THE METRIC DIMENSION OF AMALGAMATION OF CYCLES

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1 Far East Journa of Mathematca Scences (FJMS) Voume 4 Number 00 Pages 9- Ths paper s avaabe onne at 00 Pushpa Pubshng House THE METRIC DIMENSION OF AMALGAMATION OF CYCLES H. ISWADI E. T. BASKORO A. N. M. SALMAN and R. SIMANJUNTAK Combnatora Mathematcs Research Dvson Facuty of Mathematcs and Natura Scence Insttut Teknoog Bandung Jaan Ganesha 0 Bandung 40 Indonesa e-ma: hazru_swad@yahoo.com ebaskoro@math.tb.ac.d msaman@math.tb.ac.d rno@math.tb.ac.d Department of MIPA Gedung TG anta 6 Unverstas Surabaya Jaan Raya Karungkut Surabaya 609 Indonesa 00 Mathematcs Subect Cassfcaton: 05C. Keywords and phrases: resovng set bass metrc dmenson amagamaton. Partay supported by ITB Research Grant 008. Receved March 7 00 Abstract For an ordered set W = { w w... w k } of vertces and a vertex v n a connected graph G the representaton of v wth respect to W s the ordered k-tupe r ( v W ) = ( d( v w ) d( v w )... d ( v wk )) where d ( x y) represents the dstance between the vertces x and y. The set W s caed a resovng set for G f every vertex of G has a dstnct representaton. A resovng set contanng a mnmum number of vertces s caed a bass for G. The dmenson of G denoted by dm( G ) s the number of vertces

2 0 ISWADI BASKORO SALMAN and SIMANJUNTAK n a bass of G. Let { G } be a fnte coecton of graphs and each G has a fxed vertex v o caed a termna. The amagamaton Ama{ G v o } s formed by takng a of the G s and dentfyng ther termnas. In ths paper we determne the metrc dmenson of amagamaton of cyces.. Introducton In ths paper we consder fnte smpe and connected graphs. The vertex and edge sets of a graph G are denoted by V ( G) and E ( G) respectvey. For a further reference pease see Chartrand and Lesnak []. The dstance d ( u v) between two vertces u and v n a connected graph G s the ength of the shortest u v path n G. For an ordered set W = { w w... w k } V ( G) of vertces we refer to the ordered k-tupe r ( v W ) = ( d( v w ) d( v ) w... d ( v w k )) as the (metrc) representaton of v wth respect to W. The set W s caed a resovng set for G f r ( u W ) = r( v W ) mpes u = v for a u v G. A resovng set wth mnmum cardnaty s caed a mnmum resovng set or a bass. The metrc dmenson of a graph G dm( G ) s the number of vertces n a bass for G. To determne whether W s a resovng set for G we ony need to nvestgate the representatons of the vertces n V ( G) \W snce the representaton of each w W has 0 n the th-ordnate; and so t s aways unque. The nta papers dscussng the noton of a (mnmum) resovng set were wrtten by Sater n [5] and [6]. Sater ntroduced the concept of a resovng set for a connected graph G under the term ocaton set. He caed the cardnaty of a mnmum resovng set by the ocaton number of G. Independenty Harary and Meter [7] ntroduced the same concept but used the term metrc dmenson nstead. The probem of fndng a resovng set for a gven graph can be found n many dverse areas ncudng robotc navgaton [] chemstry [] or computer scence []. As descrbed n [] the navgatng agent (a pont robot) moves from node to node n a partcuar graph space. The robot can ocate tsef by the presence of dstnctvey abeed andmark nodes n the graph. Ths suggests the probem: for a gven graph what s the smaest number of andmarks needed and where shoud they be ocated so that the dstances to the andmarks unquey determne the robot s poston n the graph?

3 THE METRIC DIMENSION OF AMALGAMATION OF CYCLES In genera fndng a resovng set for arbtrary graph s a dffcut probem. In [6] t s proved that the probem of computng the metrc dmenson for genera graphs s NP-compete. Thus researchers n ths area often studed the metrc dmenson for partcuar casses of graphs or characterzed graphs havng certan metrc dmenson. Some resuts on the ont graph and cartesan product graph have been obtaned by Caceres et a. [] Khuer et a. [] and Chartrand et a. [4]. Iswad et a. obtaned some resuts on the corona product of graphs [8 9]. Saputro et a. obtaned some resuts on the decomposton product of graphs [9]. Iswad et a. determned the metrc dmenson of antpoda and pendant free graph [0]. Further Saputro et a. found some resuts on the metrc dmenson of some type of reguar graphs [7 8]. Chartrand et a. [4] have characterzed a graphs havng metrc dmensons n and n. They aso determned the metrc dmensons of some we known fames of graphs such as paths cyces compete graphs and trees. Chartrand et a. resuts are wrtten as foows: Theorem A [4]. Let G be a connected graph of order n. () dm ( G ) = f and ony f G =. () dm( G ) = n f and ony f G =. P n () For n 4 dm( G) = n f and ony f G = Kr s ( r s ) G = Kr + K s ( r s ) or G = Kr + ( K Ks ) ( r s ). (v) For n dm( G ) =. n (v) If T s a tree other than a path then dm( T ) = σ( T ) ex( T ) where σ ( T ) denotes the sum of the termna degrees of the maor vertces of T and ex ( T ) denotes the number of the exteror maor vertces of T. The foowng dentfcaton graph G = G[ G G v v] defnton s from [4]. Defnton B. Let G and G be the nontrva connected graphs where v G and v G. An dentfcaton graph G = G[ G G v v] s obtaned from G and G by dentfyng v and v such that v = v n G. K n Posson and Zhang [4] determned the ower and upper bounds of metrc

4 ISWADI BASKORO SALMAN and SIMANJUNTAK dmenson of G [ G G v v] n terms of dm( G ) and dm( G ) as stated n the foowng theorems: Theorem C. Let G and G be the nontrva connected graphs wth v G and v G and et G = G[ G G v v]. Then dm( G ) dm( G ) + dm( G ). For the upper bound we defne an equvaence cass and bnary functon frst. For a set W of vertces of G defne a reaton on V ( G) wth respect to W by urv f there exsts a Z such that r ( v W ) = r( u W ) + ( a a... a). It s easy to check that R s an equvaence reaton on V ( G). Let [ u ] W denote the equvaence cass of u wth respect to W. Then v [] v W f and ony f r ( v W ) = r( u W ) + ( a a... a) for some a Z. For a nontrva connected graph G defne a bnary functon f G : V ( G) Z wth f G dm( G) ( v) = dm( G) f v s not a otherwse. bass vertex of G Theorem D. Let G and G be the nontrva connected graphs wth v G and v G and et G = G[ G G v v]. Suppose that G contans a resovng set W such that [ v ] W = { v}. Then dm( G) W + f ( v ) G = W W + dm( G ) + dm( G ) f v s not otherwse. a bass vertex of G In partcuar f W s a bass for G then dm( G ) + dm( G ) dm( G ) dm( G ) + dm( G ) f v s not otherwse. a bass vertex of G The foowng defnton of amagamaton of graphs s taken from []. Defnton E. Let { G } be a fnte coecton of graphs and each G has a fxed

5 THE METRIC DIMENSION OF AMALGAMATION OF CYCLES vertex v o caed a termna. The amagamaton Ama{ G v o } s formed by takng of a the G s and dentfyng ther termnas. We can consder Defnton E as the dentfcaton process for a of the members n the coecton { G } consecutvey on one dentfcaton vertex. In ths paper we determne the metrc dmenson of amagamaton of cyces.. Resuts We coud consder amagamaton of cyces on n; that s Ama{ G v } where o G = C n for a. In ths partcuar amagamaton the choce of vertex v o s rreevant. So for smpfcaton we can denote ths amagamaton by ( C n ) t where t denotes the number of cyces C n. For t = the graphs ( C n ) are the cyces C n. For n = the graphs ( C ) t are caed the frendshp graphs or the Dutch t- wndms [5]. In ths paper we consder a generazaton of ( C ) where the cyces under consderaton may be of dfferent engths. We denote ths amagamaton by Ama{ Cn } t t. We ca every C n (ncudng the termna) n Ama{ C n } as a eaf and a path P n obtaned from C n by deetng the termna as a nontermna path. Throughout ths paper we w foow the foowng notatons and abes for cyces nontermna path and vertces n Ama{ C n }. For odd n n = k + k and x the termna vertex we abe a vertces n each eaf C n such that C n = xv ths w gve the nontermna path P n = v v w w w x k k k v v v k w k w k n w For even n n = k + k and x the termna vertex we defne the abes of a vertces n each eaf C n as foows: C n = xv t. v v u w w w x k k k

6 4 ISWADI BASKORO SALMAN and SIMANJUNTAK whch eads to the foowng abeng of the nontermna path P n = v v v k u w k w k w The foowng four emmas gve us some propertes of the members of a resovng set of amagamaton of cyces. Lemma. Let S be a resovng set of Ama{ C n }. Then Pn S for each. n Proof. If S has no vertex n P for some then the vertces v n. w n S P w have the same dstances to S namey d( v v) = d( w v) v. S Therefore r( v S) = r( w ) a contradcton. ~ Lemma. Let S be a resovng set of Ama{ C n }. If n s even n 4 and n S = then n S { u }. P P Proof. Suppose u S for some. Snce the remanng vertces of S w not be n P d( v v) = d( w v) v S. Therefore we have r( v S ) = r( w S ) n k k a contradcton wth S beng a resovng set. ~ Lemma. Let S be a resovng set of Ama{ C n }. For any even n n 4 ( P P ) S. n n Proof. Let n = k + n = k + and k k. By Lemma ( P Pn ) S. Suppose ( Pn P ) S =. n By consderng Lemma k k n and the symmetry property we have (( P P ) S ) = { v v } wth r k n n and s k. Then d( w v) = d( w v) v. Therefore r( w S) = S r( w S ) a contradcton; whch gves ( Pn P ) S n. ~ Now we w determne the metrc dmenson of amagamaton of cyces Ama{ C n }. r s

7 THE METRIC DIMENSION OF AMALGAMATION OF CYCLES 5 Theorem. If Ama{ C n } s an amagamaton of t cyces that conssts of t number of odd cyces and t number of even cyces then t dm( Ama{ Cn }) = t + t t = 0 otherwse. Proof. Let B be a bass of Ama{ C n }. We abe the eafs n n such a way that C wth odd engths are abeed by =... t and n even engths are abeed by = t +... t + t =. t C s of Ama{ C } n C n wth Case. For t = 0. Thus a n s t are odd et n = k + k. By usng Lemma for every resovng set S of Ama{ C n } we w have S t. Hence for every bass B of Ama{ C n } B t. Choose a set t S = S = wth k S = { v }. We w show that S s a resovng set of Ama{ C }. The representatons of vertces of Ama{ C n } that s not n S wth respect to S are r ( x S) = ( k... k ) t n r( vr S ) = ( k + r... k r... + r) coord. of S wth r k and r( wr S ) = ( k + r... k r r) wth r k. coord. of S Snce the r s are a dstnct a of these representatons are dstnct. Hence S s a resovng set of amagamaton Ama{ C n }. Snce a bass B s a mnmum resovng set B t. Therefore B = t. Case. For t. Consder an arbtrary resovng set of Ama{ C }. By Lemma every nontermna path P n wth n odd has at east one vertex of S; and by Lemma every nontermna path P n wth n even has at east two vertces of S except for one of them whom can ony contan one vertex of S. These ead to S t + t for every resovng set S; and so B t + t. n

8 6 ISWADI BASKORO SALMAN and SIMANJUNTAK Next we w show that B t + t. Choose a set S = t S wth k S = { v } wth t n = k + and k = and t + + S = { v } wth n k + and k S t + k t + = t + t + = { v u } wth t t n = k and k. + + The representatons of the other vertces of Ama{ C n } wth respect to S are r ( x S) = ( k... k k k k +... k k + ) t t + t + t + t t r( vr S ) = ( k + r... k r... + r k r t coord. of S + + r r + r + ) wth t + and r k s t + t + t + s r( v S) = ( k + s... k + s k + s k k s k s +... coord. of S + s + s + ) wth t + t and s k r( wr S ) = ( k + r... k r r k r t coord. of S + + r r + r + ) wth t + and r k s t + t + t + s r( w S) = ( k + s... k + s k + s k k s k s +... coord. of S + s + s + ) wth t + t and s k

9 THE METRIC DIMENSION OF AMALGAMATION OF CYCLES 7 and t + k t t t t + t + r( w S) = ( k + k +... k + k + k + k + t k t... ) k t k t By drect nspecton a of these representatons are dstnct. Therefore S s a resovng set. Snce a bass B s a mnmum resovng set S B t + t. ~ We ustrate both of Cases and n Fgures and. For Case we consder Ama{ C n } wth = n = n = 5 and n = 5. By choosng a bass B = { v v w } we have the coordnates of a vertces other than the bass vertces as foows: = r ( x B) = ( ) r( w B) = ( ) r( w B) ( ) = r ( v B) = ( ) r( v B) = ( ) r( v B) ( 4 ) = r ( w B) = ( ) r( w B) ( 4 ). For Case we consder Ama{ C n } wth = n = 5 n = 6 and n = 8. By choosng a bass B = { v v v w } we have the coordnates of a vertces other than the bass vertces n Ama{ C n } as beow: Fgure. The coordnates of amagamaton Ama{ C n } wth t = 0 t = t = n = n = 5 and n = 5. = r ( v B) = ( 4) r( w B) = ( 4) r( w B) ( 4 4 5) = r ( x B) = ( ) r( v B) = ( 4) r( u B) ( 5 5 6) = r ( w B) = ( 4) r( w B) = ( 4 4 5) r( v B) ( 4)

10 8 ISWADI BASKORO SALMAN and SIMANJUNTAK = r ( v B) = ( 5 5 ) r( u B) = ( 6 6 ) r( w B) ( ) = r ( w B) ( ). Fgure. The amagamaton Ama{ C n } wth t t = t = n = 5 n = 6 and n = 8. One of the natura questons we coud pose after provng Theorem s: Are there any bass other than the bass we constructed n the proof of Theorem? We w answer the queston by dentfyng a bases of Ama{ C n }. The foowng emmas are needed to fnd such bases: Lemma 4. Let B be a bass of Ama{ C n }. If Pn B = and Pn B = for some then P n { } B = v or { k w } k or { P } n B = v or k { w }. k Proof. By Lemma both B {}. v Assume that Pn = n and n cannot be even. Let n B = {} u and v a P u = wth a k and v = v b wth b k. Then d( w z) = d( w z) z B; a contradcton wth B beng a bass. The resut foows by usng symmetry property. ~ Lemma 5. Let Ama{ C n } be an amagamaton of t cyces that conssts of t number of odd cyces and t number of even cyces and B be a bass of Ama{ C n }. If n s odd and t then P n B = { v } or { w }. k k Proof. Snce n s odd by usng Lemma and Theorem B. Pn =

11 THE METRIC DIMENSION OF AMALGAMATION OF CYCLES 9 Snce t there s at east one eaf C wth even vertces such that B n Pn =. Let n B = {} u and n B = { v}. Assume that P P v a u = wth a k and v = v b wth b k. We have d( w z) = d( w z) z B. By symmetry property the resut foows. ~ Lemma 6. Let B be a bass of Ama{ C n }. If n s even and Pn B = { a b} then nether { a b} { v v... v } nor { a b} { w... }. k w w k Proof. By Theorem there s at east one eaf C n wth n even and such that n B =. Let n B = { u}. For the contrary assume that { a b} P k P k { v v... v } and u { v v... v }. Then d( w z) = d( w ) z B; a z contradcton wth B beng a bass. By usng symmetry property the resut foows. ~ Now we are ready to dentfy a bases of Ama{ C n }. We w consder exacty two cases; frst t = 0 and second t. Theorem. If Ama{ C n } s an amagamaton of t cyces that conssts of t number of odd cyces and t number of even cyces then t t ( ) n t = 0 { } = Ama C = n t t ( nt + ) ( ( ) ( )) C n C k otherwse = t + where C ( b a) s the tota number of combnatons of b obects taken a. Proof. Let t be a number of odd cyces and t be number of even cyces of Ama{ C n }. Case. t = 0. By usng Lemma and Theorem every nontermna path P n w contan ony one bass vertex. By usng Lemma 4 for every par of odd eaves P n and P n one of them say P n contans a bass vertex v or k w. Then for t k = we can dentfy the bass B of Ama{ Cn } B = { a b} where

12 0 ISWADI BASKORO SALMAN and SIMANJUNTAK a { v k w } k and b V ( Pn ) or a V ( Pn ) and b { v k w }. k Hence the number of dfferent bases of Ama{ C n } s ( n ) + ( n ) 4. By usng smar reason we can generaze for t = t > and we have the number of t t dfferent bases of Ama{ C n } s ( n ). = Case. t. By usng Lemma Lemma and Theorem every nontermna path wth even vertces w have one bass vertex and every nontermna path wth odd vertces w have two bass vertces except one ony have one bass vertex. We abe the nontermna path P n s of Ama{ C n } n such a way that P n wth even engths are abeed by =... t P n wth odd engths and contans two bass vertces are abeed by = t +... t + t = t and P n wth odd ength and contans one bass vertex are abeed by = t +. By usng Lemma 5 the bass vertces of nontermna paths wth even vertces havng ether s or w s. A nontermna path P wth odd vertces havng one bass vertex v k k n can have every z V ( P n )\{ u } as ts bass vertex. By usng Lemma 6 every nontermna path P n wth odd vertces havng two bass vertces cannot have both ther bass vertces n ether { v v... v } or { w w... w }. Hence we can count the number of dfferent bases of Ama{ C n } as foows: k t t ( nt + ) ( C( n ) C( k )) = t + where C ( b a) s the tota number of combnatons of b obects taken a. ~ References [] J. Caceres C. Hernando M. Mora M. L. Puertas I. M. Peayo and C. Seara On the metrc dmenson of some fames of graphs Eectronc Notes n Dscrete Math. (005) 9-. [] K. Carson Generazed books and Cm -snakes are prme graphs Ars. Combn. 80 (006) 5-. [] G. Chartrand and L. Lesnak Graphs and Dgraphs rd ed. Chapman and Ha/CRC 000. k

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