)ΞΆ(πœ…π‘Ž ΞΆ(πœ„π‘Ž)ΞΆ(πœ…π‘Ž NOUN cana-1301 271 5 ) ) PUNCT cana-1301 271 6 ) ) PUNCT cana-1301 271 7 , , PUNCT cana-1301 271 8 π‘ƒβ„Žoψra π‘ƒβ„Žoψra PROPN cana-1301 271 9 βˆ’ βˆ’ PROPN cana-1301 271 10 ( ( PUNCT cana-1301 271 11 πœ„π‘Žπœ…π‘Ž πœ„π‘Žπœ…π‘Ž PROPN cana-1301 271 12 ) ) PUNCT cana-1301 271 13 = = PUNCT cana-1301 272 1 π‘ƒβ„Žoψrb π‘ƒβ„Žoψrb PROPN cana-1301 272 2 βˆ’ βˆ’ PROPN cana-1301 272 3 ( ( PUNCT cana-1301 272 4 ΞΆ(πœ„π‘Ž)ΞΆ(πœ…π‘Ž ΞΆ(πœ„π‘Ž)ΞΆ(πœ…π‘Ž NOUN cana-1301 272 5 ) ) PUNCT cana-1301 272 6 ) ) PUNCT cana-1301 272 7 for for ADP cana-1301 272 8 all all DET cana-1301 272 9 πœ„π‘Žπœ…π‘Ž πœ„π‘Žπœ…π‘Ž NOUN cana-1301 272 10 ∈ ∈ PROPN cana-1301 273 1 π‘‰π‘Ž π‘‰π‘Ž PROPN cana-1301 273 2 2 2 PROPN cana-1301 273 3 ⃑ ⃑ NOUN cana-1301 273 4 . . PUNCT cana-1301 274 1 ( ( PUNCT cana-1301 274 2 ii ii NOUN cana-1301 274 3 ) ) PUNCT cana-1301 274 4 a a DET cana-1301 274 5 weak weak ADJ cana-1301 274 6 isomorphism isomorphism NOUN cana-1301 274 7 between between ADP cana-1301 274 8 πΊπ‘Ž πΊπ‘Ž NOUN cana-1301 274 9 and and CCONJ cana-1301 274 10 𝐺𝑏is 𝐺𝑏is PROPN cana-1301 274 11 a a DET cana-1301 274 12 bijective bijective ADJ cana-1301 274 13 transformation transformation NOUN cana-1301 274 14 ΞΆ ΞΆ NOUN cana-1301 274 15 : : PUNCT cana-1301 274 16 π‘‰π‘Ž π‘‰π‘Ž PROPN cana-1301 274 17 β†’ β†’ SYM cana-1301 274 18 𝑉𝑏 𝑉𝑏 PROPN cana-1301 274 19 such such ADJ cana-1301 274 20 that that PRON cana-1301 274 21 for for ADP cana-1301 274 22 each each DET cana-1301 274 23 β„Ž β„Ž PROPN cana-1301 274 24 = = SYM cana-1301 274 25 1 1 NUM cana-1301 274 26 , , PUNCT cana-1301 274 27 2 2 NUM cana-1301 274 28 , , PUNCT cana-1301 274 29 β‹― β‹― PROPN cana-1301 274 30 , , PUNCT cana-1301 274 31 π‘š π‘š X cana-1301 274 32 ( ( PUNCT cana-1301 274 33 a a PRON cana-1301 274 34 ) ) PUNCT cana-1301 274 35 ΞΆ ΞΆ NOUN cana-1301 274 36 is be AUX cana-1301 274 37 a a DET cana-1301 274 38 homomorphism homomorphism NOUN cana-1301 274 39 ( ( PUNCT cana-1301 274 40 b b NOUN cana-1301 274 41 ) ) PUNCT cana-1301 274 42 π‘ƒβ„Žoψqa π‘ƒβ„Žoψqa PROPN cana-1301 274 43 + + CCONJ cana-1301 274 44 ( ( PUNCT cana-1301 274 45 πœ„π‘Ž πœ„π‘Ž NOUN cana-1301 274 46 ) ) PUNCT cana-1301 275 1 = = SYM cana-1301 275 2 π‘ƒβ„Žoψqb π‘ƒβ„Žoψqb PROPN cana-1301 275 3 + + CCONJ cana-1301 275 4 ( ( PUNCT cana-1301 275 5 ΞΆ(πœ„π‘Ž ΞΆ(πœ„π‘Ž NOUN cana-1301 275 6 ) ) PUNCT cana-1301 275 7 ) ) PUNCT cana-1301 275 8 , , PUNCT cana-1301 275 9 π‘ƒβ„Žoψqa π‘ƒβ„Žoψqa PROPN cana-1301 275 10 βˆ’ βˆ’ PROPN cana-1301 275 11 ( ( PUNCT cana-1301 275 12 πœ„π‘Ž πœ„π‘Ž NOUN cana-1301 275 13 ) ) PUNCT cana-1301 275 14 = = SYM cana-1301 275 15 π‘ƒβ„Žoψqb π‘ƒβ„Žoψqb PROPN cana-1301 275 16 βˆ’ βˆ’ PROPN cana-1301 275 17 ( ( PUNCT cana-1301 275 18 ΞΆ(πœ„π‘Ž))for ΞΆ(πœ„π‘Ž))for ADP cana-1301 275 19 all all PRON cana-1301 275 20 πœ„π‘Ž πœ„π‘Ž ADP cana-1301 275 21 ∈ ∈ PROPN cana-1301 275 22 π‘‰π‘Ž. π‘‰π‘Ž. PROPN cana-1301 275 23 ( ( PUNCT cana-1301 275 24 iii iii NOUN cana-1301 275 25 ) ) PUNCT cana-1301 275 26 a a DET cana-1301 275 27 co co ADJ cana-1301 275 28 - - ADJ cana-1301 275 29 weak weak ADJ cana-1301 275 30 isomorphism isomorphism NOUN cana-1301 275 31 betweenπΊπ‘Ž betweenπΊπ‘Ž NOUN cana-1301 275 32 and and CCONJ cana-1301 275 33 𝐺𝑏 𝐺𝑏 PROPN cana-1301 275 34 is be AUX cana-1301 275 35 a a DET cana-1301 275 36 bijective bijective ADJ cana-1301 275 37 transformation transformation NOUN cana-1301 275 38 ΞΆ ΞΆ NOUN cana-1301 275 39 : : PUNCT cana-1301 275 40 π‘‰π‘Ž π‘‰π‘Ž PROPN cana-1301 275 41 β†’ β†’ SYM cana-1301 275 42 𝑉𝑏such 𝑉𝑏such PROPN cana-1301 275 43 that that SCONJ cana-1301 275 44 for for ADP cana-1301 275 45 each each DET cana-1301 275 46 β„Ž β„Ž PROPN cana-1301 275 47 = = SYM cana-1301 275 48 1 1 NUM cana-1301 275 49 , , PUNCT cana-1301 275 50 2 2 NUM cana-1301 275 51 , , PUNCT cana-1301 275 52 β‹― β‹― PROPN cana-1301 275 53 , , PUNCT cana-1301 275 54 π‘š π‘š X cana-1301 275 55 ( ( PUNCT cana-1301 275 56 a a PRON cana-1301 275 57 ) ) PUNCT cana-1301 275 58 ΞΆ ΞΆ NOUN cana-1301 275 59 is be AUX cana-1301 275 60 a a DET cana-1301 275 61 homomorphism homomorphism NOUN cana-1301 275 62 ( ( PUNCT cana-1301 275 63 b b NOUN cana-1301 275 64 ) ) PUNCT cana-1301 275 65 π‘ƒβ„Žoψra π‘ƒβ„Žoψra PROPN cana-1301 275 66 + + CCONJ cana-1301 275 67 ( ( PUNCT cana-1301 275 68 πœ„π‘Žπœ…π‘Ž πœ„π‘Žπœ…π‘Ž PROPN cana-1301 275 69 ) ) PUNCT cana-1301 275 70 = = PUNCT cana-1301 276 1 π‘ƒβ„Žoψrb π‘ƒβ„Žoψrb NOUN cana-1301 276 2 + + CCONJ cana-1301 276 3 ( ( PUNCT cana-1301 276 4 ΞΆ(πœ„π‘Ž)ΞΆ(πœ…π‘Ž ΞΆ(πœ„π‘Ž)ΞΆ(πœ…π‘Ž NOUN cana-1301 276 5 ) ) PUNCT cana-1301 276 6 ) ) PUNCT cana-1301 276 7 , , PUNCT cana-1301 276 8 π‘ƒβ„Žoψra π‘ƒβ„Žoψra PROPN cana-1301 276 9 βˆ’ βˆ’ PROPN cana-1301 276 10 ( ( PUNCT cana-1301 276 11 πœ„π‘Žπœ…π‘Ž πœ„π‘Žπœ…π‘Ž PROPN cana-1301 276 12 ) ) PUNCT cana-1301 276 13 = = PUNCT cana-1301 277 1 π‘ƒβ„Žoψrb π‘ƒβ„Žoψrb PROPN cana-1301 277 2 βˆ’ βˆ’ PROPN cana-1301 277 3 ( ( PUNCT cana-1301 277 4 ΞΆ(πœ„π‘Ž)ΞΆ(πœ…π‘Ž ΞΆ(πœ„π‘Ž)ΞΆ(πœ…π‘Ž NOUN cana-1301 277 5 ) ) PUNCT cana-1301 277 6 ) ) PUNCT cana-1301 277 7 for for ADP cana-1301 277 8 all all DET cana-1301 277 9 πœ„π‘Žπœ…π‘Ž πœ„π‘Žπœ…π‘Ž NOUN cana-1301 277 10 ∈ ∈ PROPN cana-1301 278 1 π‘‰π‘Ž π‘‰π‘Ž PROPN cana-1301 278 2 2 2 PROPN cana-1301 278 3 ⃑ ⃑ NOUN cana-1301 278 4 . . PUNCT cana-1301 279 1 that that SCONJ cana-1301 279 2 isomorphism isomorphism NOUN cana-1301 279 3 between between ADP cana-1301 279 4 two two NUM cana-1301 279 5 m m NOUN cana-1301 279 6 - - PUNCT cana-1301 279 7 bpfpgs bpfpgs NOUN cana-1301 279 8 is be AUX cana-1301 279 9 an an DET cana-1301 279 10 equivalence equivalence NOUN cana-1301 279 11 relation relation NOUN cana-1301 279 12 is be AUX cana-1301 279 13 simple simple ADJ cana-1301 279 14 to to PART cana-1301 279 15 confirm confirm VERB cana-1301 279 16 . . PUNCT cana-1301 280 1 however however ADV cana-1301 280 2 , , PUNCT cana-1301 280 3 if if SCONJ cana-1301 280 4 two two NUM cana-1301 280 5 m m NOUN cana-1301 280 6 - - PUNCT cana-1301 280 7 bpfgs bpfgs PROPN cana-1301 280 8 are be AUX cana-1301 280 9 isomorphic isomorphic ADJ cana-1301 280 10 so so SCONJ cana-1301 280 11 that that SCONJ cana-1301 280 12 one one NOUN cana-1301 280 13 is be AUX cana-1301 280 14 m m NOUN cana-1301 280 15 - - NOUN cana-1301 280 16 bpfpg bpfpg NOUN cana-1301 280 17 , , PUNCT cana-1301 280 18 the the DET cana-1301 280 19 other other ADJ cana-1301 280 20 will will AUX cana-1301 280 21 be be AUX cana-1301 280 22 m m NOUN cana-1301 280 23 - - PUNCT cana-1301 280 24 bpfg bpfg ADJ cana-1301 280 25 . . PUNCT cana-1301 281 1 as as ADP cana-1301 281 2 evidence evidence NOUN cana-1301 281 3 , , PUNCT cana-1301 281 4 consider consider VERB cana-1301 281 5 the the DET cana-1301 281 6 following following NOUN cana-1301 281 7 . . PUNCT cana-1301 282 1 communications communication NOUN cana-1301 282 2 on on ADP cana-1301 282 3 applied apply VERB cana-1301 282 4 nonlinear nonlinear ADJ cana-1301 282 5 analysis analysis NOUN cana-1301 282 6 issn issn NOUN cana-1301 282 7 : : PUNCT cana-1301 282 8 1074 1074 NUM cana-1301 282 9 - - PUNCT cana-1301 282 10 133x 133x NUM cana-1301 282 11 vol vol NOUN cana-1301 282 12 31 31 NUM cana-1301 282 13 no no NOUN cana-1301 282 14 . . PUNCT cana-1301 283 1 7s 7 NOUN cana-1301 283 2 ( ( PUNCT cana-1301 283 3 2024 2024 NUM cana-1301 283 4 ) ) PUNCT cana-1301 283 5 274 274 NUM cana-1301 283 6 https://internationalpubls.com https://internationalpubls.com X cana-1301 283 7 theorem theorem VERB cana-1301 283 8 4.3 4.3 NUM cana-1301 283 9 : : PUNCT cana-1301 283 10 let let VERB cana-1301 283 11 πΊπ‘Ž πΊπ‘Ž NOUN cana-1301 283 12 be be AUX cana-1301 283 13 an an DET cana-1301 283 14 m m NOUN cana-1301 283 15 - - NOUN cana-1301 283 16 bpfpg bpfpg NOUN cana-1301 283 17 and and CCONJ cana-1301 283 18 let let VERB cana-1301 283 19 𝐺𝑏 𝐺𝑏 PROPN cana-1301 283 20 be be AUX cana-1301 283 21 an an DET cana-1301 283 22 m m NOUN cana-1301 283 23 - - PUNCT cana-1301 283 24 bpfg bpfg ADJ cana-1301 283 25 . . PUNCT cana-1301 284 1 if if SCONJ cana-1301 284 2 there there PRON cana-1301 284 3 exists exist VERB cana-1301 284 4 an an DET cana-1301 284 5 isomorphismΞΆ isomorphismΞΆ NOUN cana-1301 284 6 : : PUNCT cana-1301 284 7 πΊπ‘Ž πΊπ‘Ž NOUN cana-1301 284 8 β†’ β†’ SYM cana-1301 284 9 𝐺𝑏 𝐺𝑏 PROPN cana-1301 284 10 , , PUNCT cana-1301 284 11 𝐺𝑏 𝐺𝑏 PROPN cana-1301 284 12 can can AUX cana-1301 284 13 be be AUX cana-1301 284 14 represented represent VERB cana-1301 284 15 as as ADP cana-1301 284 16 m m NOUN cana-1301 284 17 - - NOUN cana-1301 284 18 bpfpg bpfpg NOUN cana-1301 284 19 with with ADP cana-1301 284 20 the the DET cana-1301 284 21 same same ADJ cana-1301 284 22 planarity planarity NOUN cana-1301 284 23 value value NOUN cana-1301 284 24 of of ADP cana-1301 284 25 πΊπ‘Ž πΊπ‘Ž NOUN cana-1301 284 26 . . PUNCT cana-1301 285 1 proof proof NOUN cana-1301 285 2 : : PUNCT cana-1301 285 3 line line NOUN cana-1301 285 4 and and CCONJ cana-1301 285 5 node node ADJ cana-1301 285 6 weights weight NOUN cana-1301 285 7 are be AUX cana-1301 285 8 preserved preserve VERB cana-1301 285 9 through through ADP cana-1301 285 10 isomorphism isomorphism NOUN cana-1301 285 11 . . PUNCT cana-1301 286 1 additionally additionally ADV cana-1301 286 2 , , PUNCT cana-1301 286 3 isomorphic isomorphic ADJ cana-1301 286 4 mbpfgs mbpfgs NOUN cana-1301 286 5 maintain maintain VERB cana-1301 286 6 their their PRON cana-1301 286 7 order order NOUN cana-1301 286 8 and and CCONJ cana-1301 286 9 size size NOUN cana-1301 286 10 . . PUNCT cana-1301 287 1 𝐺𝑏 𝐺𝑏 PROPN cana-1301 287 2 will will AUX cana-1301 287 3 therefore therefore ADV cana-1301 287 4 have have VERB cana-1301 287 5 the the DET cana-1301 287 6 same same ADJ cana-1301 287 7 size size NOUN cana-1301 287 8 and and CCONJ cana-1301 287 9 order order NOUN cana-1301 287 10 as as ADP cana-1301 287 11 πΊπ‘Ž. πΊπ‘Ž. PROPN cana-1301 287 12 after after ADP cana-1301 287 13 then then ADV cana-1301 287 14 , , PUNCT cana-1301 287 15 𝐺𝑏 𝐺𝑏 PROPN cana-1301 287 16 can can AUX cana-1301 287 17 be be AUX cana-1301 287 18 represented represent VERB cana-1301 287 19 similarly similarly ADV cana-1301 287 20 to to ADP cana-1301 287 21 πΊπ‘Ž. πΊπ‘Ž. PROPN cana-1301 287 22 as as ADP cana-1301 287 23 a a DET cana-1301 287 24 result result NOUN cana-1301 287 25 , , PUNCT cana-1301 287 26 𝐺𝑏 𝐺𝑏 PROPN cana-1301 287 27 will will AUX cana-1301 287 28 have have VERB cana-1301 287 29 the the DET cana-1301 287 30 same same ADJ cana-1301 287 31 number number NOUN cana-1301 287 32 of of ADP cana-1301 287 33 junctions junction NOUN cana-1301 287 34 where where SCONJ cana-1301 287 35 lines line NOUN cana-1301 287 36 cross cross VERB cana-1301 287 37 and and CCONJ cana-1301 287 38 the the DET cana-1301 287 39 same same ADJ cana-1301 287 40 planarity planarity NOUN cana-1301 287 41 value value NOUN cana-1301 287 42 as as ADP cana-1301 287 43 πΊπ‘Ž. πΊπ‘Ž. PROPN cana-1301 287 44 accordingly accordingly ADV cana-1301 287 45 , , PUNCT cana-1301 287 46 𝐺𝑏may 𝐺𝑏may PROPN cana-1301 287 47 be be AUX cana-1301 287 48 represented represent VERB cana-1301 287 49 as as ADP cana-1301 287 50 m m NOUN cana-1301 287 51 - - NOUN cana-1301 287 52 bpfpg bpfpg NOUN cana-1301 287 53 with with ADP cana-1301 287 54 the the DET cana-1301 287 55 same same ADJ cana-1301 287 56 planarity planarity NOUN cana-1301 287 57 value value NOUN cana-1301 287 58 as as SCONJ cana-1301 287 59 πΊπ‘Ž. πΊπ‘Ž. PROPN cana-1301 287 60 theorem theorem VERB cana-1301 287 61 4.4 4.4 NUM cana-1301 287 62 : : PUNCT cana-1301 287 63 let let VERB cana-1301 287 64 𝐺𝑏 𝐺𝑏 PROPN cana-1301 287 65 be be AUX cana-1301 287 66 the the DET cana-1301 287 67 m m NOUN cana-1301 287 68 - - PUNCT cana-1301 287 69 bpfdg bpfdg NOUN cana-1301 287 70 of of ADP cana-1301 287 71 m m PROPN cana-1301 287 72 - - PUNCT cana-1301 287 73 bpfdg bpfdg NOUN cana-1301 287 74 of of ADP cana-1301 287 75 a a DET cana-1301 287 76 strong strong ADJ cana-1301 287 77 m m PROPN cana-1301 287 78 - - PUNCT cana-1301 287 79 bpfdg bpfdg PROPN cana-1301 287 80 𝐺 𝐺 PROPN cana-1301 287 81 without without ADP cana-1301 287 82 weak weak ADJ cana-1301 287 83 lines line NOUN cana-1301 287 84 . . PUNCT cana-1301 288 1 then then ADV cana-1301 288 2 there there PRON cana-1301 288 3 exists exist VERB cana-1301 288 4 a a DET cana-1301 288 5 co co ADJ cana-1301 288 6 - - ADJ cana-1301 288 7 weak weak ADJ cana-1301 288 8 isomorphism isomorphism NOUN cana-1301 288 9 connecting connect VERB cana-1301 288 10 𝐺 𝐺 NOUN cana-1301 288 11 and and CCONJ cana-1301 288 12 𝐺𝑏. 𝐺𝑏. PROPN cana-1301 288 13 proof proof NOUN cana-1301 288 14 : : PUNCT cana-1301 288 15 assume assume VERB cana-1301 288 16 that that SCONJ cana-1301 288 17 𝐺𝑏is 𝐺𝑏is PROPN cana-1301 288 18 an an DET cana-1301 288 19 m m NOUN cana-1301 288 20 - - PUNCT cana-1301 288 21 bpfdg bpfdg NOUN cana-1301 288 22 of of ADP cana-1301 288 23 πΊπ‘Žand πΊπ‘Žand PROPN cana-1301 288 24 that that DET cana-1301 288 25 πΊπ‘Žis πΊπ‘Žis PROPN cana-1301 288 26 an an DET cana-1301 288 27 m m NOUN cana-1301 288 28 - - PUNCT cana-1301 288 29 bpfdg bpfdg NOUN cana-1301 288 30 of of ADP cana-1301 288 31 𝐺. 𝐺. NOUN cana-1301 288 32 the the DET cana-1301 288 33 no no NOUN cana-1301 288 34 . . PUNCT cana-1301 288 35 of of ADP cana-1301 288 36 nodes node NOUN cana-1301 288 37 of of ADP cana-1301 288 38 𝐺𝑏 𝐺𝑏 PROPN cana-1301 288 39 are be AUX cana-1301 288 40 equal equal ADJ cana-1301 288 41 to to ADP cana-1301 288 42 the the DET cana-1301 288 43 strong strong ADJ cana-1301 288 44 m m NOUN cana-1301 288 45 - - PUNCT cana-1301 288 46 bpf bpf NOUN cana-1301 288 47 faces face NOUN cana-1301 288 48 of of ADP cana-1301 288 49 πΊπ‘Žand πΊπ‘Žand NOUN cana-1301 288 50 the the DET cana-1301 288 51 no no NOUN cana-1301 288 52 . . PUNCT cana-1301 288 53 of of ADP cana-1301 288 54 strong strong ADJ cana-1301 288 55 m m NOUN cana-1301 288 56 - - PUNCT cana-1301 288 57 bpf bpf NOUN cana-1301 288 58 faces face NOUN cana-1301 288 59 in in ADP cana-1301 288 60 πΊπ‘Žis πΊπ‘Žis PROPN cana-1301 288 61 equal equal ADJ cana-1301 288 62 to to ADP cana-1301 288 63 the the DET cana-1301 288 64 no no NOUN cana-1301 288 65 . . PUNCT cana-1301 288 66 of of ADP cana-1301 288 67 nodes node NOUN cana-1301 288 68 in in ADP cana-1301 288 69 𝐺. 𝐺. NOUN cana-1301 288 70 the the DET cana-1301 288 71 no no NOUN cana-1301 288 72 . . PUNCT cana-1301 288 73 of of ADP cana-1301 288 74 nodes node NOUN cana-1301 288 75 in in ADP cana-1301 288 76 𝐺𝑏 𝐺𝑏 PROPN cana-1301 288 77 and and CCONJ cana-1301 288 78 𝐺 𝐺 PROPN cana-1301 288 79 are be AUX cana-1301 288 80 therefore therefore ADV cana-1301 288 81 equal equal ADJ cana-1301 288 82 . . PUNCT cana-1301 289 1 additionally additionally ADV cana-1301 289 2 , , PUNCT cana-1301 289 3 an an DET cana-1301 289 4 m m NOUN cana-1301 289 5 - - NOUN cana-1301 289 6 bpfpg bpfpg NOUN cana-1301 289 7 and and CCONJ cana-1301 289 8 its its PRON cana-1301 289 9 dual dual ADJ cana-1301 289 10 have have VERB cana-1301 289 11 the the DET cana-1301 289 12 same same ADJ cana-1301 289 13 number number NOUN cana-1301 289 14 of of ADP cana-1301 289 15 lines line NOUN cana-1301 289 16 . . PUNCT cana-1301 290 1 the the DET cana-1301 290 2 line line NOUN cana-1301 290 3 relationship relationship NOUN cana-1301 290 4 value value NOUN cana-1301 290 5 of of ADP cana-1301 290 6 a a DET cana-1301 290 7 line line NOUN cana-1301 290 8 in in ADP cana-1301 290 9 a a DET cana-1301 290 10 dual dual ADJ cana-1301 290 11 graph graph NOUN cana-1301 290 12 is be AUX cana-1301 290 13 identical identical ADJ cana-1301 290 14 to to ADP cana-1301 290 15 the the DET cana-1301 290 16 line line NOUN cana-1301 290 17 relationship relationship NOUN cana-1301 290 18 value value NOUN cana-1301 290 19 of of ADP cana-1301 290 20 a a DET cana-1301 290 21 line line NOUN cana-1301 290 22 in in ADP cana-1301 290 23 m m NOUN cana-1301 290 24 - - NOUN cana-1301 290 25 bpfg bpfg ADJ cana-1301 290 26 , , PUNCT cana-1301 290 27 according accord VERB cana-1301 290 28 to to ADP cana-1301 290 29 the the DET cana-1301 290 30 definition definition NOUN cana-1301 290 31 of of ADP cana-1301 290 32 m m PROPN cana-1301 290 33 - - PUNCT cana-1301 290 34 bpfdg bpfdg NOUN cana-1301 290 35 . . PUNCT cana-1301 291 1 the the DET cana-1301 291 2 co co ADJ cana-1301 291 3 - - ADJ cana-1301 291 4 weak weak ADJ cana-1301 291 5 isomorphism isomorphism NOUN cana-1301 291 6 between between ADP cana-1301 291 7 𝐺 𝐺 PROPN cana-1301 291 8 and and CCONJ cana-1301 291 9 𝐺𝑏can 𝐺𝑏can PROPN cana-1301 291 10 be be AUX cana-1301 291 11 created create VERB cana-1301 291 12 . . PUNCT cana-1301 292 1 hence hence ADV cana-1301 292 2 the the DET cana-1301 292 3 outcome outcome NOUN cana-1301 292 4 . . PUNCT cana-1301 293 1 with with ADP cana-1301 293 2 the the DET cana-1301 293 3 same same ADJ cana-1301 293 4 number number NOUN cana-1301 293 5 of of ADP cana-1301 293 6 nodes node NOUN cana-1301 293 7 , , PUNCT cana-1301 293 8 two two NUM cana-1301 293 9 m m NOUN cana-1301 293 10 - - PUNCT cana-1301 293 11 bpfpgs bpfpgs NOUN cana-1301 293 12 may may AUX cana-1301 293 13 be be AUX cana-1301 293 14 isomorphic isomorphic ADJ cana-1301 293 15 . . PUNCT cana-1301 294 1 however however ADV cana-1301 294 2 , , PUNCT cana-1301 294 3 the the DET cana-1301 294 4 following follow VERB cana-1301 294 5 relationships relationship NOUN cana-1301 294 6 may may AUX cana-1301 294 7 exist exist VERB cana-1301 294 8 between between ADP cana-1301 294 9 the the DET cana-1301 294 10 m m PROPN cana-1301 294 11 - - PUNCT cana-1301 294 12 bpf bpf NOUN cana-1301 294 13 planarity planarity NOUN cana-1301 294 14 values value NOUN cana-1301 294 15 of of ADP cana-1301 294 16 two two NUM cana-1301 294 17 m m ADJ cana-1301 294 18 - - PUNCT cana-1301 294 19 bpfpgs bpfpgs NOUN cana-1301 294 20 . . PUNCT cana-1301 295 1 theorem theorem VERB cana-1301 295 2 4.5 4.5 NUM cana-1301 295 3 : : PUNCT cana-1301 295 4 let let VERB cana-1301 295 5 πΊπ‘Žand πΊπ‘Žand PROPN cana-1301 295 6 𝐺𝑏be 𝐺𝑏be PROPN cana-1301 295 7 two two NUM cana-1301 295 8 isomorphic isomorphic ADJ cana-1301 295 9 m m NOUN cana-1301 295 10 - - NOUN cana-1301 295 11 bpfgs bpfgs ADJ cana-1301 295 12 with with ADP cana-1301 295 13 the the DET cana-1301 295 14 corresponding correspond VERB cana-1301 295 15 m m PROPN cana-1301 295 16 - - PUNCT cana-1301 295 17 bpf bpf NOUN cana-1301 295 18 planarity planarity NOUN cana-1301 295 19 values value NOUN cana-1301 295 20 Ξ³πΊπ‘Ž Ξ³πΊπ‘Ž ADJ cana-1301 295 21 and and CCONJ cana-1301 295 22 γ𝐺𝑏 γ𝐺𝑏 NOUN cana-1301 295 23 . . PUNCT cana-1301 296 1 thenΞ³πΊπ‘Ž thenΞ³πΊπ‘Ž PRON cana-1301 296 2 = = PUNCT cana-1301 296 3 γ𝐺𝑏 γ𝐺𝑏 NOUN cana-1301 296 4 . . PUNCT cana-1301 297 1 proof proof NOUN cana-1301 297 2 : : PUNCT cana-1301 297 3 obvious obvious ADJ cana-1301 297 4 . . PUNCT cana-1301 298 1 we we PRON cana-1301 298 2 then then ADV cana-1301 298 3 make make VERB cana-1301 298 4 the the DET cana-1301 298 5 following follow VERB cana-1301 298 6 claims claim NOUN cana-1301 298 7 without without ADP cana-1301 298 8 providing provide VERB cana-1301 298 9 any any DET cana-1301 298 10 proof proof NOUN cana-1301 298 11 . . PUNCT cana-1301 299 1 theorem theorem VERB cana-1301 299 2 4.6 4.6 NUM cana-1301 299 3 : : PUNCT cana-1301 299 4 let let VERB cana-1301 299 5 πΊπ‘Ž πΊπ‘Ž NOUN cana-1301 299 6 and and CCONJ cana-1301 299 7 𝐺𝑏 𝐺𝑏 PROPN cana-1301 299 8 be be AUX cana-1301 299 9 two two NUM cana-1301 299 10 weak weak ADJ cana-1301 299 11 isomorphic isomorphic ADJ cana-1301 299 12 m m NOUN cana-1301 299 13 - - PUNCT cana-1301 299 14 bpfgs bpfgs ADJ cana-1301 299 15 with with ADP cana-1301 299 16 the the DET cana-1301 299 17 corresponding correspond VERB cana-1301 299 18 m m PROPN cana-1301 299 19 - - PUNCT cana-1301 299 20 bpf bpf NOUN cana-1301 299 21 planarity planarity NOUN cana-1301 299 22 values value NOUN cana-1301 299 23 of of ADP cana-1301 299 24 Ξ³πΊπ‘Ž Ξ³πΊπ‘Ž NOUN cana-1301 299 25 and and CCONJ cana-1301 299 26 γ𝐺𝑏 γ𝐺𝑏 NOUN cana-1301 299 27 . . PUNCT cana-1301 300 1 if if SCONJ cana-1301 300 2 the the DET cana-1301 300 3 line line NOUN cana-1301 300 4 positive positive ADJ cana-1301 300 5 relationship relationship NOUN cana-1301 300 6 and and CCONJ cana-1301 300 7 a a DET cana-1301 300 8 line line NOUN cana-1301 300 9 negative negative ADJ cana-1301 300 10 relationship relationship NOUN cana-1301 300 11 values value NOUN cana-1301 300 12 of of ADP cana-1301 300 13 the the DET cana-1301 300 14 respective respective ADJ cana-1301 300 15 intersecting intersecting ADJ cana-1301 300 16 lines line NOUN cana-1301 300 17 are be AUX cana-1301 300 18 the the DET cana-1301 300 19 same same ADJ cana-1301 300 20 , , PUNCT cana-1301 300 21 then then ADV cana-1301 300 22 Ξ³πΊπ‘Ž Ξ³πΊπ‘Ž NOUN cana-1301 300 23 = = PUNCT cana-1301 300 24 γ𝐺𝑏 γ𝐺𝑏 X cana-1301 300 25 . . PUNCT cana-1301 301 1 theorem theorem VERB cana-1301 301 2 4.7 4.7 NUM cana-1301 301 3 : : PUNCT cana-1301 301 4 let let VERB cana-1301 301 5 πΊπ‘Žand πΊπ‘Žand PROPN cana-1301 301 6 𝐺𝑏 𝐺𝑏 PROPN cana-1301 301 7 be be AUX cana-1301 301 8 two two NUM cana-1301 301 9 co co ADJ cana-1301 301 10 - - ADJ cana-1301 301 11 weak weak ADJ cana-1301 301 12 isomorphic isomorphic ADJ cana-1301 301 13 m m NOUN cana-1301 301 14 - - PUNCT cana-1301 301 15 bpfgs bpfgs ADJ cana-1301 301 16 with with ADP cana-1301 301 17 the the DET cana-1301 301 18 corresponding correspond VERB cana-1301 301 19 m m PROPN cana-1301 301 20 - - PUNCT cana-1301 301 21 bpf bpf NOUN cana-1301 301 22 planarity planarity NOUN cana-1301 301 23 values value NOUN cana-1301 301 24 of of ADP cana-1301 301 25 Ξ³πΊπ‘Ž Ξ³πΊπ‘Ž NOUN cana-1301 301 26 and and CCONJ cana-1301 301 27 γ𝐺𝑏 γ𝐺𝑏 NOUN cana-1301 301 28 . . PUNCT cana-1301 302 1 if if SCONJ cana-1301 302 2 the the DET cana-1301 302 3 terminal terminal ADJ cana-1301 302 4 nodes node NOUN cana-1301 302 5 of of ADP cana-1301 302 6 the the DET cana-1301 302 7 respective respective ADJ cana-1301 302 8 intersecting intersecting ADJ cana-1301 302 9 lines line NOUN cana-1301 302 10 have have VERB cana-1301 302 11 the the DET cana-1301 302 12 same same ADJ cana-1301 302 13 minimum minimum NOUN cana-1301 302 14 , , PUNCT cana-1301 302 15 positive positive ADJ cana-1301 302 16 relationship relationship NOUN cana-1301 302 17 and and CCONJ cana-1301 302 18 maximum maximum ADJ cana-1301 302 19 negative negative ADJ cana-1301 302 20 relationship relationship NOUN cana-1301 302 21 values value NOUN cana-1301 302 22 , , PUNCT cana-1301 302 23 then then ADV cana-1301 302 24 Ξ³πΊπ‘Ž Ξ³πΊπ‘Ž NOUN cana-1301 302 25 = = PRON cana-1301 302 26 γ𝐺𝑏 γ𝐺𝑏 ADV cana-1301 302 27 . . PUNCT cana-1301 303 1 conclusions conclusion NOUN cana-1301 303 2 unpredictable unpredictable ADJ cana-1301 303 3 behavior behavior NOUN cana-1301 303 4 and and CCONJ cana-1301 303 5 opinions opinion NOUN cana-1301 303 6 of of ADP cana-1301 303 7 voters voter NOUN cana-1301 303 8 leads lead VERB cana-1301 303 9 to to ADP cana-1301 303 10 vagueness vagueness NOUN cana-1301 303 11 and and CCONJ cana-1301 303 12 uncertainty uncertainty NOUN cana-1301 303 13 when when SCONJ cana-1301 303 14 traditional traditional ADJ cana-1301 303 15 graph graph NOUN cana-1301 303 16 theory theory NOUN cana-1301 303 17 is be AUX cana-1301 303 18 used use VERB cana-1301 303 19 for for ADP cana-1301 303 20 analyzing analyze VERB cana-1301 303 21 the the DET cana-1301 303 22 voter voter NOUN cana-1301 303 23 sentiment sentiment NOUN cana-1301 303 24 and and CCONJ cana-1301 303 25 shifting shift VERB cana-1301 303 26 loyalties loyalty NOUN cana-1301 303 27 . . PUNCT cana-1301 304 1 we we PRON cana-1301 304 2 already already ADV cana-1301 304 3 know know VERB cana-1301 304 4 that that SCONJ cana-1301 304 5 bipolar bipolar ADJ cana-1301 304 6 fuzzy fuzzy ADJ cana-1301 304 7 sets set NOUN cana-1301 304 8 are be AUX cana-1301 304 9 capable capable ADJ cana-1301 304 10 of of ADP cana-1301 304 11 handling handle VERB cana-1301 304 12 fuzzy fuzzy ADJ cana-1301 304 13 sets set NOUN cana-1301 304 14 with with ADP cana-1301 304 15 vague vague ADJ cana-1301 304 16 and and CCONJ cana-1301 304 17 uncertainty uncertainty NOUN cana-1301 304 18 data datum NOUN cana-1301 304 19 to to ADP cana-1301 304 20 some some DET cana-1301 304 21 extent extent NOUN cana-1301 304 22 . . PUNCT cana-1301 305 1 by by ADP cana-1301 305 2 modifying modify VERB cana-1301 305 3 them they PRON cana-1301 305 4 to to PART cana-1301 305 5 include include VERB cana-1301 305 6 multiple multiple ADJ cana-1301 305 7 parameters parameter NOUN cana-1301 305 8 on on ADP cana-1301 305 9 bipolar bipolar ADJ cana-1301 305 10 . . PUNCT cana-1301 306 1 in in ADP cana-1301 306 2 this this DET cana-1301 306 3 research research NOUN cana-1301 306 4 , , PUNCT cana-1301 306 5 we we PRON cana-1301 306 6 apply apply VERB cana-1301 306 7 the the DET cana-1301 306 8 idea idea NOUN cana-1301 306 9 of of ADP cana-1301 306 10 m m NOUN cana-1301 306 11 - - ADJ cana-1301 306 12 bipolar bipolar ADJ cana-1301 306 13 fuzzy fuzzy ADJ cana-1301 306 14 set set NOUN cana-1301 306 15 to to PART cana-1301 306 16 multigraph multigraph NOUN cana-1301 306 17 and and CCONJ cana-1301 306 18 planar planar ADJ cana-1301 306 19 graph graph NOUN cana-1301 306 20 because because SCONJ cana-1301 306 21 m m ADJ cana-1301 306 22 - - ADJ cana-1301 306 23 bipolar bipolar ADJ cana-1301 306 24 fuzzy fuzzy ADJ cana-1301 306 25 sets set NOUN cana-1301 306 26 have have AUX cana-1301 306 27 shown show VERB cana-1301 306 28 advantages advantage NOUN cana-1301 306 29 over over ADP cana-1301 306 30 bipolar bipolar ADJ cana-1301 306 31 fuzzy fuzzy ADJ cana-1301 306 32 sets set NOUN cana-1301 306 33 in in ADP cana-1301 306 34 addressing address VERB cana-1301 306 35 ambiguity ambiguity NOUN cana-1301 306 36 and and CCONJ cana-1301 306 37 uncertainty uncertainty NOUN cana-1301 306 38 to to ADP cana-1301 306 39 some some DET cana-1301 306 40 extent extent NOUN cana-1301 306 41 . . PUNCT cana-1301 307 1 we we PRON cana-1301 307 2 can can AUX cana-1301 307 3 generate generate VERB cana-1301 307 4 m m NOUN cana-1301 307 5 - - PUNCT cana-1301 307 6 bpfgs bpfgs NOUN cana-1301 307 7 to to PART cana-1301 307 8 provide provide VERB cana-1301 307 9 better well ADJ cana-1301 307 10 analysis analysis NOUN cana-1301 307 11 . . PUNCT cana-1301 308 1 recent recent ADJ cana-1301 308 2 developments development NOUN cana-1301 308 3 in in ADP cana-1301 308 4 the the DET cana-1301 308 5 fields field NOUN cana-1301 308 6 of of ADP cana-1301 308 7 artificial artificial ADJ cana-1301 308 8 intelligence intelligence NOUN cana-1301 308 9 , , PUNCT cana-1301 308 10 electrical electrical ADJ cana-1301 308 11 engineering engineering NOUN cana-1301 308 12 and and CCONJ cana-1301 308 13 quantum quantum NOUN cana-1301 308 14 mechanics mechanic NOUN cana-1301 308 15 expand expand VERB cana-1301 308 16 the the DET cana-1301 308 17 scope scope NOUN cana-1301 308 18 and and CCONJ cana-1301 308 19 the the DET cana-1301 308 20 application application NOUN cana-1301 308 21 of of ADP cana-1301 308 22 m m PROPN cana-1301 308 23 - - PUNCT cana-1301 308 24 bpfgs bpfgs NOUN cana-1301 308 25 . . PUNCT cana-1301 309 1 communications communication NOUN cana-1301 309 2 on on ADP cana-1301 309 3 applied apply VERB cana-1301 309 4 nonlinear nonlinear ADJ cana-1301 309 5 analysis analysis NOUN cana-1301 309 6 issn issn NOUN cana-1301 309 7 : : PUNCT cana-1301 309 8 1074 1074 NUM cana-1301 309 9 - - PUNCT cana-1301 309 10 133x 133x NUM cana-1301 309 11 vol vol NOUN cana-1301 309 12 31 31 NUM cana-1301 309 13 no no NOUN cana-1301 309 14 . . PUNCT cana-1301 310 1 7s 7 NOUN cana-1301 310 2 ( ( PUNCT cana-1301 310 3 2024 2024 NUM cana-1301 310 4 ) ) PUNCT cana-1301 310 5 275 275 NUM cana-1301 310 6 https://internationalpubls.com https://internationalpubls.com X cana-1301 310 7 references reference NOUN cana-1301 310 8 [ [ X cana-1301 310 9 1 1 NUM cana-1301 310 10 ] ] PUNCT cana-1301 310 11 abdul abdul PROPN cana-1301 310 12 - - PUNCT cana-1301 310 13 jabbar jabbar PROPN cana-1301 310 14 , , PUNCT cana-1301 310 15 n. n. NOUN cana-1301 310 16 , , PUNCT cana-1301 310 17 naoom naoom PROPN cana-1301 310 18 , , PUNCT cana-1301 310 19 j.h j.h PROPN cana-1301 310 20 . . PROPN cana-1301 310 21 , , PUNCT cana-1301 310 22 ouda ouda NOUN cana-1301 310 23 , , PUNCT cana-1301 310 24 e.h e.h PROPN cana-1301 310 25 . . PROPN cana-1301 310 26 : : PUNCT cana-1301 310 27 fuzzy fuzzy ADJ cana-1301 310 28 dual dual ADJ cana-1301 310 29 graph graph NOUN cana-1301 310 30 . . PUNCT cana-1301 311 1 j. j. PROPN cana-1301 311 2 al al PROPN cana-1301 311 3 - - PUNCT cana-1301 311 4 nahrain nahrain PROPN cana-1301 311 5 univ univ PROPN cana-1301 311 6 . . PUNCT cana-1301 312 1 12(4),168 12(4),168 NUM cana-1301 312 2 - - SYM cana-1301 312 3 171 171 NUM cana-1301 312 4 ( ( PUNCT cana-1301 312 5 2009 2009 NUM cana-1301 312 6 ) ) PUNCT cana-1301 313 1 [ [ X cana-1301 313 2 2 2 NUM cana-1301 313 3 ] ] X cana-1301 313 4 akram akram NOUN cana-1301 313 5 , , PUNCT cana-1301 313 6 m. m. NOUN cana-1301 313 7 : : PUNCT cana-1301 313 8 bipolar bipolar ADJ cana-1301 313 9 fuzzy fuzzy ADJ cana-1301 313 10 graphs graph NOUN cana-1301 313 11 . . PUNCT cana-1301 313 12 inf inf PROPN cana-1301 313 13 . . PUNCT cana-1301 314 1 sci sci PROPN cana-1301 314 2 . . PROPN cana-1301 315 1 181 181 NUM cana-1301 315 2 , , PUNCT cana-1301 315 3 5548 5548 NUM cana-1301 315 4 - - SYM cana-1301 315 5 5564 5564 NUM cana-1301 315 6 ( ( PUNCT cana-1301 315 7 2011 2011 NUM cana-1301 315 8 ) ) PUNCT cana-1301 316 1 [ [ X cana-1301 316 2 3 3 NUM cana-1301 316 3 ] ] X cana-1301 316 4 akram akram NOUN cana-1301 316 5 , , PUNCT cana-1301 316 6 m. m. NOUN cana-1301 316 7 : : PUNCT cana-1301 316 8 bipolar bipolar ADJ cana-1301 316 9 fuzzy fuzzy ADJ cana-1301 316 10 graphs graph NOUN cana-1301 316 11 with with ADP cana-1301 316 12 applications application NOUN cana-1301 316 13 . . PUNCT cana-1301 317 1 knowl knowl PROPN cana-1301 317 2 . . PUNCT cana-1301 318 1 -based -based ADJ cana-1301 318 2 syst syst NOUN cana-1301 318 3 . . PUNCT cana-1301 318 4 39 39 NUM cana-1301 318 5 , , PUNCT cana-1301 318 6 1 1 NUM cana-1301 318 7 - - SYM cana-1301 318 8 8 8 NUM cana-1301 318 9 ( ( PUNCT cana-1301 318 10 2013 2013 NUM cana-1301 318 11 ) ) PUNCT cana-1301 319 1 [ [ X cana-1301 319 2 4 4 NUM cana-1301 319 3 ] ] X cana-1301 319 4 akram akram NOUN cana-1301 319 5 , , PUNCT cana-1301 319 6 m. m. NOUN cana-1301 319 7 , , PUNCT cana-1301 319 8 dudek dudek PROPN cana-1301 319 9 , , PUNCT cana-1301 319 10 w.a w.a PROPN cana-1301 319 11 . . PROPN cana-1301 319 12 : : PUNCT cana-1301 319 13 regular regular ADJ cana-1301 319 14 bipolar bipolar ADJ cana-1301 319 15 fuzzy fuzzy ADJ cana-1301 319 16 graphs graph NOUN cana-1301 319 17 . . PUNCT cana-1301 320 1 neural neural ADJ cana-1301 320 2 comput comput NOUN cana-1301 320 3 . . PUNCT cana-1301 321 1 appl appl PROPN cana-1301 321 2 . . PROPN cana-1301 322 1 21 21 NUM cana-1301 322 2 , , PUNCT cana-1301 322 3 197 197 NUM cana-1301 322 4 - - SYM cana-1301 322 5 205 205 NUM cana-1301 322 6 ( ( PUNCT cana-1301 322 7 2012 2012 NUM cana-1301 322 8 ) ) PUNCT cana-1301 323 1 [ [ X cana-1301 323 2 5 5 NUM cana-1301 323 3 ] ] X cana-1301 323 4 akram akram NOUN cana-1301 323 5 , , PUNCT cana-1301 323 6 m. m. NOUN cana-1301 323 7 , , PUNCT cana-1301 323 8 karunambigai karunambigai PROPN cana-1301 323 9 , , PUNCT cana-1301 323 10 m.g m.g PROPN cana-1301 323 11 . . PROPN cana-1301 323 12 : : PUNCT cana-1301 323 13 metric metric ADJ cana-1301 323 14 in in ADP cana-1301 323 15 bipolar bipolar ADJ cana-1301 323 16 fuzzy fuzzy ADJ cana-1301 323 17 graphs graph NOUN cana-1301 323 18 . . PUNCT cana-1301 324 1 world world NOUN cana-1301 324 2 appl appl PROPN cana-1301 324 3 . . PUNCT cana-1301 325 1 sci sci PROPN cana-1301 325 2 . . PUNCT cana-1301 325 3 j. j. PROPN cana-1301 325 4 14 14 NUM cana-1301 325 5 , , PUNCT cana-1301 325 6 1920 1920 NUM cana-1301 325 7 - - SYM cana-1301 325 8 1927 1927 NUM cana-1301 325 9 ( ( PUNCT cana-1301 325 10 2011 2011 NUM cana-1301 325 11 ) ) PUNCT cana-1301 326 1 [ [ X cana-1301 326 2 6 6 NUM cana-1301 326 3 ] ] X cana-1301 326 4 akram akram PROPN cana-1301 326 5 , , PUNCT cana-1301 326 6 m. m. NOUN cana-1301 326 7 , , PUNCT cana-1301 326 8 dudek dudek PROPN cana-1301 326 9 , , PUNCT cana-1301 326 10 w.a w.a PROPN cana-1301 326 11 . . PROPN cana-1301 326 12 , , PUNCT cana-1301 326 13 sarwar sarwar PROPN cana-1301 326 14 , , PUNCT cana-1301 326 15 s. s. PROPN cana-1301 326 16 : : PUNCT cana-1301 326 17 properties property NOUN cana-1301 326 18 of of ADP cana-1301 326 19 bipolar bipolar ADJ cana-1301 326 20 fuzzy fuzzy ADJ cana-1301 326 21 hypergraphs hypergraph NOUN cana-1301 326 22 . . PUNCT cana-1301 327 1 ital ital PROPN cana-1301 327 2 . . PUNCT cana-1301 328 1 j. j. PROPN cana-1301 328 2 pure pure PROPN cana-1301 328 3 appl appl PROPN cana-1301 328 4 . . PUNCT cana-1301 328 5 math math NOUN cana-1301 328 6 . . PUNCT cana-1301 329 1 31 31 NUM cana-1301 329 2 . . X cana-1301 329 3 426 426 NUM cana-1301 329 4 - - SYM cana-1301 329 5 458 458 NUM cana-1301 329 6 ( ( PUNCT cana-1301 329 7 2013 2013 NUM cana-1301 329 8 ) ) PUNCT cana-1301 330 1 [ [ X cana-1301 330 2 7 7 NUM cana-1301 330 3 ] ] X cana-1301 330 4 akram akram NOUN cana-1301 330 5 , , PUNCT cana-1301 330 6 m. m. NOUN cana-1301 330 7 , , PUNCT cana-1301 330 8 alshehri alshehri PROPN cana-1301 330 9 , , PUNCT cana-1301 330 10 n. n. NOUN cana-1301 330 11 , , PUNCT cana-1301 330 12 davvaz davvaz PROPN cana-1301 330 13 , , PUNCT cana-1301 330 14 b. b. PROPN cana-1301 330 15 , , PUNCT cana-1301 330 16 ashraf ashraf PROPN cana-1301 330 17 , , PUNCT cana-1301 330 18 a. a. NOUN cana-1301 330 19 : : PUNCT cana-1301 330 20 bipolar bipolar ADJ cana-1301 330 21 fuzzy fuzzy ADJ cana-1301 330 22 digraphs digraph VERB cana-1301 330 23 in in ADP cana-1301 330 24 decision decision NOUN cana-1301 330 25 support support NOUN cana-1301 330 26 systems system NOUN cana-1301 330 27 . . PUNCT cana-1301 331 1 j. j. PROPN cana-1301 331 2 multi multi PROPN cana-1301 331 3 - - ADJ cana-1301 331 4 valued value VERB cana-1301 331 5 logic logic NOUN cana-1301 331 6 soft soft ADJ cana-1301 331 7 comput comput NOUN cana-1301 331 8 . . PUNCT cana-1301 332 1 1 1 NUM cana-1301 332 2 - - SYM cana-1301 332 3 21 21 NUM cana-1301 332 4 ( ( PUNCT cana-1301 332 5 2016 2016 NUM cana-1301 332 6 ) ) PUNCT cana-1301 332 7 ( ( PUNCT cana-1301 332 8 in in ADP cana-1301 332 9 press press NOUN cana-1301 332 10 ) ) PUNCT cana-1301 333 1 [ [ X cana-1301 333 2 8 8 NUM cana-1301 333 3 ] ] X cana-1301 333 4 akram akram NOUN cana-1301 333 5 , , PUNCT cana-1301 333 6 m. m. NOUN cana-1301 333 7 , , PUNCT cana-1301 333 8 samantha samantha PROPN cana-1301 333 9 , , PUNCT cana-1301 333 10 s. s. PROPN cana-1301 333 11 , , PUNCT cana-1301 333 12 pal pal NOUN cana-1301 333 13 , , PUNCT cana-1301 333 14 m. m. NOUN cana-1301 333 15 : : PUNCT cana-1301 333 16 application application NOUN cana-1301 333 17 of of ADP cana-1301 333 18 bipolar bipolar ADJ cana-1301 333 19 fuzzy fuzzy ADJ cana-1301 333 20 sets set NOUN cana-1301 333 21 in in ADP cana-1301 333 22 planar planar ADJ cana-1301 333 23 graphs graph NOUN cana-1301 333 24 . . PUNCT cana-1301 334 1 int int NOUN cana-1301 334 2 . . PUNCT cana-1301 335 1 j. j. PROPN cana-1301 335 2 appl appl PROPN cana-1301 335 3 . . PUNCT cana-1301 336 1 comput comput PROPN cana-1301 336 2 . . PUNCT cana-1301 337 1 math math NOUN cana-1301 337 2 . . PUNCT cana-1301 338 1 3 3 NUM cana-1301 338 2 , , PUNCT cana-1301 338 3 ( ( PUNCT cana-1301 338 4 2017 2017 NUM cana-1301 338 5 ) ) PUNCT cana-1301 339 1 [ [ X cana-1301 339 2 9 9 NUM cana-1301 339 3 ] ] PUNCT cana-1301 339 4 alshehri alshehri PROPN cana-1301 339 5 , , PUNCT cana-1301 339 6 n. n. NOUN cana-1301 339 7 , , PUNCT cana-1301 339 8 akram akram PROPN cana-1301 339 9 , , PUNCT cana-1301 339 10 m. m. NOUN cana-1301 339 11 : : PUNCT cana-1301 339 12 intuitionistic intuitionistic ADJ cana-1301 339 13 fuzzy fuzzy ADJ cana-1301 339 14 planar planar ADJ cana-1301 339 15 graphs graph NOUN cana-1301 339 16 . . PUNCT cana-1301 340 1 discrete discrete ADJ cana-1301 340 2 dyn dyn NOUN cana-1301 340 3 . . PUNCT cana-1301 341 1 nat nat PROPN cana-1301 341 2 . . PUNCT cana-1301 342 1 soc soc PROPN cana-1301 342 2 . . PUNCT cana-1301 343 1 article article NOUN cana-1301 343 2 i i PROPN cana-1301 343 3 d d PROPN cana-1301 343 4 397823 397823 NUM cana-1301 343 5 [ [ X cana-1301 343 6 10 10 NUM cana-1301 343 7 ] ] X cana-1301 343 8 bhattacharya bhattacharya NOUN cana-1301 343 9 , , PUNCT cana-1301 343 10 p. p. NOUN cana-1301 343 11 : : PUNCT cana-1301 343 12 some some DET cana-1301 343 13 remarks remark NOUN cana-1301 343 14 on on ADP cana-1301 343 15 fuzzy fuzzy ADJ cana-1301 343 16 graphs graph NOUN cana-1301 343 17 . . PUNCT cana-1301 344 1 pattern pattern NOUN cana-1301 344 2 recognit recognit VERB cana-1301 344 3 . . PUNCT cana-1301 345 1 lett lett PROPN cana-1301 345 2 . . PROPN cana-1301 346 1 6 6 NUM cana-1301 346 2 , , PUNCT cana-1301 346 3 297 297 NUM cana-1301 346 4 - - SYM cana-1301 346 5 302 302 NUM cana-1301 346 6 ( ( PUNCT cana-1301 346 7 1987 1987 NUM cana-1301 346 8 ) ) PUNCT cana-1301 347 1 [ [ X cana-1301 347 2 11 11 NUM cana-1301 347 3 ] ] PUNCT cana-1301 347 4 cerruti cerruti NOUN cana-1301 347 5 , , PUNCT cana-1301 347 6 u. u. NOUN cana-1301 347 7 : : PUNCT cana-1301 347 8 graphs graph NOUN cana-1301 347 9 and and CCONJ cana-1301 347 10 fuzzy fuzzy ADJ cana-1301 347 11 graphs graph NOUN cana-1301 347 12 . . PUNCT cana-1301 348 1 fuzzy fuzzy ADJ cana-1301 348 2 information information NOUN cana-1301 348 3 and and CCONJ cana-1301 348 4 decision decision NOUN cana-1301 348 5 processes process NOUN cana-1301 348 6 , , PUNCT cana-1301 348 7 pp.123 pp.123 PROPN cana-1301 348 8 - - PUNCT cana-1301 348 9 131 131 NUM cana-1301 348 10 ( ( PUNCT cana-1301 348 11 1982 1982 NUM cana-1301 348 12 ) ) PUNCT cana-1301 349 1 [ [ X cana-1301 349 2 12 12 NUM cana-1301 349 3 ] ] X cana-1301 349 4 dubois dubois PROPN cana-1301 349 5 , , PUNCT cana-1301 349 6 d. d. PROPN cana-1301 349 7 , , PUNCT cana-1301 349 8 kaci kaci PROPN cana-1301 349 9 , , PUNCT cana-1301 349 10 s. s. PROPN cana-1301 349 11 , , PUNCT cana-1301 349 12 prade prade NOUN cana-1301 349 13 , , PUNCT cana-1301 349 14 h. h. NOUN cana-1301 349 15 : : PUNCT cana-1301 349 16 bipolarity bipolarity NOUN cana-1301 349 17 in in ADP cana-1301 349 18 reasoning reasoning NOUN cana-1301 349 19 and and CCONJ cana-1301 349 20 decision decision NOUN cana-1301 349 21 , , PUNCT cana-1301 349 22 an an DET cana-1301 349 23 introduction introduction NOUN cana-1301 349 24 . . PUNCT cana-1301 350 1 in in ADP cana-1301 350 2 : : PUNCT cana-1301 350 3 international international ADJ cana-1301 350 4 conference conference NOUN cana-1301 350 5 on on ADP cana-1301 350 6 information information NOUN cana-1301 350 7 processing processing NOUN cana-1301 350 8 and and CCONJ cana-1301 350 9 management management NOUN cana-1301 350 10 of of ADP cana-1301 350 11 uncertainty uncertainty NOUN cana-1301 350 12 , , PUNCT cana-1301 350 13 ipmu’04 ipmu’04 PROPN cana-1301 350 14 , , PUNCT cana-1301 350 15 pp pp ADJ cana-1301 350 16 . . PUNCT cana-1301 351 1 959 959 NUM cana-1301 351 2 - - SYM cana-1301 351 3 966 966 NUM cana-1301 351 4 ( ( PUNCT cana-1301 351 5 2004 2004 NUM cana-1301 351 6 ) ) PUNCT cana-1301 352 1 [ [ X cana-1301 352 2 13 13 NUM cana-1301 352 3 ] ] PUNCT cana-1301 352 4 eslahchi eslahchi PROPN cana-1301 352 5 , , PUNCT cana-1301 352 6 c. c. PROPN cana-1301 352 7 , , PUNCT cana-1301 352 8 onaghe onaghe NOUN cana-1301 352 9 , , PUNCT cana-1301 352 10 b.n b.n PROPN cana-1301 352 11 . . PROPN cana-1301 352 12 : : PUNCT cana-1301 353 1 vertex vertex NOUN cana-1301 353 2 strength strength NOUN cana-1301 353 3 of of ADP cana-1301 353 4 fuzzy fuzzy ADJ cana-1301 353 5 graphs graph NOUN cana-1301 353 6 . . PUNCT cana-1301 354 1 int int NOUN cana-1301 354 2 . . PUNCT cana-1301 355 1 j. j. PROPN cana-1301 355 2 math math PROPN cana-1301 355 3 . . PUNCT cana-1301 356 1 math math NOUN cana-1301 356 2 . . PUNCT cana-1301 357 1 sci sci PROPN cana-1301 357 2 . . PROPN cana-1301 358 1 1 1 NUM cana-1301 358 2 - - SYM cana-1301 358 3 9 9 NUM cana-1301 358 4 . . PUNCT cana-1301 358 5 article article NOUN cana-1301 358 6 i i PROPN cana-1301 358 7 d d PROPN cana-1301 358 8 43614 43614 NUM cana-1301 359 1 [ [ X cana-1301 359 2 14 14 NUM cana-1301 359 3 ] ] PUNCT cana-1301 359 4 ghorai ghorai PROPN cana-1301 359 5 , , PUNCT cana-1301 359 6 g. g. PROPN cana-1301 359 7 , , PUNCT cana-1301 359 8 pal pal NOUN cana-1301 359 9 , , PUNCT cana-1301 359 10 m. m. NOUN cana-1301 359 11 : : PUNCT cana-1301 359 12 faces face NOUN cana-1301 359 13 and and CCONJ cana-1301 359 14 dual dual ADJ cana-1301 359 15 of of ADP cana-1301 359 16 m m ADJ cana-1301 359 17 - - ADJ cana-1301 359 18 polar polar ADJ cana-1301 359 19 fuzzy fuzzy ADJ cana-1301 359 20 planar planar ADJ cana-1301 359 21 graphs graph NOUN cana-1301 359 22 . . PUNCT cana-1301 360 1 journal journal NOUN cana-1301 360 2 of of ADP cana-1301 360 3 intelligent intelligent ADJ cana-1301 360 4 & & CCONJ cana-1301 360 5 fuzzy fuzzy ADJ cana-1301 360 6 systems system NOUN cana-1301 360 7 , , PUNCT cana-1301 360 8 31(3 31(3 NUM cana-1301 360 9 ) ) PUNCT cana-1301 360 10 , , PUNCT cana-1301 360 11 2043 2043 NUM cana-1301 360 12 - - SYM cana-1301 360 13 2049 2049 NUM cana-1301 360 14 ( ( PUNCT cana-1301 360 15 2016 2016 NUM cana-1301 360 16 ) ) PUNCT cana-1301 361 1 [ [ X cana-1301 361 2 15 15 NUM cana-1301 361 3 ] ] X cana-1301 361 4 kauffman kauffman PROPN cana-1301 361 5 , , PUNCT cana-1301 361 6 a. a. NOUN cana-1301 361 7 : : PUNCT cana-1301 361 8 introduction introduction NOUN cana-1301 361 9 a a DET cana-1301 361 10 la la X cana-1301 361 11 theorie theorie PROPN cana-1301 361 12 des des X cana-1301 361 13 sous sous PROPN cana-1301 361 14 - - ADJ cana-1301 361 15 emsemblesflous emsemblesflous ADJ cana-1301 361 16 , , PUNCT cana-1301 361 17 vol vol NOUN cana-1301 361 18 . . PROPN cana-1301 361 19 1 1 NUM cana-1301 361 20 . . X cana-1301 361 21 masson masson PROPN cana-1301 361 22 et et PROPN cana-1301 361 23 cie cie PROPN cana-1301 361 24 , , PUNCT cana-1301 361 25 paris paris PROPN cana-1301 361 26 ( ( PUNCT cana-1301 361 27 1973 1973 NUM cana-1301 361 28 ) ) PUNCT cana-1301 362 1 [ [ X cana-1301 362 2 16 16 NUM cana-1301 362 3 ] ] X cana-1301 362 4 mathew mathew PROPN cana-1301 362 5 , , PUNCT cana-1301 362 6 s. s. PROPN cana-1301 362 7 , , PUNCT cana-1301 362 8 sunitha sunitha VERB cana-1301 362 9 , , PUNCT cana-1301 362 10 m. m. NOUN cana-1301 362 11 s. s. PROPN cana-1301 362 12 : : PUNCT cana-1301 362 13 types type NOUN cana-1301 362 14 of of ADP cana-1301 362 15 arcs arc NOUN cana-1301 362 16 in in ADP cana-1301 362 17 a a DET cana-1301 362 18 fuzzy fuzzy ADJ cana-1301 362 19 graph graph NOUN cana-1301 362 20 . . PUNCT cana-1301 362 21 inf inf PROPN cana-1301 362 22 . . PUNCT cana-1301 362 23 sci sci PROPN cana-1301 362 24 . . PUNCT cana-1301 362 25 179(11 179(11 NUM cana-1301 362 26 ) ) PUNCT cana-1301 362 27 , , PUNCT cana-1301 362 28 1760 1760 NUM cana-1301 362 29 - - SYM cana-1301 362 30 1768 1768 NUM cana-1301 362 31 ( ( PUNCT cana-1301 362 32 2009 2009 NUM cana-1301 362 33 ) ) PUNCT cana-1301 363 1 [ [ X cana-1301 363 2 17 17 NUM cana-1301 363 3 ] ] X cana-1301 363 4 mordeson mordeson NOUN cana-1301 363 5 , , PUNCT cana-1301 363 6 j.n j.n PROPN cana-1301 363 7 . . PROPN cana-1301 363 8 , , PUNCT cana-1301 363 9 peng peng PROPN cana-1301 363 10 , , PUNCT cana-1301 363 11 c. c. PROPN cana-1301 363 12 s. s. PROPN cana-1301 363 13 : : PUNCT cana-1301 363 14 operations operation NOUN cana-1301 363 15 on on ADP cana-1301 363 16 fuzzy fuzzy ADJ cana-1301 363 17 graphs graph NOUN cana-1301 363 18 . . PUNCT cana-1301 364 1 inf inf PROPN cana-1301 364 2 . . PUNCT cana-1301 365 1 sci sci PROPN cana-1301 365 2 . . PROPN cana-1301 365 3 79 79 NUM cana-1301 365 4 , , PUNCT cana-1301 365 5 159 159 NUM cana-1301 365 6 - - SYM cana-1301 365 7 170 170 NUM cana-1301 365 8 ( ( PUNCT cana-1301 365 9 1994 1994 NUM cana-1301 365 10 ) ) PUNCT cana-1301 366 1 [ [ X cana-1301 366 2 18 18 NUM cana-1301 366 3 ] ] SYM cana-1301 366 4 mordeson mordeson NOUN cana-1301 366 5 , , PUNCT cana-1301 366 6 j.n j.n PROPN cana-1301 366 7 . . PROPN cana-1301 366 8 , , PUNCT cana-1301 366 9 nair nair PROPN cana-1301 366 10 , , PUNCT cana-1301 366 11 p.s p.s PROPN cana-1301 366 12 . . PUNCT cana-1301 366 13 : : PUNCT cana-1301 366 14 fuzzy fuzzy ADJ cana-1301 366 15 graphs graph NOUN cana-1301 366 16 and and CCONJ cana-1301 366 17 fuzzy fuzzy ADJ cana-1301 366 18 hypergraphs hypergraph NOUN cana-1301 366 19 , , PUNCT cana-1301 366 20 second second ADJ cana-1301 366 21 edition edition NOUN cana-1301 366 22 2001 2001 NUM cana-1301 366 23 . . PUNCT cana-1301 367 1 physica physica PROPN cana-1301 367 2 verlag verlag PROPN cana-1301 367 3 , , PUNCT cana-1301 367 4 heidelberg heidelberg PROPN cana-1301 367 5 ( ( PUNCT cana-1301 367 6 1998 1998 NUM cana-1301 367 7 ) ) PUNCT cana-1301 368 1 [ [ X cana-1301 368 2 19 19 NUM cana-1301 368 3 ] ] X cana-1301 368 4 pal pal NOUN cana-1301 368 5 , , PUNCT cana-1301 368 6 a. a. NOUN cana-1301 368 7 , , PUNCT cana-1301 368 8 samantha samantha PROPN cana-1301 368 9 , , PUNCT cana-1301 368 10 s. s. PROPN cana-1301 368 11 , , PUNCT cana-1301 368 12 pal pal NOUN cana-1301 368 13 , , PUNCT cana-1301 368 14 m. m. NOUN cana-1301 368 15 : : PUNCT cana-1301 368 16 concept concept NOUN cana-1301 368 17 of of ADP cana-1301 368 18 fuzzy fuzzy ADJ cana-1301 368 19 planar planar ADJ cana-1301 368 20 graphs graph NOUN cana-1301 368 21 . . PUNCT cana-1301 369 1 in in ADP cana-1301 369 2 : : PUNCT cana-1301 369 3 proceedings proceeding NOUN cana-1301 369 4 of of ADP cana-1301 369 5 science science NOUN cana-1301 369 6 and and CCONJ cana-1301 369 7 information information NOUN cana-1301 369 8 conference conference NOUN cana-1301 369 9 2013 2013 NUM cana-1301 369 10 , , PUNCT cana-1301 369 11 october october PROPN cana-1301 369 12 7 7 NUM cana-1301 369 13 - - SYM cana-1301 369 14 9 9 NUM cana-1301 369 15 , , PUNCT cana-1301 369 16 2013 2013 NUM cana-1301 369 17 , , PUNCT cana-1301 369 18 london london PROPN cana-1301 369 19 , , PUNCT cana-1301 369 20 uk uk PROPN cana-1301 369 21 , , PUNCT cana-1301 369 22 pp pp ADJ cana-1301 369 23 . . PUNCT cana-1301 370 1 557 557 NUM cana-1301 370 2 - - SYM cana-1301 370 3 563 563 NUM cana-1301 370 4 ( ( PUNCT cana-1301 370 5 2013 2013 NUM cana-1301 370 6 ) ) PUNCT cana-1301 371 1 [ [ X cana-1301 371 2 20 20 NUM cana-1301 371 3 ] ] X cana-1301 371 4 prem prem PROPN cana-1301 371 5 kumar kumar PROPN cana-1301 371 6 singh singh PROPN cana-1301 371 7 : : PUNCT cana-1301 371 8 bipolarity bipolarity NOUN cana-1301 371 9 in in ADP cana-1301 371 10 multi multi ADJ cana-1301 371 11 - - ADJ cana-1301 371 12 way way ADJ cana-1301 371 13 fuzzy fuzzy ADJ cana-1301 371 14 context context NOUN cana-1301 371 15 and and CCONJ cana-1301 371 16 its its PRON cana-1301 371 17 analysis analysis NOUN cana-1301 371 18 using use VERB cana-1301 371 19 m m NOUN cana-1301 371 20 - - PUNCT cana-1301 371 21 way way NOUN cana-1301 371 22 granulation granulation NOUN cana-1301 371 23 : : PUNCT cana-1301 371 24 granular granular ADJ cana-1301 371 25 computing computing NOUN cana-1301 371 26 , , PUNCT cana-1301 371 27 march march PROPN cana-1301 371 28 , , PUNCT cana-1301 371 29 2021 2021 NUM cana-1301 371 30 . . PUNCT cana-1301 372 1 [ [ X cana-1301 372 2 21 21 NUM cana-1301 372 3 ] ] X cana-1301 372 4 riaz riaz PROPN cana-1301 372 5 , , PUNCT cana-1301 372 6 f. f. PROPN cana-1301 372 7 , , PUNCT cana-1301 372 8 ali ali PROPN cana-1301 372 9 , , PUNCT cana-1301 372 10 k.m k.m PROPN cana-1301 372 11 . . PROPN cana-1301 372 12 : : PUNCT cana-1301 373 1 applications application NOUN cana-1301 373 2 of of ADP cana-1301 373 3 graph graph NOUN cana-1301 373 4 theory theory NOUN cana-1301 373 5 in in ADP cana-1301 373 6 computer computer NOUN cana-1301 373 7 science science NOUN cana-1301 373 8 . . PUNCT cana-1301 374 1 in in ADP cana-1301 374 2 : : PUNCT cana-1301 374 3 cicsyn cicsyn NOUN cana-1301 374 4 , , PUNCT cana-1301 374 5 2011 2011 NUM cana-1301 374 6 third third ADJ cana-1301 374 7 international international ADJ cana-1301 374 8 conference conference NOUN cana-1301 374 9 on on ADP cana-1301 374 10 computational computational ADJ cana-1301 374 11 intelligence intelligence NOUN cana-1301 374 12 , , PUNCT cana-1301 374 13 communication communication NOUN cana-1301 374 14 systems system NOUN cana-1301 374 15 and and CCONJ cana-1301 374 16 networks network NOUN cana-1301 374 17 , , PUNCT cana-1301 374 18 pp pp ADJ cana-1301 374 19 . . PUNCT cana-1301 375 1 142 142 NUM cana-1301 375 2 - - SYM cana-1301 375 3 145 145 NUM cana-1301 375 4 ( ( PUNCT cana-1301 375 5 2011 2011 NUM cana-1301 375 6 ) ) PUNCT cana-1301 376 1 [ [ X cana-1301 376 2 22 22 NUM cana-1301 376 3 ] ] X cana-1301 376 4 rosenfeld rosenfeld PROPN cana-1301 376 5 , , PUNCT cana-1301 376 6 a. a. NOUN cana-1301 376 7 : : PUNCT cana-1301 376 8 fuzzy fuzzy ADJ cana-1301 376 9 graphs graph NOUN cana-1301 376 10 . . PUNCT cana-1301 377 1 in in ADP cana-1301 377 2 : : PUNCT cana-1301 377 3 zadeh zadeh PROPN cana-1301 377 4 , , PUNCT cana-1301 377 5 l.a l.a PROPN cana-1301 377 6 . . PROPN cana-1301 377 7 , , PUNCT cana-1301 377 8 fu fu PROPN cana-1301 377 9 , , PUNCT cana-1301 377 10 k.s k.s PROPN cana-1301 377 11 . . PROPN cana-1301 377 12 , , PUNCT cana-1301 377 13 shimura shimura PROPN cana-1301 377 14 , , PUNCT cana-1301 377 15 m. m. NOUN cana-1301 377 16 ( ( PUNCT cana-1301 377 17 eds ed NOUN cana-1301 377 18 ) ) PUNCT cana-1301 377 19 fuzzy fuzzy ADJ cana-1301 377 20 sets set NOUN cana-1301 377 21 and and CCONJ cana-1301 377 22 their their PRON cana-1301 377 23 applications application NOUN cana-1301 377 24 , , PUNCT cana-1301 377 25 pp pp CCONJ cana-1301 377 26 , , PUNCT cana-1301 377 27 77 77 NUM cana-1301 377 28 - - SYM cana-1301 377 29 95 95 NUM cana-1301 377 30 . . PUNCT cana-1301 378 1 academic academic ADJ cana-1301 378 2 press press NOUN cana-1301 378 3 , , PUNCT cana-1301 378 4 new new PROPN cana-1301 378 5 york york PROPN cana-1301 378 6 ( ( PUNCT cana-1301 378 7 1975 1975 NUM cana-1301 378 8 ) ) PUNCT cana-1301 379 1 [ [ X cana-1301 379 2 23 23 NUM cana-1301 379 3 ] ] X cana-1301 379 4 samantha samantha PROPN cana-1301 379 5 , , PUNCT cana-1301 379 6 s. s. PROPN cana-1301 379 7 , , PUNCT cana-1301 379 8 pal pal NOUN cana-1301 379 9 , , PUNCT cana-1301 379 10 m. m. NOUN cana-1301 379 11 , , PUNCT cana-1301 379 12 pal pal NOUN cana-1301 379 13 , , PUNCT cana-1301 379 14 a. a. NOUN cana-1301 379 15 : : PUNCT cana-1301 379 16 new new ADJ cana-1301 379 17 concepts concept NOUN cana-1301 379 18 of of ADP cana-1301 379 19 fuzzy fuzzy ADJ cana-1301 379 20 planar planar ADJ cana-1301 379 21 graph graph NOUN cana-1301 379 22 . . PUNCT cana-1301 380 1 int int NOUN cana-1301 380 2 . . PUNCT cana-1301 381 1 j. j. PROPN cana-1301 381 2 adv adv PROPN cana-1301 381 3 . . PUNCT cana-1301 381 4 res.artif res.artif PUNCT cana-1301 381 5 . . PUNCT cana-1301 382 1 intell intell PROPN cana-1301 382 2 . . PUNCT cana-1301 383 1 3(1 3(1 NUM cana-1301 383 2 ) ) PUNCT cana-1301 383 3 , , PUNCT cana-1301 383 4 52 52 NUM cana-1301 383 5 - - SYM cana-1301 383 6 59 59 NUM cana-1301 383 7 ( ( PUNCT cana-1301 383 8 2014 2014 NUM cana-1301 383 9 ) ) PUNCT cana-1301 384 1 [ [ X cana-1301 384 2 24 24 NUM cana-1301 384 3 ] ] PUNCT cana-1301 384 4 samantha samantha PROPN cana-1301 384 5 , , PUNCT cana-1301 384 6 s. s. PROPN cana-1301 384 7 , , PUNCT cana-1301 384 8 pal pal NOUN cana-1301 384 9 , , PUNCT cana-1301 384 10 m. m. NOUN cana-1301 384 11 : : PUNCT cana-1301 384 12 bipolar bipolar ADJ cana-1301 384 13 fuzzy fuzzy ADJ cana-1301 384 14 hypergraphs hypergraph NOUN cana-1301 384 15 . . PUNCT cana-1301 385 1 int int NOUN cana-1301 385 2 . . PUNCT cana-1301 386 1 j. j. PROPN cana-1301 386 2 fuzzy fuzzy PROPN cana-1301 386 3 logic logic PROPN cana-1301 386 4 syst syst NOUN cana-1301 386 5 . . PUNCT cana-1301 387 1 2(1 2(1 NUM cana-1301 387 2 ) ) PUNCT cana-1301 388 1 , , PUNCT cana-1301 388 2 17 17 NUM cana-1301 388 3 - - SYM cana-1301 388 4 28 28 NUM cana-1301 388 5 ( ( PUNCT cana-1301 388 6 2012 2012 NUM cana-1301 388 7 ) ) PUNCT cana-1301 389 1 [ [ X cana-1301 389 2 25 25 NUM cana-1301 389 3 ] ] PUNCT cana-1301 389 4 samantha samantha PROPN cana-1301 389 5 , , PUNCT cana-1301 389 6 s. s. PROPN cana-1301 389 7 , , PUNCT cana-1301 389 8 pal pal NOUN cana-1301 389 9 , , PUNCT cana-1301 389 10 m. m. NOUN cana-1301 389 11 : : PUNCT cana-1301 389 12 fuzzy fuzzy ADJ cana-1301 389 13 planar planar ADJ cana-1301 389 14 graphs graph NOUN cana-1301 389 15 . . PUNCT cana-1301 390 1 ieee ieee NOUN cana-1301 390 2 transactions transaction NOUN cana-1301 390 3 on on ADP cana-1301 390 4 fuzzy fuzzy ADJ cana-1301 390 5 systems system NOUN cana-1301 390 6 . . PUNCT cana-1301 391 1 23 23 NUM cana-1301 391 2 , , PUNCT cana-1301 391 3 1936 1936 NUM cana-1301 391 4 - - SYM cana-1301 391 5 1942 1942 NUM cana-1301 391 6 ( ( PUNCT cana-1301 391 7 2015 2015 NUM cana-1301 391 8 ) ) PUNCT cana-1301 392 1 [ [ X cana-1301 392 2 26 26 NUM cana-1301 392 3 ] ] SYM cana-1301 392 4 shinoj shinoj PROPN cana-1301 392 5 , , PUNCT cana-1301 392 6 t.k t.k PROPN cana-1301 392 7 . . PROPN cana-1301 392 8 , , PUNCT cana-1301 392 9 john john PROPN cana-1301 392 10 , , PUNCT cana-1301 392 11 s.j s.j PROPN cana-1301 392 12 . . PROPN cana-1301 392 13 : : PUNCT cana-1301 393 1 intuitionistic intuitionistic ADJ cana-1301 393 2 fuzzy fuzzy ADJ cana-1301 393 3 multisets multiset NOUN cana-1301 393 4 and and CCONJ cana-1301 393 5 its its PRON cana-1301 393 6 application application NOUN cana-1301 393 7 in in ADP cana-1301 393 8 medical medical ADJ cana-1301 393 9 diagnosis diagnosis NOUN cana-1301 393 10 . . PUNCT cana-1301 394 1 world world NOUN cana-1301 394 2 acad acad PROPN cana-1301 394 3 . . PUNCT cana-1301 395 1 sci sci PROPN cana-1301 395 2 . . PUNCT cana-1301 395 3 eng eng PROPN cana-1301 395 4 . . PROPN cana-1301 395 5 technol technol PROPN cana-1301 395 6 . . PUNCT cana-1301 395 7 6(1 6(1 NUM cana-1301 395 8 ) ) PUNCT cana-1301 395 9 , , PUNCT cana-1301 395 10 1418 1418 NUM cana-1301 395 11 - - SYM cana-1301 395 12 1421 1421 NUM cana-1301 395 13 ( ( PUNCT cana-1301 395 14 2012 2012 NUM cana-1301 395 15 ) ) PUNCT cana-1301 396 1 [ [ X cana-1301 396 2 27 27 NUM cana-1301 396 3 ] ] X cana-1301 396 4 sunitha sunitha VERB cana-1301 396 5 , , PUNCT cana-1301 396 6 m.s m.s PROPN cana-1301 396 7 . . PROPN cana-1301 396 8 , , PUNCT cana-1301 396 9 vijayakumar vijayakumar PROPN cana-1301 396 10 , , PUNCT cana-1301 396 11 a. a. NOUN cana-1301 396 12 : : PUNCT cana-1301 396 13 complement complement NOUN cana-1301 396 14 of of ADP cana-1301 396 15 a a DET cana-1301 396 16 fuzzy fuzzy ADJ cana-1301 396 17 graph graph NOUN cana-1301 396 18 . . PUNCT cana-1301 397 1 indian indian PROPN cana-1301 397 2 j. j. PROPN cana-1301 397 3 pure pure PROPN cana-1301 397 4 appl appl PROPN cana-1301 397 5 . . PUNCT cana-1301 397 6 math math NOUN cana-1301 397 7 . . PUNCT cana-1301 398 1 33 33 NUM cana-1301 398 2 , , PUNCT cana-1301 398 3 1451 1451 NUM cana-1301 398 4 - - SYM cana-1301 398 5 1464 1464 NUM cana-1301 398 6 ( ( PUNCT cana-1301 398 7 2002 2002 NUM cana-1301 398 8 ) ) PUNCT cana-1301 399 1 [ [ X cana-1301 399 2 28 28 NUM cana-1301 399 3 ] ] X cana-1301 399 4 wang wang PROPN cana-1301 399 5 , , PUNCT cana-1301 399 6 s. s. PROPN cana-1301 399 7 , , PUNCT cana-1301 399 8 zhang zhang PROPN cana-1301 399 9 , , PUNCT cana-1301 399 10 y. y. PROPN cana-1301 399 11 , , PUNCT cana-1301 399 12 yang yang PROPN cana-1301 399 13 , , PUNCT cana-1301 399 14 x. x. PROPN cana-1301 399 15 , , PUNCT cana-1301 399 16 sun sun PROPN cana-1301 399 17 , , PUNCT cana-1301 399 18 p. p. PROPN cana-1301 399 19 , , PUNCT cana-1301 399 20 dong dong PROPN cana-1301 399 21 , , PUNCT cana-1301 399 22 z. z. PROPN cana-1301 399 23 , , PUNCT cana-1301 399 24 liu liu PROPN cana-1301 399 25 , , PUNCT cana-1301 399 26 a. a. PROPN cana-1301 399 27 , , PUNCT cana-1301 399 28 yuan yuan NOUN cana-1301 399 29 , , PUNCT cana-1301 399 30 t. t. PROPN cana-1301 399 31 : : PUNCT cana-1301 399 32 pathological pathological ADJ cana-1301 399 33 brain brain NOUN cana-1301 399 34 detection detection NOUN cana-1301 399 35 by by ADP cana-1301 399 36 a a DET cana-1301 399 37 novel novel ADJ cana-1301 399 38 image image NOUN cana-1301 399 39 feature feature NOUN cana-1301 399 40 - - PUNCT cana-1301 399 41 fractional fractional ADJ cana-1301 399 42 fourier fourier NOUN cana-1301 399 43 entropy entropy PROPN cana-1301 399 44 . . PUNCT cana-1301 399 45 entropy entropy PROPN cana-1301 399 46 17(12 17(12 NUM cana-1301 399 47 ) ) PUNCT cana-1301 399 48 , , PUNCT cana-1301 399 49 82788296 82788296 NUM cana-1301 399 50 ( ( PUNCT cana-1301 399 51 2015 2015 NUM cana-1301 399 52 ) ) PUNCT cana-1301 400 1 [ [ X cana-1301 400 2 29 29 NUM cana-1301 400 3 ] ] X cana-1301 400 4 yager yager NOUN cana-1301 400 5 , , PUNCT cana-1301 400 6 r.r r.r PROPN cana-1301 400 7 . . PUNCT cana-1301 400 8 : : PUNCT cana-1301 400 9 on on ADP cana-1301 400 10 the the DET cana-1301 400 11 theory theory NOUN cana-1301 400 12 of of ADP cana-1301 400 13 bags bag NOUN cana-1301 400 14 . . PUNCT cana-1301 401 1 int int NOUN cana-1301 401 2 . . PUNCT cana-1301 402 1 j. j. PROPN cana-1301 402 2 gen gen PROPN cana-1301 402 3 syst syst PROPN cana-1301 402 4 . . PUNCT cana-1301 403 1 13 13 NUM cana-1301 403 2 , , PUNCT cana-1301 403 3 23 23 NUM cana-1301 403 4 - - SYM cana-1301 403 5 37 37 NUM cana-1301 403 6 ( ( PUNCT cana-1301 403 7 1986 1986 NUM cana-1301 403 8 ) ) PUNCT cana-1301 403 9 communications communication NOUN cana-1301 403 10 on on ADP cana-1301 403 11 applied apply VERB cana-1301 403 12 nonlinear nonlinear ADJ cana-1301 403 13 analysis analysis NOUN cana-1301 403 14 issn issn NOUN cana-1301 403 15 : : PUNCT cana-1301 403 16 1074 1074 NUM cana-1301 403 17 - - PUNCT cana-1301 403 18 133x 133x NUM cana-1301 403 19 vol vol NOUN cana-1301 403 20 31 31 NUM cana-1301 403 21 no no NOUN cana-1301 403 22 . . PUNCT cana-1301 404 1 7s 7 NOUN cana-1301 404 2 ( ( PUNCT cana-1301 404 3 2024 2024 NUM cana-1301 404 4 ) ) PUNCT cana-1301 404 5 276 276 NUM cana-1301 404 6 https://internationalpubls.com https://internationalpubls.com X cana-1301 405 1 [ [ X cana-1301 405 2 30 30 NUM cana-1301 405 3 ] ] X cana-1301 405 4 zadeh zadeh PROPN cana-1301 405 5 , , PUNCT cana-1301 405 6 l.a l.a PROPN cana-1301 405 7 . . PROPN cana-1301 405 8 : : PUNCT cana-1301 405 9 fuzzy fuzzy ADJ cana-1301 405 10 sets set NOUN cana-1301 405 11 . . PUNCT cana-1301 406 1 inf inf PROPN cana-1301 406 2 . . PUNCT cana-1301 406 3 control control PROPN cana-1301 406 4 8 8 NUM cana-1301 406 5 , , PUNCT cana-1301 406 6 338 338 NUM cana-1301 406 7 - - SYM cana-1301 406 8 353 353 NUM cana-1301 406 9 ( ( PUNCT cana-1301 406 10 1965 1965 NUM cana-1301 406 11 ) ) PUNCT cana-1301 407 1 [ [ X cana-1301 407 2 31 31 NUM cana-1301 407 3 ] ] SYM cana-1301 407 4 zadeh zadeh PROPN cana-1301 407 5 , , PUNCT cana-1301 407 6 l.a l.a PROPN cana-1301 407 7 . . PROPN cana-1301 407 8 : : PUNCT cana-1301 407 9 similarity similarity NOUN cana-1301 407 10 relations relation NOUN cana-1301 407 11 and and CCONJ cana-1301 407 12 fuzzy fuzzy ADJ cana-1301 407 13 orderings ordering NOUN cana-1301 407 14 . . PUNCT cana-1301 408 1 inf inf PROPN cana-1301 408 2 . . PUNCT cana-1301 409 1 sci sci PROPN cana-1301 409 2 . . PUNCT cana-1301 410 1 3(2 3(2 NUM cana-1301 410 2 ) ) PUNCT cana-1301 410 3 , , PUNCT cana-1301 410 4 177 177 NUM cana-1301 410 5 - - SYM cana-1301 410 6 200 200 NUM cana-1301 410 7 ( ( PUNCT cana-1301 410 8 1971 1971 NUM cana-1301 410 9 ) ) PUNCT cana-1301 411 1 [ [ X cana-1301 411 2 32 32 NUM cana-1301 411 3 ] ] X cana-1301 411 4 zhang zhang PROPN cana-1301 411 5 , , PUNCT cana-1301 411 6 w.r w.r PROPN cana-1301 411 7 . . PUNCT cana-1301 411 8 : : PUNCT cana-1301 411 9 bipolar bipolar ADJ cana-1301 411 10 fuzzy fuzzy ADJ cana-1301 411 11 sets set NOUN cana-1301 411 12 . . PUNCT cana-1301 412 1 in in ADP cana-1301 412 2 : : PUNCT cana-1301 412 3 proceedings proceeding NOUN cana-1301 412 4 of of ADP cana-1301 412 5 fuzz fuzz NOUN cana-1301 412 6 - - PUNCT cana-1301 412 7 ieee ieee NOUN cana-1301 412 8 , , PUNCT cana-1301 412 9 pp pp ADJ cana-1301 412 10 . . PUNCT cana-1301 412 11 835 835 NUM cana-1301 412 12 - - SYM cana-1301 412 13 840 840 NUM cana-1301 412 14 ( ( PUNCT cana-1301 412 15 1998 1998 NUM cana-1301 412 16 ) ) PUNCT cana-1301 413 1 [ [ X cana-1301 413 2 33 33 NUM cana-1301 413 3 ] ] X cana-1301 413 4 zhang zhang PROPN cana-1301 413 5 . . PUNCT cana-1301 414 1 w.r w.r PROPN cana-1301 414 2 . . PUNCT cana-1301 414 3 : : PUNCT cana-1301 415 1 from from ADP cana-1301 415 2 equilibrium equilibrium NOUN cana-1301 415 3 - - PUNCT cana-1301 415 4 based base VERB cana-1301 415 5 business business NOUN cana-1301 415 6 intelligence intelligence NOUN cana-1301 415 7 to to AUX cana-1301 415 8 information information NOUN cana-1301 415 9 conservational conservational ADJ cana-1301 415 10 quantum quantum NOUN cana-1301 415 11 - - PUNCT cana-1301 415 12 fuzzy fuzzy ADJ cana-1301 415 13 cryptography cryptography NOUN cana-1301 415 14 - - PUNCT cana-1301 415 15 a a DET cana-1301 415 16 cellular cellular ADJ cana-1301 415 17 transformation transformation NOUN cana-1301 415 18 of of ADP cana-1301 415 19 bipolar bipolar ADJ cana-1301 415 20 fuzzy fuzzy ADJ cana-1301 415 21 sets set NOUN cana-1301 415 22 to to ADP cana-1301 415 23 quantum quantum ADJ cana-1301 415 24 intelligence intelligence NOUN cana-1301 415 25 machinery machinery NOUN cana-1301 415 26 . . PUNCT cana-1301 416 1 ieee ieee NOUN cana-1301 416 2 transactions transaction NOUN cana-1301 416 3 on on ADP cana-1301 416 4 fuzzy fuzzy ADJ cana-1301 416 5 systems system NOUN cana-1301 416 6 ( ( PUNCT cana-1301 416 7 2017 2017 NUM cana-1301 416 8 ) ) PUNCT cana-1301 417 1 [ [ X cana-1301 417 2 34 34 NUM cana-1301 417 3 ] ] X cana-1301 417 4 zhang zhang PROPN cana-1301 417 5 . . PUNCT cana-1301 418 1 w.r w.r PROPN cana-1301 418 2 . . PUNCT cana-1301 418 3 : : PUNCT cana-1301 419 1 science science NOUN cana-1301 419 2 vs vs ADP cana-1301 419 3 sophistrya sophistrya ADJ cana-1301 419 4 historical historical ADJ cana-1301 419 5 debate debate NOUN cana-1301 419 6 on on ADP cana-1301 419 7 bipolar bipolar ADJ cana-1301 419 8 fuzzy fuzzy ADJ cana-1301 419 9 sets set NOUN cana-1301 419 10 and and CCONJ cana-1301 419 11 equilibrium equilibrium NOUN cana-1301 419 12 - - PUNCT cana-1301 419 13 based base VERB cana-1301 419 14 mathematics mathematic NOUN cana-1301 419 15 for for ADP cana-1301 419 16 ai&qi ai&qi ADJ cana-1301 419 17 . . PUNCT cana-1301 419 18 journal journal NOUN cana-1301 419 19 of of ADP cana-1301 419 20 intelligent intelligent ADJ cana-1301 419 21 & & CCONJ cana-1301 419 22 fuzzy fuzzy ADJ cana-1301 419 23 systems system NOUN cana-1301 419 24 . . PUNCT cana-1301 420 1 41 41 NUM cana-1301 420 2 , , PUNCT cana-1301 420 3 6781 6781 NUM cana-1301 420 4 - - SYM cana-1301 420 5 6799 6799 NUM cana-1301 420 6 ( ( PUNCT cana-1301 420 7 2021 2021 NUM cana-1301 420 8 ) ) PUNCT cana-1301 421 1 [ [ X cana-1301 421 2 35 35 NUM cana-1301 421 3 ] ] X cana-1301 421 4 zhang zhang PROPN cana-1301 421 5 . . PUNCT cana-1301 422 1 w.r w.r PROPN cana-1301 422 2 . . PUNCT cana-1301 422 3 : : PUNCT cana-1301 423 1 ground-0 ground-0 NUM cana-1301 423 2 axioms axiom NOUN cana-1301 423 3 vs. vs. ADP cana-1301 423 4 first first ADJ cana-1301 423 5 principles principle NOUN cana-1301 423 6 and and CCONJ cana-1301 423 7 second second ADJ cana-1301 423 8 law law NOUN cana-1301 423 9 : : PUNCT cana-1301 423 10 from from ADP cana-1301 423 11 the the DET cana-1301 423 12 geometry geometry NOUN cana-1301 423 13 of of ADP cana-1301 423 14 light light NOUN cana-1301 423 15 and and CCONJ cana-1301 423 16 logic logic NOUN cana-1301 423 17 of of ADP cana-1301 423 18 photon photon NOUN cana-1301 423 19 to to ADP cana-1301 423 20 mind mind VERB cana-1301 423 21 - - PUNCT cana-1301 423 22 light light ADJ cana-1301 423 23 - - PUNCT cana-1301 423 24 matter matter NOUN cana-1301 423 25 unity unity NOUN cana-1301 423 26 - - PUNCT cana-1301 423 27 ai&qi ai&qi ADV cana-1301 423 28 . . PUNCT cana-1301 423 29 ieee ieee PROPN cana-1301 423 30 / / SYM cana-1301 423 31 caa caa PROPN cana-1301 423 32 journal journal PROPN cana-1301 423 33 of of ADP cana-1301 423 34 automaticasinica automaticasinica PROPN cana-1301 423 35 . . PUNCT cana-1301 424 1 8(3 8(3 NUM cana-1301 424 2 ) ) PUNCT cana-1301 424 3 , , PUNCT cana-1301 424 4 534 534 NUM cana-1301 424 5 - - SYM cana-1301 424 6 553 553 NUM cana-1301 424 7 ( ( PUNCT cana-1301 424 8 2021 2021 NUM cana-1301 424 9 ) ) PUNCT cana-1301 425 1 [ [ X cana-1301 425 2 36 36 NUM cana-1301 425 3 ] ] X cana-1301 425 4 zhang zhang PROPN cana-1301 425 5 , , PUNCT cana-1301 425 6 y. y. PROPN cana-1301 425 7 , , PUNCT cana-1301 425 8 chen chen PROPN cana-1301 425 9 , , PUNCT cana-1301 425 10 s. s. PROPN cana-1301 425 11 , , PUNCT cana-1301 425 12 wang wang PROPN cana-1301 425 13 , , PUNCT cana-1301 425 14 s. s. PROPN cana-1301 425 15 , , PUNCT cana-1301 425 16 yang yang PROPN cana-1301 425 17 , , PUNCT cana-1301 425 18 j. j. PROPN cana-1301 425 19 , , PUNCT cana-1301 425 20 philips philips PROPN cana-1301 425 21 , , PUNCT cana-1301 425 22 p. p. NOUN cana-1301 425 23 : : PUNCT cana-1301 425 24 magnetic magnetic ADJ cana-1301 425 25 resonance resonance NOUN cana-1301 425 26 brain brain NOUN cana-1301 425 27 image image NOUN cana-1301 425 28 classification classification NOUN cana-1301 425 29 based base VERB cana-1301 425 30 on on ADP cana-1301 425 31 weighted weight VERB cana-1301 425 32 - - PUNCT cana-1301 425 33 type type NOUN cana-1301 425 34 fractional fractional ADJ cana-1301 425 35 fourier fourier NOUN cana-1301 425 36 transform transform NOUN cana-1301 425 37 and and CCONJ cana-1301 425 38 nonparallel nonparallel ADJ cana-1301 425 39 support support NOUN cana-1301 425 40 vector vector NOUN cana-1301 425 41 machine machine NOUN cana-1301 425 42 . . PUNCT cana-1301 426 1 int int NOUN cana-1301 426 2 . . PUNCT cana-1301 427 1 j. j. PROPN cana-1301 427 2 imaging imaging PROPN cana-1301 427 3 syst syst PROPN cana-1301 427 4 . . PUNCT cana-1301 428 1 technol technol NOUN cana-1301 428 2 . . PUNCT cana-1301 429 1 24(4 24(4 NOUN cana-1301 429 2 ) ) PUNCT cana-1301 429 3 , , PUNCT cana-1301 429 4 317 317 NUM cana-1301 429 5 - - SYM cana-1301 429 6 327 327 NUM cana-1301 429 7 ( ( PUNCT cana-1301 429 8 2015 2015 NUM cana-1301 429 9 ) ) PUNCT cana-1301 429 10 . . PUNCT cana-1301 430 1 [ [ X cana-1301 430 2 37 37 NUM cana-1301 430 3 ] ] PUNCT cana-1301 430 4 g. g. PROPN cana-1301 430 5 srinivasa srinivasa PROPN cana-1301 430 6 rao rao PROPN cana-1301 430 7 , , PUNCT cana-1301 430 8 d. d. PROPN cana-1301 430 9 madhusudhanarao madhusudhanarao PROPN cana-1301 430 10 and and CCONJ cana-1301 430 11 p. p. PROPN cana-1301 430 12 siva siva PROPN cana-1301 430 13 prasad prasad PROPN cana-1301 430 14 , , PUNCT cana-1301 430 15 simple simple ADJ cana-1301 430 16 ternary ternary ADJ cana-1301 430 17 semi semi NOUN cana-1301 430 18 - - NOUN cana-1301 430 19 rings ring NOUN cana-1301 430 20 , , PUNCT cana-1301 430 21 the the DET cana-1301 430 22 global global ADJ cana-1301 430 23 journal journal NOUN cana-1301 430 24 of of ADP cana-1301 430 25 mathematics mathematics PROPN cana-1301 430 26 & & CCONJ cana-1301 430 27 mathematical mathematical PROPN cana-1301 430 28 sciences sciences PROPN cana-1301 430 29 , , PUNCT cana-1301 430 30 9(2 9(2 NUM cana-1301 430 31 ) ) PUNCT cana-1301 430 32 ( ( PUNCT cana-1301 430 33 2016 2016 NUM cana-1301 430 34 ) ) PUNCT cana-1301 430 35 , , PUNCT cana-1301 430 36 185 185 NUM cana-1301 430 37 - - SYM cana-1301 430 38 196 196 NUM cana-1301 430 39 . . PUNCT cana-1301 431 1 [ [ X cana-1301 431 2 38 38 NUM cana-1301 431 3 ] ] X cana-1301 431 4 d. d. PROPN cana-1301 431 5 madhusudhana madhusudhana PROPN cana-1301 431 6 rao rao PROPN cana-1301 431 7 , , PUNCT cana-1301 431 8 g. g. PROPN cana-1301 431 9 srinivasa srinivasa PROPN cana-1301 431 10 rao rao PROPN cana-1301 431 11 , , PUNCT cana-1301 431 12 special special ADJ cana-1301 431 13 elements element NOUN cana-1301 431 14 in in ADP cana-1301 431 15 ternary ternary ADJ cana-1301 431 16 semi semi ADJ cana-1301 431 17 rings ring NOUN cana-1301 431 18 , , PUNCT cana-1301 431 19 international international ADJ cana-1301 431 20 journal journal NOUN cana-1301 431 21 of of ADP cana-1301 431 22 engineering engineering NOUN cana-1301 431 23 research research NOUN cana-1301 431 24 and and CCONJ cana-1301 431 25 applications application NOUN cana-1301 431 26 , , PUNCT cana-1301 431 27 4(11 4(11 NUM cana-1301 431 28 ) ) PUNCT cana-1301 431 29 ( ( PUNCT cana-1301 431 30 2014 2014 NUM cana-1301 431 31 ) ) PUNCT cana-1301 431 32 , , PUNCT cana-1301 431 33 123 123 NUM cana-1301 431 34 - - SYM cana-1301 431 35 130 130 NUM cana-1301 431 36 . . PUNCT cana-1301 432 1 [ [ X cana-1301 432 2 39 39 NUM cana-1301 432 3 ] ] PUNCT cana-1301 432 4 g. g. PROPN cana-1301 432 5 srinivasa srinivasa PROPN cana-1301 432 6 rao rao PROPN cana-1301 432 7 , , PUNCT cana-1301 432 8 d. d. PROPN cana-1301 432 9 madhusudhana madhusudhana PROPN cana-1301 432 10 rao rao PROPN cana-1301 432 11 , , PUNCT cana-1301 432 12 a a DET cana-1301 432 13 study study NOUN cana-1301 432 14 on on ADP cana-1301 432 15 ternary ternary ADJ cana-1301 432 16 semi semi ADJ cana-1301 432 17 rings ring NOUN cana-1301 432 18 , , PUNCT cana-1301 432 19 int int NOUN cana-1301 432 20 . . PUNCT cana-1301 433 1 j. j. PROPN cana-1301 433 2 of of ADP cana-1301 433 3 math math PROPN cana-1301 433 4 . . PUNCT cana-1301 434 1 archive archive NOUN cana-1301 434 2 , , PUNCT cana-1301 434 3 5(12 5(12 NUM cana-1301 434 4 ) ) PUNCT cana-1301 434 5 ( ( PUNCT cana-1301 434 6 2014 2014 NUM cana-1301 434 7 ) ) PUNCT cana-1301 434 8 , , PUNCT cana-1301 434 9 24 24 NUM cana-1301 434 10 - - SYM cana-1301 434 11 30 30 NUM cana-1301 434 12 . . PUNCT cana-1301 435 1 [ [ X cana-1301 435 2 40 40 NUM cana-1301 435 3 ] ] PUNCT cana-1301 435 4 g. g. PROPN cana-1301 435 5 srinivasa srinivasa PROPN cana-1301 435 6 rao rao PROPN cana-1301 435 7 , , PUNCT cana-1301 435 8 d. d. PROPN cana-1301 435 9 madhusudhana madhusudhana PROPN cana-1301 435 10 rao rao PROPN cana-1301 435 11 , , PUNCT cana-1301 435 12 characteristics characteristic NOUN cana-1301 435 13 of of ADP cana-1301 435 14 ternary ternary ADJ cana-1301 435 15 semi semi ADJ cana-1301 435 16 rings ring NOUN cana-1301 435 17 , , PUNCT cana-1301 435 18 int.j int.j PROPN cana-1301 435 19 . . PROPN cana-1301 435 20 of of ADP cana-1301 435 21 engg engg PROPN cana-1301 435 22 . . PUNCT cana-1301 436 1 res re NOUN cana-1301 436 2 . . PUNCT cana-1301 436 3 and and CCONJ cana-1301 436 4 mgt mgt PROPN cana-1301 436 5 . . PUNCT cana-1301 436 6 , , PUNCT cana-1301 436 7 2(1 2(1 NUM cana-1301 436 8 ) ) PUNCT cana-1301 436 9 ( ( PUNCT cana-1301 436 10 2015 2015 NUM cana-1301 436 11 ) ) PUNCT cana-1301 436 12 , , PUNCT cana-1301 436 13 3 3 NUM cana-1301 436 14 - - SYM cana-1301 436 15 6 6 NUM cana-1301 436 16 . . PUNCT cana-1301 437 1 [ [ X cana-1301 437 2 41 41 NUM cana-1301 437 3 ] ] PUNCT cana-1301 437 4 g. g. PROPN cana-1301 437 5 srinivasa srinivasa PROPN cana-1301 437 6 rao rao PROPN cana-1301 437 7 , , PUNCT cana-1301 437 8 a. a. PROPN cana-1301 437 9 nagamalleswara nagamalleswara PROPN cana-1301 437 10 rao rao PROPN cana-1301 437 11 , , PUNCT cana-1301 437 12 p.l.n p.l.n PROPN cana-1301 437 13 . . PROPN cana-1301 437 14 varma varma PROPN cana-1301 437 15 , , PUNCT cana-1301 437 16 d.madhusudhana d.madhusudhana PROPN cana-1301 437 17 rao rao PROPN cana-1301 437 18 , , PUNCT cana-1301 437 19 ch ch NOUN cana-1301 437 20 . . PROPN cana-1301 437 21 ramprasad ramprasad ADJ cana-1301 437 22 , , PUNCT cana-1301 437 23 prime prime ADJ cana-1301 437 24 biinterior biinterior PROPN cana-1301 437 25 ideals ideal NOUN cana-1301 437 26 in in ADP cana-1301 437 27 tgsr tgsr ADJ cana-1301 437 28 , , PUNCT cana-1301 437 29 malaya malaya PROPN cana-1301 437 30 journal journal PROPN cana-1301 437 31 of of ADP cana-1301 437 32 mathematika mathematika NOUN cana-1301 437 33 , , PUNCT cana-1301 437 34 vol.9 vol.9 PROPN cana-1301 437 35 , , PUNCT cana-1301 437 36 no.1 no.1 NUM cana-1301 437 37 , , PUNCT cana-1301 437 38 pp:542 pp:542 ADV cana-1301 437 39 - - PUNCT cana-1301 437 40 546 546 NUM cana-1301 437 41 , , PUNCT cana-1301 437 42 2021 2021 NUM cana-1301 437 43 . . PUNCT cana-1301 438 1 [ [ X cana-1301 438 2 42 42 NUM cana-1301 438 3 ] ] PUNCT cana-1301 438 4 g. g. PROPN cana-1301 438 5 srinivasa srinivasa PROPN cana-1301 438 6 rao rao PROPN cana-1301 438 7 , , PUNCT cana-1301 438 8 a. a. PROPN cana-1301 438 9 nagamalleswara nagamalleswara PROPN cana-1301 438 10 rao rao PROPN cana-1301 438 11 , , PUNCT cana-1301 438 12 p.l.n p.l.n PROPN cana-1301 438 13 . . PROPN cana-1301 438 14 varma varma PROPN cana-1301 438 15 , , PUNCT cana-1301 438 16 d. d. PROPN cana-1301 438 17 madhusudhana madhusudhana PROPN cana-1301 438 18 rao rao PROPN cana-1301 438 19 , , PUNCT cana-1301 438 20 ch ch NOUN cana-1301 438 21 . . PROPN cana-1301 438 22 ramprasad ramprasad ADJ cana-1301 438 23 , , PUNCT cana-1301 438 24 bi bi ADJ cana-1301 438 25 - - ADJ cana-1301 438 26 interior interior ADJ cana-1301 438 27 ideals ideal NOUN cana-1301 438 28 in in ADP cana-1301 438 29 tgsr tgsr ADJ cana-1301 438 30 , , PUNCT cana-1301 438 31 advances advance NOUN cana-1301 438 32 in in ADP cana-1301 438 33 mathematics mathematics NOUN cana-1301 438 34 scientific scientific ADJ cana-1301 438 35 journal journal NOUN cana-1301 438 36 , , PUNCT cana-1301 438 37 10 10 NUM cana-1301 438 38 ( ( PUNCT cana-1301 438 39 2021 2021 NUM cana-1301 438 40 ) ) PUNCT cana-1301 438 41 , , PUNCT cana-1301 438 42 no.3 no.3 VERB cana-1301 438 43 , , PUNCT cana-1301 438 44 pp pp CCONJ cana-1301 438 45 : : PUNCT cana-1301 438 46 1183 1183 NUM cana-1301 438 47 - - SYM cana-1301 438 48 1195 1195 NUM cana-1301 438 49 . . PUNCT