On Metric Dimension of Two Constructed Families from Antiprism Graph

Transcription

1 Mah S Le 2, No, ) Mahemaal Sees Leers A Ieraoal 203 NSP Naural Sees Publhg Cor O Mer Dmeso of Two Cosrued Famles from Aprm Graph M Al,2, G Al,2 ad M T Rahm 2 Cere for Mahemaal Imagg Tehques ad Dep Mah Sees, Uversy of Lverpool, UK 2 Dep of Mahemas, Naoal Uversy of Compuer & Emergg Sees, Peshawar, Paa Emal: muraza_psh@yahooom, arqrahm@uedup, goharal@uedup Reeved: Ja 202, Reved: 3 May 202, Aeped: 2 Jul 202 Publhed ole: Ja 203 Absra: I h paper we ompue he mer dmeso of wo famles of graphs osrued from aprm graph Keywords: Mers dmeso, bas, resolvg se, aprm Iroduo For a oeed graph G, he dae d u, v) bewee wo veres u, v V G) he legh of a shores pah bewee hem G Le W w, w,, w } be a ordered se of veres of G ad le v 2 be a verex of G, he represeao of he verex v wh respe o W, deoed by r v W) he - uple d v, w ), d v, w ),, d v, w )) If d veres of G have d represeaos wh 2 respe o W, he W alled a resolvg se or loag se for G [2] A resolvg se of mmum ardaly alled a mer bas for G ad h ardaly he mer dmeso of G dm G), For a gve ordered se of veres W w, w,, w } of a graph G, he h ompoe of r v W) 2 0 f ad oly f v w Thus, o show ha W a resolvg se suffes o verfy ha r x W) r y W) for eah par of d veres x, y V G) \ W Movaed by he problem of uquely deermg he loao of a ruder a ewor, he oep of mer dmeso was rodued by Slaer [0,] ad suded depedely by Harary ad Meler [3] Applaos of h vara o he avgao of robos ewors are dussed [8] ad applaos o hemry [2] whle applaos o problems of paer reogo ad mage proessg, some of whh volve he use of herarhal daa sruures are gve [9] I [4,5,6] Imra e al proved he mer dmeso of some famles of ovex polyopes I [2] Charrad e al proved ha a graph has mer dmeso f ad oly f a pah, hee pahs o veres osue a famly of graphs wh osa mer dmeso Smlarly, yle wh 3 veres also osue suh a famly of graphs as her mer dmeso 2 I [] J Caeres e al proved ha: dm p m C 2, ) 3, f, odd; oher we

2 2 M Al e al: O mer dmeso of wo osrued famles Se prms D are he rvale plae graphs obaed by he aresa produ of he pah P 2 wh a yle C ; hey also osue a famly of 3 - regular graphs wh osa mer dmeso Javad e al proved [7] ha he graph of aprm A osues a famly of regular graphs wh osa mer dmeso ad dm A ) 3 for every 5 I h paper, we exed h sudy by osderg wo famles of graph whh are osrued from aprm The aprm A, 3, oss of a ouer -yle a a2 a, a er -yle b b2 b, ad a se of spoes b ad b a,,2,3,, where ae modulo a The graph H osrued from he graph A as follows: We delee he edges a a from A For eah,2,,, we rodue ew veres ad d for a ad b respevely For eah,2,,, rodue ew edges b, a d, d ad b, where ae modulo The graph R osrued from he graph,2,,, we rodue ew veres ad rodue ew edges b, a d, d, d d ad 2 Ma Resuls d for a ad b where A as follows: We delee he edges a a from A For eah b respevely For eah,2,, ae modulo Theorem: Le 6 be a eger he dm H )3 Proof We dguh wo ases Case ): 2, 3, IN We osder W b, b, b } V H ) We show ha W a 2 resolvg se for V H ) For h we fd he represeaos of he veres of V H )\W wh respe o W The represeaos of he veres are as follows;, 2, ), for 3 ; r b W ) 2,2 2, ), 2 r r a,2, ), W ),, 2 ), 2 2,3 2, ),,, ),,, ), W ),,), 2,2 2, ), for ; for 2 ; for 2 ; 2 ; ; 2 2,2, ), ;,, 2 ), 2 ; r d W ),,2), ; 2 2,2 3, ), 2 Noe ha here are o wo veres havg he same represeaos mplyg ha dm ) 3 H

3 M Al e al: O mer dmeso of wo osrued famles 3 Now we show ha dm H ) 3, by provg ha here o resolvg se W wh W 2 We have he followg possbles; ) Boh veres belog o b :,2,, } V H ) Whou loss of geeraly we suppose ha oe resolvg verex b ad he oher b, 2 ) For 2 r b, b }) r a b, b }), ) ad for +, r a b, b }) r a b, b }),), a orado 2) Boh veres belog o :,2,, } V H ) Whou loss of geeraly we suppose ha oe resolvg verex, ad he oher, 2 ) The for 2 r b, }) r a, }) 2, ) for r a, }) r a, }),), a orado 3) Boh veres belog o a :,2,, } V H) We suppose ha oe resolvg verex a ad he oher a, 2 ) The for 2 r a, a }) r b a, a }) 2, ) d :,2,, } V H We suppose ha oe resolvg verex For 2 r a d, d}) r b d, d}) 3, 2) r d, d}) r d, d}), a orado :,2,, } V H ad for, r b a, a }) r b a, a }),), a orado 4) Boh veres belog o ) d ad he oher d, 2 ) ad for, 2,) 5) Oe verex belog o b ) ad aoher verex belog o :,2,, } V H) Whou loss of geeraly we suppose ha oe resolvg verex b ad he oher, ) For r a b, }) r b b, }), ) ad for, r a b, }) r a b, }),2), a orado 6) Oe verex belog o b :,2,, } V H ) ad aoher verex belog o a :,2,, } V H) Whou loss of geeraly we suppose ha oe resolvg verex b ad he oher a, ) The for r a b, a}) r b, a}), ) ad for, r b b, a }) r a b, a }),2), a orado 2 :,2,, } V H 7) Oe verex belog o ) ad aoher verex belog o a :,2,, } V H) Whou loss of geeraly we suppose ha oe resolvg verex ad he oher a, ) For r a, a}) r b, a}) 2, ) ad for, r a, a}) r, a}),2), smlarly for +, r a, a }) r, a }),2), a orado 2 8) Oe verex belog o :,2,, } V H) o :,2,, } V H) he oher d, ) For r a a, d}) r b a, d}) 2, he represeao r a, d}) r 2 a, d}) 2,), a orado 9) Oe verex belog o :,2,, } V H) :,2,, } V H) a ad aoher verex belog d Whou loss of geeraly we suppose ha oe resolvg verex a ad 2) ad for b ad aoher verex belog o d Whou loss of geeraly we suppose ha oe resolvg verex b ad he oher d, ) For r a b, d}) r b b, d}), 2) ad for he represeao

4 4 M Al e al: O mer dmeso of wo osrued famles r a b, d}) r b, d}),), smlarly for, r a b, d }) r b, d }),), a orado 2 :,2,, } V H :,2,, } V H) 0) Oe verex belog o ) ad aoher verex belog o d Whou loss of geeraly we suppose ha oe resolvg verex ad he oher d, ) For r a, d}) r b, d}) 2, 2) ad for, r a, d}) r, d}),), smlarly for, r a, d }) r, d }),), a orado 2 Hee, from above follows ha here o resolvg se wh wo veres for V H ) mplyg ha dm H) 3 Case): 2, 3, IN Le W b, b, b } V H ) We show ha W a 2 2 resolvg se for V H) For h we fd he represeaos of veres of V H ) \ W wh respe o W The represeaos of he veres are as follow;,, ), for ; r a W ),, 2 ), for 2 ; 2 2,3 2, ), for 2 r b r, 2,2 ), W ) 2 2,2 3, 2 ),,2, ),,, 3 ), W ),,), 2 3,2 4, ), for 3 ; 3 ; 2 ; 2; 3 2,2, 2), for ; r d W ),, 3 ), for 2 ; 2 3,4 2, ), for 2 Proeedg o same le as ase) we oe ha here are o wo veres havg he same represeaos, mplyg ha dm H ) 3 Also as ase), a be show ha here o se W wh W 2, suh ha W a resolvg se for V H ) for 6 ad odd Thus, dm ) 3 Hee dm H) 3 From ase) ad ase) we ge dm H) 3 Theorem: Le 6 be a eger he dm R ) 3 Proof We dguh wo ases: Case) 2, 3, IN Suppose W b, b, b } V R ) We show ha W a resolvg se H 2 for V R ) For h we fd he represeaos of he veres of V R ) \ W wh respe o W The represeaos of he veres are as follows;

5 M Al e al: O mer dmeso of wo osrued famles 5 r b r, 2, ), W ) 2,2 2, ),,2, ), W ),, 2 ), 2 2,3 2, ), for 3 ; 2 for ; for 2 ; for 2 r a,, ),,, ), W ),,), 2,2 2, ), ; 2 ; ; 2 2,2, ), ;,, 2 ), 2 ; r d W ),,2), ; 2 2,2 3, ), 2 We oe ha here are o wo veres havg he same represeaos mplyg ha dm ) 3 Now we show ha dm R ) 3, by provg ha here o resolvg se W wh W 2 We have he followg possbles, ) Boh veres belog o b :,2,, } V R ) Whou loss of geeraly we suppose he resolvg veres b ad b, 2 ) For 2 r b, b }) r a b, b }), ) ad for, r a b, b }) r a b, b }),), a orado 2) Boh veres belog o :,2,, } V R ) We suppose ha oe resolvg verex ad he oher, 2 ) For 2 r b, }) r a, }) 2, ) ad for, r b, }) r a, }) 2, ), a orado :,2,, } V R 3) Boh veres belog o a ) We suppose ha oe resolvg verex a ad he oher a, 2 ) For 2 r a, a }) r b a, a}) 2, ) ad for r b a, a }) r b a, a }),), a orado 4) Boh veres belog o d :,2,, } V R ) We suppose ha oe resolvg verex d ad he oher d, 2 ) For 2 r a d, d}) r b d, d}) 3, 2) ad for, r d, d }) r d, d }) 2,), a orado :,2,, } V R 5) Oe verex belog o b ) ad aoher belog o :,2,, } V R ) R Whou loss of geeraly we suppose ha oe resolvg verex b ad he oher, ) For r a b, }) r b b, }), ) ad for +, r a b, }) r a b, }),2), a orado 6) Oe verex belog o b :,2,, } V R ) ad aoher belog o a :,2,, } V R ) Whou loss of geeraly we suppose ha oe resolvg verex b ad he oher a, )

6 6 M Al e al: O mer dmeso of wo osrued famles For r r a b, a}) r b, a}), ) ad for b, a }) r a b, a }),2), a orado b 2 :,2,, } V R 7) Oe verex belog o ) ad aoher belog o a :,2,, } V R ) Whou loss of geeraly we suppose ha oe resolvg verex ad he oher a, ) For r a, a}) r b, a}) 2, ) ad for, r a, a}) r, a}),2), smlarly for r a, a }) r, a }),2), a orado 2 :,2,, } V R 8) Oe verex belog o a ) ad aoher belog o d :,2,, } V R ) Whou loss of geeraly we suppose ha oe resolvg verex a ad he oher d, ) For r a a, d}) r b a, d}) 2, 2) ad for, r a, d }) r a, d }) 2,), a orado 2 :,2,, } V R 9) Oe verex belog o b ) ad aoher belog o d :,2,, } V R ) Whou loss of geeraly we suppose ha oe resolvg verex b ad he oher d, ) For r a, }), }), b d r b b d 2) ad for r a b, d}) r b, d}),), smlarly for ha r a b, d }) r b, d }),), a orado 2 :,2,, } V R 0) Oe verex belog o ) ad aoher belog o d :,2,, } V R ) Whou loss of geeraly we suppose ha oe resolvg verex ad he oher d, ) For r a, d}) r b, d}) 2, 2) ad for r a, d}) r, d}),), smlarly for ha r a, d }) r, d }),), a orado 2 Hee, from above follows ha here o resolvg se wh wo veres for V R ) mplyg ha dm R ) 3 Case ): 2, 3, IN Cosder W b, b, b } V R ) We show ha W a 2 2 resolvg se for V R ) For h we fd he represeaos of veres of V R ) \ W wh respe o W The represeaos of he veres are as follow;,, ), for ; r a W ),, 2 ), for 2 ; 2 2,3 2, ), for 2 r b r, 2,2 ), W ) 2 2,2 3, 2 ),,2, ),,, 3 ), W ),,), 2 3,2 4, ), for 3 ; 3 ; 2 ; 2; 3

7 M Al e al: O mer dmeso of wo osrued famles 7 2,2, 2), for ; r d W ),, 3 ), for 2 ; 2 3,4 2, ), for 2 Proeedg o same le as ase) we observe ha here are o wo veres havg he same represeaos, mplyg ha dm R ) 3 Also as ase), a be show ha here o se W wh W 2, suh ha W a resolvg se for V R ) for 6 ad odd Thus, dm ) 3 Hee dm R ) 3 From ase) ad ase) we ge dm R ) 3 3 Coluso I h paper suded he mer dmeso of wo famles of graphs whh are he exeso of he aprm graph We have see ha he mer dmeso of hese graphs fe ad does o deped o he order of he graph ad oly hree veres appropraely hose suffe o resolve all he veres of hese graphs R Referees [] J Caeres, C Herado, M Mora, I M Pelayo, M L Pueras, C Seara, D R Wood, O he mer dmeso of aresa produ of graphs, SIAM J D Mah ) [2] G Charrad, L Eroh, M A Johso, O R Oellerma, Resolvably graphs ad mer dmeso of a graph, D Appl Mah ) 99-3 [3] F Harary, R A Meler, O he mer dmeso of a graph, Ars Comb 2 976) 9-95 [4] M Imra, A Q Bag, A Ahmad, Famles of plae graphs wh osa mer dmeso, o appear Ulas Mah [5] M Imra, S A Bohary, A Q Bag, O famles of ovex polyopes wh osa mer dmeso, Compu Mah Appl ) [6] M Imra, A Q Bag, M K Shafq, Adrea Feovova, Classes of ovex polyopes wh osa mer dmeso, Ulas, Mah, press [7] I Javad, M T Rahm, K Al, Famles of regular graphs wh osa mer dmeso, Ulas Mah ) 2-33 [8] K Karlraj, J V Verold, O equaable olorg of helm ad gear graphs, Ieraoal J Mah Comb, 4 200) [9] R A Meler, I Tomesu, Mer bases dgal geomery, Compuer Vo, Graphs, ad Image Proessg, ) 3-2 [0] P J Slaer, Leaves of rees, Cogress Numer 4 975) [] P J Slaer, Domag ad referee ses graphs, J Mah Phys S )