Some Algorithms on Cactus Graphs

Transcription

1 Annals of Pure and Appled Mahemacs Vol, No, 01, ISSN: X (P), (onlne) Publshed on 4 December 01 wwwresearchmahscorg Annals of Some Algorhms on Cacus Graphs Kalyan Das Ramnagar College, Depal, Purba Mednpur Mdnapore 7110, Inda emal: kalyand380@gmalcom Receved 3 Sepember 01; acceped 18 December 01 Absrac A cacus graph s a conneced graph n whch every block s eher an edge or a cycle In hs paper we gve a bref dea how o desgn some opmal algorhms on cacus graphs n O(n) me, where n s he oal number of verces of he graph The cacus graph has many applcaons n real lfe problems, specally n rado communcaon sysem Keywords: Desgn of algorhms, analyss of algorhms, all-par shores pahs, domnang se, -nbd coverng se, ndependen se, spannng ree, colorng, cacus graph AMS Mahemacs Subec Classfcaon (010): 05C78, 68R10 1 Inroducon In graph heory, a cacus graph s a conneced graph n whch any wo smple cycles have a mos one verex n common Equvalenly, every edge n such a graph belongs o a mos one smple cycle Equvalenly, every block (maxmal subgraph whou a cuverex) s an edge or cycle Le G = ( V, be a fne, conneced, undreced smple graph of n verces m edges, where V s he se of verces and E s he se of edges A verex u s called a cuverex f removal of u and all edges ncden on u dsconnec he graph A conneced graph whou a cuverex s called a non-separable graph A block of a graph s a maxmal non-separable subgraph A cycle s a conneced graph (or subgraph) n whch every verex s of degree wo A block whch s a cycle s called a cyclced block A cacus graph s a conneced graph n whch every block s eher an edge or a cycle A weghed graph G s a graph n whch every edge s assocaes wh a wegh Whou loss of generaly we assume ha all weghs are posve A weghed cacus graph s a weghed, conneced graph n whch every block conanng wo verces s an edge and hree or more verces s a cycle Cacus graph were frs suded under he name of Husm rees, besowed on hem by Frank Harary and George Eugene Unlenbeck n honour of prevous work of 114

2 Some Algorhms on Cacus Graphs hese graphs by Kod Husm Cacus graph has many applcaons These graphs can be used o model physcal seng where a ree would be napproprae Examples of such seng arse n elecommuncaons when consderng feeder for rural, suburban and lgh urban regons [33] and n maeral handlng nework when auomaed guded vehcles are used [34] Moreover, rng and bus srucures are ofen used n local area neworks The combnaon of local area nework forms a cacus graph Because of varous applcaons n real lfe suaon and elecommuncaon problem, cacus graphs have exensve suded durng las decade Some well known problems lke all-par shores pah problem, domnaon problem, colorng and labelng problems, coverng problems ec are solved n polynomal me on cacus graphs effcenly Lo of algorhms have been desgn o solve varous graph heorec problems, some of hem are avalable n [48-59] To solve some problems on cacus graphs, a ree s consruced, called ree, whch s descrbed below T Fgure 1: A weghed cacus graph G Formaon of he ree T In hs hess we use a mehod n whch blocks and cuverces of he graph G are deermned usng DFS echnque and here afer form an nermedae graph G e, G = ( V, E ) where V { B, B, K, B } and E = {( B, B ) : =,, = 1,, K, N, B and = 1 N B are adacen blocks } 115 / Now he ree T s consruced from G as follows: We dscard some suable edges from G n such a way ha he resulan graph becomes a ree The procedure for such reducon s gven below: Le us ake any arbrary verex of G, conanng a leas wo cuverces of G, as roo of he ree T and mark All he adacen verces of hs roo are aken as chldren of level one and are marked If here are edges beween he verces of hs level, hen hose edges are dscarded Each verces of level one s consdered one by

3 Kalyan Das one o fnd he verces whch are adacen o hem bu unmarked These verces are aken as chldren of he correspondng verces of level one and are placed a level wo These chldren a level wo are marked and f here be any edge beween hem hen hey are dscarded Ths process s connued unl all he verces are marked Thus he ree T = ( V, E ) where V { B, B, K, B } and E E s obaned For convenence, we refer he verces of 116 = 1 N T as nodes We noe ha each B n he node of hs ree s a block of he graph G = ( V, The paren of he node ree T wll be denoed by Paren( B ) 3 Dfferen Problems on Cacus Graphs and s Soluons 31 Compuaon of All-Pars Shores Pahs on Weghed Cacus Graphs Le G = ( V, be a fne, conneced, undreced, smple graph of n verces and m edges, where V s he se of verces and E s he se of edges A pah of a graph G s an alernang sequence of dsnc verces and edges whch begns and ends wh verces n G The lengh of a pah s he sum of he weghs of he edges n he pah A pah from verex u o v s a shores pah f here s no oher pah from u o v wh lower lengh The dsance d ( u, beween verces u and v s he lengh of shores pah beween u and v n G For any general graph wh n verces, soluon o he all-par shores pah problem akes O ( n 3 ) me [1] A lo of work have been done n mprovng hs runnng me usng randomzaon and probablsc mehods for general as well as specal knds of graphs Ahua eal [] have gven a faser sequenal usng Radx heap and Fbonacc heap for sngle source shores pah problem n O ( m + n logc ) me for a nework wh n verces and m edges and non-negave negers are coss bounded by C In [47], Sedal has gven an O ( M ( n) logn) me sequenal algorhm for all-par shores pah problem for an undreced and unweghed arbrary graph wh n verces, where M (n) s he me (bes value of M (n) s O ( n 376 ) necessary o mulply wo n n marces of small negers Alon e al [3] have repored a sub-cubc algorhm for compung APSP on γ dreced graph wh edge lengh whch requre O( Mn ) me, where γ = (3+ ω)/, ω < 3 and M s he larges edge lengh Gall and Margal [16] have mproved he ω+ 1)/ ω dependence of M and have also gven an O( M n logn) algorhm for undreced graph Rav e al [4] have gven a sequenal algorhm o solve all-par shores pah (APSP)on nerval graph n O( n ) me Pal and Bhaacharee n [41] have gven an O ( n ) me algorhm for fndng he dsance beween all par of verces on nerval graphs In hs problem, we selec a specfed verex x and fnd a block whch conans x We consruc a ree T akng hs block as roo Afer consrucng he ree we

4 Some Algorhms on Cacus Graphs frs compue he dsance beween x and all oher verces n hs roo Then we compue he dsance from x o verces (oher han x ) of he blocks correspondng o he nodes n level one as follows Le B be a node a level one Then e s s enry pon We compue he dsances of every verex v of B from e and addng d x, e ) wh hese dsances we oban he dsance and so shores pah of he ( verces of he block B from x Smlarly, we compue he dsance from x o oher verces of he blocks a level one For he nodes, e, blocks of he remanng levels, he dsance from x can be compued by he same process In general, le us consder a block B a level and assume ha he dsance beween x and all verces of he blocks a level 1 have been calculaed The enry pon of B s e Then d ( x, e ) s known as e belongs o a block a level 1 We now compue d( e, for all v B Then d( x, u) = d( x, e ) + d( e, u) for all u B Ths deermne he dsance beween x and any verex of any block a level Ths procedure akes O (n) me Thus o compue all-par shores pah on weghed cacus graphs, O ( n ) me s requred 3 Fndng a Mnmum Domnang Se Le G = ( V, be a fne, conneced, undreced, smple graph of n verces and m edges, where V s he se of verces and E s he se of edges A subse D of V s sad o domnae V f every verex n V D s adacen o a leas one verex n D In hs case D s called a domnang se of he graph G The se D s called a mnmum domnang se f he cardnaly of D s mnmum among all domnang ses of he graph G The problem of deermnng a mnmum cardnaly domnang se has been dscussed n [11], and has obvous applcaon n he opmum locaon of facles n a nework When resrced o nerval graphs, he mnmum domnang se problem along wh several relaed varans, becomes polynomal me solvable [6, 7] Krasch e al [3] frs presened polynomal me algorhm whch akes O ( n 6 ) me for domnaon problems on cocomparably graphs These algorhms are vald for he cardnaly case only In [43], we ge a fas algorhm for domnaon problems on permuaon graphs whch akes O ( m + n) me In hs problem we consruc a ree T usng he blocks and cuverces and applyng he Euler Tour we oban a sequence of nodes Consder he nodes of ha sequence one by one and fnd he domnang verces from each node Hence we form an algorhm whch descrbes a raversng from leaf node o he roo If here exs any subree on he way of raversng hen we raverse all he branches of he subree from leaf o roo excep s roo and mee he roo when all he branches are raversed For any leaf node and for an neror node we apply dfferen 117

5 Kalyan Das mehod and oban he domnang ses for each node Ths process akes O (n) me for compung mnmum domnang se on cacus graph G 33 Fndng a Mnmum -Neghbourhood Coverng se The k-neghbourhood-coverng ( k -NC) problem s a varan of he domnaon problem A verex x k -domnaes anoher verex y f d( x, y) k A verex z s k -NC of an edge ( x, y) f d( x, z) k and d( y, z) k e, he verex z k -domnaes boh x and y Conversely f d( x, z) k and d( y, z) k hen he edge ( x, y) s sad o be k -neghbourhood covered by he verex z A se of verces C V s a k -NC se f every edge n E s k -NC by some verces n C The k -NC number ρ ( G, k) of G s he mnmum cardnaly of all k -NC ses For k = 1, Lehel e al [35] have presened a lnear me algorhm for compung ρ (G,1) for an nerval graph G Chang e al [1] and Hwang e al [], have presened lnear me algorhms for compung ρ (G,1) for a srongly chordal graph provded ha srong elmnaon orderng s known Hwang e al [] also proved ha k NC problem s NP-complee for chordal graphs Mondal e al [38] have presened a lnear me algorhm for compung -NC problem for an nerval graph Also a lnear me algorhm for rapezod graph has presened by Ghosh e al [17] In hs Problem we consruc a ree T usng blocks and cuverces of G Thereafer applyng Euler our on ha ree we oban a sequence of nodes There are wo ypes of nodes, some are leaf nodes and some are neror nodes Dependng upon he number of verces of cycles and pahs we deermne he number of coverng verces from each node as well as he graph G Thus he algorhm whch fnds he -neghbourhood coverng (-NC) se of he graph G n O (n) me The algorhm also akes O (n) space 34 Fndng a Maxmum Wegh -Colour Se on Weghed Cacus Graphs The graph colourng problem (GCP) plays a cenral role n graph heory and has drec applcaons n real lfe problems [5], and s relaed o many oher problems such as meablng [13, 37], frequency assgnmen [19] ec A k-colourng (assgnmen) of an undreced graph G = ( V,, where V s he se of V = n verces and E V V he se of edges, s a mappng α : V {1,, K, k} ha assgns a posve neger from {1,, K, k} (represenng he colours) o each verex We say ha a colourng s feasble f he end nodes of every edge n E have assgned dfferen colours, e, for all ( u, E, α( u) α( We call conflc he suaon when wo nodes beween whch an edge exss have he same colour assocaed o hem We say ha a colourng s nfeasble f a leas one conflc occurs Alernavely o he formulaon as an assgnmen problem, he GCP can also be represened as a paronng problem, n whch a feasble k-colourng corresponds o a 118

6 Some Algorhms on Cacus Graphs paron of he se of nodes no k ses C, C, 1 K, Ck such ha no edge exss beween wo nodes from he same colour class The graph colourng problem s NP-complee Hence, we need o use approxmae algorhmc mehods o oban soluons close o he absolue mnmum n a reasonable execuon me The maxmum wegh k-colourable Subgraph (MWKC) problem s relaed o he followng problem The npu o hs problem conss of an neger number k and an undreced graph G = ( V,, where each verex v has a non-negave wegh w v The goal s o pck a subse V V, such ha here exss a colourng c of G [V ] wh k colours, and among all such subses, he value w v V v, w v s maxmum Ths problem s NP-hard for general graph even for spl graph [0] The maxmum wegh k-colourng problem s same as he maxmum wegh k-ndependen se (MWKIS) problem The maxmum k-ndependen se problem on G s o deermne k dson ndependen ses S, S, 1 K, Sk n G such ha S1 US UKS k s maxmum The MWKIS problem s NP-complee for general graphs [0] Many work on colourng problem has been done prevously Local search n large neghbour and eraed local search for GCP are descrbed n [9, 4] The maxmum wegh -colourng problem or he maxmum wegh -ndependen se (MWIS) problem, whch s a specal case of he (MWKIS) problem, s also NP-complee for general graphs and applcaons have been suded n he las decade [3, 4, 36] In [3], Hsao eal have solved he wo-rack assgnmen problem by solvng he MIS problem on crcular arc graph In [36], Lou e al have solved he maxmum -chan problem on a gven pon se, whch s he same as he MWIS problem on permuaon graph In hs problem we fnd odd and even blocks and form block-cuverex graph G usng he odd blocks only from he graph G Nex represen G n erms of edge wegh(wegh of he cuverex) and verex wegh(wegh of he mnmum wegh verex) and form a ree T The mehod of fndng maxmum wegh -coloured se BQ s o delee such a verex from each odd block so ha mnmum wegh s dcarded Also n hs problem we fnd mnmum wegh feedback verex se Here we selec mnmum wegh verces or cuverex from boh even and odd blocks Thus he algorhm whch fnds he -coloured se as well as he mnmum feedback verex of he graph G akes O (n) me The algorhm also akes O (n) space 35 Fndng a Maxmum Independen Se and Maxmum -Independen Se Le G = ( V, be a fne, conneced, undreced, smple graph of n verces and m edges, where V s he se of verces and E s he se of edges A subse of he verces of a graph G = ( V, s an ndependen se f no wo verces n hs subse 119

7 Kalyan Das are adacen The maxmum ndependen se (MIS) problem on G s o deermne a maxmum sze ndependen se on G The MIS problem s NP-complee for general graphs [18], bu can be solved n polynomal me for many specal graphs [8] The maxmum k-ndependen se (MKIS) problem on G s o deermne k dson ndependen ses S, S, 1 K, Sk n G such ha S1 US UKSk s maxmum The MKIS problem s NP-complee for general graphs [0] The maxmum -ndependen se (MIS) problem, whch s a specal case of he MKIS problem, s also NP-complee for general graphs and applcaons have been suded n he las decade [3, 36] In [3], Hsao e al have solved he wo-rack assgnmen problem by solvng he MIS problem on crcular arc graph In [36], Lou e al have solved he maxmum -chan problem on a gven pon se, whch s he same as he MIS problem on permuaon graph In hs problem, MKIS problem s consdered on a non-weghed cacus graph for k = 1 and k = In hs problem a ree T s consruced usng blocks and cuverces of he graph G There afer apply Euler Tour o fnd a sequence of nodes o consder one by one from leaf o roo node For leaf nodes and neror nodes separae echnques are used o fnd verces for ndependen se For he -ndependen se problem one verex from each odd cycle s removed so ha alernae verces from cycles and pahs of he graph G form -ndependen se Thus he algorhms for he above wo problems ake O (n) me 36 Fndng Maxmum and Mnmum Hegh Spannng Trees Le G = ( V, be a fne, conneced, undreced, smple graph of n verces and m edges, where V s he se of verces and E s he se of edges A ree s a conneced graph whou any crcus A ree T s sad o be a spannng ree of a conneced graph G f T s a subgraph of G and T conans all verces of G The longes dsance ld ( u, and dsance d ( u, beween wo verces u and v are he lengh lp ( u, and ρ ( u, n G f such pahs exs Noe ha ld ( u, u) = 0, ld ( u, = ld( v, u) and ld ( u, ld( u, w) + ld( w, Also d ( u, u) = 0, d ( u, = d( v, u) and d ( u, d( u, w) + d( w, The elongaon of a verex u n a graph G s he longes dsance from verex u o a verex furhes from u e, el( u) = max{ ld( u, : v V} Verex v s sad o be a furhes verex of u f ld ( u, = el( u) The eccenrcy of a verex u n a graph G s he longes dsance from he verex u o a verex furhes from u e, e( u) = max{ d( u, : v V} In a ree, a verex v s sad o be a level l f v s a a dsance l from he roo The hegh of a ree s he maxmum level whch s occurred n he ree A graph may have more han one spannng ree The hegh of a spannng ree T of a graph G s denoed by H ( T, G) A maxmum hegh spannng ree s a 10

8 Some Algorhms on Cacus Graphs spannng ree whose hegh s maxmum among all spannng rees of a graph The hegh of he maxmum hegh spannng ree of a graph G s denoed by H max ( G) = max{ el( u) : u V} Suppose v be he verex for whch H max (G) s aaned and v s furhes verex, hen he longes pah e, lp ( v, v ) s called as maxmum hegh pah ( v, v ) and denoed by MHP ( v, v ) A mnmum hegh spannng ree s a spannng ree whose hegh s mnmum among all spannng ree of a graph The hegh of he mnmum hegh spannng ree of a graph G s denoed by H mn ( G) = mn{ e( u) : u V} The verex x for whch H mn ( G) = e( x) s called he cener of G Some relaed works are dscussed here: In [44], a spannng ree of maxmal wegh and bounded radus s deermned from a complee non-orened graph G = ( V, wh verex se V and edge se E wh edge wegh n O ( n ) me, n s he oal number of verces n G In [39], he mnmum spannng ree problem s consdered for a graph wh n verces and m edges They nroduced randomzed search heurscs o fnd mnmum spannng ree n polynomal me wh ou employng global echnques of greedy algorhms In [6], he auhors fnd a spannng ree T ha mnmzes DT = Max(, ) EdT (, ) where d T (, ) s he dsance beween and n a graph G = ( V, The mnmum resrced dameer spannng ree problem s o fnd spannng ree T such ha he resrced dameer s mnmzed I s solved n O ( logn) me In [7], he mnmum dameer spannng ree problem on graphs wh non-negave edge lenghs s deermned whch s equvalen for fndng shores pahs ree from absolue 1-cener problem of he general graph s solvable n O ( mn + n log n) me [34] In hs Problem, we fnd he maxmum hegh spannng ree by fndng he elongaon and he longes pah MHP ( u, Then deleng one edge from each cycle whch s no consder durng he calculaon of elongaon and MHP ( u, he maxmum hegh of he spannng ree s obaned whose hegh s equal o MHP ( u, Also we fnd he mnmum hegh spannng ree by fndng he eccenrcy and he radus of he graph G and deleng one edge from each cycle so ha he radus s he mnmum hegh of he spannng ree These algorhms fnd he maxmum hegh spannng ree and mnmum hegh spannng ree n O (n) me 37 L(,1)-labellng of cacus graphs The L(,1) -labellng of a graph G s an absracon of assgnng neger frequences o rado ransmers such ha he ransmers ha are one un of dsance apar receve frequences ha dffer by a leas wo, and ransmers ha are wo uns of dsance apar receve frequences ha dffer by a leas one The span of an L(,1) 11

9 Kalyan Das -labellng s he dfference beween he larges and he smalles frequences assgned o he verces The L(,1) -labellng number of a graph G, denoed by λ (G), s he leas neger k such ha G has an L(,1) -labellng of span k Several resuls are known for L(,1) -labellng of graphs, bu, o he bes of our knowledge no resul s known for cacus graph The lower bound for λ (G) s + 1, whch s acheved for he sar K 1, Grggs and Yeh [15] prove ha λ ( G) + for general graph and mprove hs upper bound o λ ( G) + 3 when G s 3- conneced and λ ( G) when G s dameer (dameer graph s a graph where all nodes have eher dsance 1 or each oher) Jonas [9] mproves he upper bound o λ ( G) + 4 f, by consrucve labellng schemes Chang and Kuo [10] furher decrease he bound o + Furher, Kral and Skrekovsk [31] mproves hs bound λ ( G) + 1 for any graph G The bes known resul ll dae s λ ( G) + due o Goncalves [14] To label he verces of a cacus graph, we frs label he verces of all nduced subgraphs of he cacus graph We obaned he followng resuls Le H be a subgraph of G, hen obvously λ( H) λ( G) [10] If G and H are wo graphs and f V I = φ hen G V H λ( GU H) = max{ λ( G), λ( H )} and λ ( G + H ) = max{ VH 1, λ( G)} + max{ VH 1, λ( H )} + [10] Also, λ( GU v H ) max{ λ( G), λ( H )}, where { v} = V G I V H For any sar graph K, λ ( K1, ) = + 1, whch s equal o n, where n s he 1, number of verces For any cycle C of lengh n, λ ( ) = 4 = + [15] Suppose a graph G conans wo cycles λ ( CnU Cm) = 5 = + 1 v 0 n C n C n and 1 C m oned by a cuverex v 0, hen Le a graph G 1 conans n number of rangles wh a common cuverex Then λ ( G1 ) = + 1 or + accordng as n s even or odd, where s he degree of he cuverex Le a graph G conans n number of cycles of lengh 3 and m number of cycles of lengh 4 If hey have a common cuverex wh degree, hen λ ( G) = + 1 Le G be a graph whch conans fne number of cycles of any lengh and fne number of edges If v 0 be he common cuverex wh degree hen λ ( G) = + 1

10 Some Algorhms on Cacus Graphs Le G be a graph, conans a cycle of any lengh and fne number of edges, hey have a common cuverex v 0 If be he degree of he cuverex hen, λ ( G) = + 1 Le G be a graph conans a cycle of any lengh and each verex of he cycle has anoher cycle of lengh hree If s he degree of G hen λ ( G) = + 3 For any caerpllar graph he value of λ les beween + 1 and + Le G 1 and G be wo cacus graphs If λ ( G1) and + 1 λ ( G) + 3, hen, + 1 λ ( G) + 3, where G = G1U G v The me complexy of he proposed algorhm o label a cacus graph usng L(,1)-labellng echnque akes O (n) me, where n s he oal number of verces of he cacus graph 38 L(0,1)-labellng of cacus graphs An L(0,1) -labellng of a graph G s an assgnmen of nonnegave negers o he verces of G such ha he dfference beween he labels assgned o any wo adacen verces s a leas zero and he dfference beween he labels assgned o any wo verces whch are a dsance wo s a leas one The span of an L(0,1) -labellng s he maxmum label number assgned o any verex of G The L(0,1) -labellng number of a graph G, denoed by λ ( ), s he leas neger k such ha G has 0,1 G 13 an L(0,1) -labellng of span k Ths labellng has an applcaon o a compuer code assgnmen problem The ask s o assgn neger conrol codes o a nework of compuer saons wh dsance resrcons Some resuls are avalable on L ( h, k) -labellng problem Here we dscuss some parcular cases When h = 0 and k = 1 hen we ge L(0,1) -labellng problem Several resuls are known for L(0,1) -labellng of graphs, bu, o he bes of our knowledge no resul s known for cacus graph In hs secon, he known resul for general graphs and some relaed graphs of cacus graph are presened The upper bound for λ ( ) of any graph G s λ (G [30], 0,1 G 0,1 ) where s he degree of he graph Here we label he verces of a cacus graph by L(0,1) -labellng and have shown ha, 1 λ ( G 0,1 ) for a cacus graph, where s he degree of he graph G Here we sar he labellng by he labellng he subgraphs of he cacus graph And we obaned some resuls whch are saed below If we label a sar graph K by L(0,1) -labellng, hen we ge 1, λ 0,1( K1, ) = 1 For any cycle C n of lengh n, λ 0,1( Cn ) = 1, when n = 4k, where k s a posve neger, and λ ( C ) = for oher cases [8] 0,1 n Le G be a graph whch conans wo cycles and hey have a common

11 Kalyan Das cuverex If be he degree of G, hen, λ 0,1( G) =, when wo cycles are of lengh 3 and 1, for ohers Ths resul s rue for he graph conans n number of cycles of any lenghs, oned wh a common cuverex For he graph whch conans fne number of cycles of any lengh and fne number of edges, hen λ ( G) = 1 If he graph s a sun graph wh n verces, 0,1 0,1 ( n hen we proved ha λ S ) = = 1 Suppose G conans a cycle of any lengh and each verex of he cycle has anoher cycle of any lengh, hen 1 λ ( G 0,1 ) I s proved for caerpllar, lobser and ree he value of λ 0,1 s 1 Fnally, by arrangng all he resuls, we can conclude ha for a cacus graph 1 λ ( G 0,1 ) 39 (,1) -oal labellng of he cacus graph A (,1)-oal labellng of a graph G = ( V, s an assgnmen of negers o each verex and edge such ha: () any wo adacen verces of G receve dsnc negers, () any wo adacen edges of G receve dsnc negers, and () a verex and s ncden edge receve negers ha dffer by a leas The span of a (,1)-oal labellng s he maxmum dfference beween wo labels The mnmum span of a (,1)-oal labellng of G s called he (,1)-oal number and denoed by λ ( G ) Movaed by frequency channel assgnmen problem Grggs and Yeh [15] nroduced he L(,1) -labellng of graphs The noaon was subsequenly generalzed o he L( p, q) -labellng problem of graphs Le p and q be wo non-negave negers An L( p, q) -labellng of a graph G s a funcon c from s verex se V (G) o he se {0,1, K, k} such ha c( x) c( y) p f x and y are adacen and c( x) c( y) q f x and y are a dsance The L( p, q) -labellng number λ ( ) of G s he smalles k such ha G has an L( p, q) p, q G -labellng c wh max{ c ( v V ( G)} = k Ths labellng s called (,1)-oal labellng of graphs whch nroduced by Have and Yu [1] and generalzed o he (d,1)-oal labellng, where d 1 be an neger A k - (d,1)-oal labellng of a graph G s a funcon c from V ( G) E( G) o he se {0,1, K, k} such ha c( u) c( f u and v are adacen and c( u) c( e) d f a verex u s ncden o an edge e The (d,1)-oal number, denoed by λ d (G), s he leas neger k such ha G has a k - (d,1) -oal labellng I s shown n [40] ha for any cacus graphs, + 1 λ,1 + 3 Now n hs secon, we label he verces and edges of a cacus graphs G by (,1) -oal 14

12 Some Algorhms on Cacus Graphs labellng and s shown ha + 1 λ + [1] We label he verces and edges of a cacus graph by (,1)-oal labellng procedure and have shown ha, + 1 λ ( G) + for a cacus graph, where s he degree of he graph G Frs we label he verces of dfferen subgraphs of cacus graph by (,1) -oal labellng If H s a subgraph of G, hen λ H ) λ ( ) For any sar graph K 1,, ( G λ ( K1, ) = + If we label he cycle C n, hen we ge, λ ( Cn ) = 4 When a graph conans wo or more cycles oned wh a common cuverex, hen he value of λ equal o +, f all cycles are of even lenghs and + 1, for ohers Le G be a graph, conans a cycle of any lengh and fne number of edges and hey have a common cuverex v 0 If be he degree of he cuverex, hen λ ( G) =, f he cycle s of even lengh and + 1, for oher cases + For any sun S n, he value of λ s + If graph s obaned from S n by addng an edge o each of he penden verex of S n, hen λ = + for ha graph For a graph whch conans a cycle of any lengh and each verex of he cycle conan anoher cycle of any lengh, hen λ equal o + The λ value of he pah, caerpllar graph and lobser are same and equal o + One of he mporan resul of (,1) -oal labellng of cacus graph s descrbed below Le G 1 and G be wo cacus graphs If λ ( G1 ) 1 + and + 1 λ ( G) +, hen + 1 λ ( G) +, G s he unon of wo graphs G 1 and G, hey have only one common verex v and max { 1, } 1 + Combnng all he resuls, we conclude ha If s he degree of a cacus graph G, hen + 1 λ ( G) + REFERENCES 1 Aho, A, Hopcrof, J and Ullman, J, The Desgn and Analyss of Compuer Algorhms, Addson-Wesley, Readng, MA, 1974 Ahua, R K, Mehlhorn, K, Orln, JB, and Taran, R E, Faser algorhms for he shores pah problem, J ACM, 37 () (1990) Alon, N, Gall, Z, and Margal, O, On he exponen of he all-pars shores pah problem, proc 3h IEEE FOCS, IEEE (1991) Ahua, RK, Magnan, TL and Sharma, D, Very large-scale neghbourhood search Inernaonal Transacons n Operaonal Research, 7 (000) Allen, M, Kumaran, G and Lu, T, A combned algorhm for graph-colourng n regser allocaon, Proceedng of he Compuaonal Symposum on Graph 15

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