An Investigation on Some theorems on K-Path Vertex Cover

Size: px

Start display at page:

Download "An Investigation on Some theorems on K-Path Vertex Cover"

Brandon Cummings
5 years ago
Views:

1 Global Journal of Pure and Appled Mathematcs. ISSN Volume 1, Number (016), pp Research Inda Publcatons An Inestgaton on Some theorems on K-Path Vertex Coer P. Durga Bhaan Assocate Professor of Mathematcs, Amrta Sa Insttute of Scence & Technology, Partala, Krshna Dstrct, Andhra Pradesh, Inda. K. Vjay Kumar Assocate Pof Mathematcs, Samsruth Engneerng & Technology, Ghateswar, Hyderabad, Andhra Pradesh, Inda. S. Satyanarayana Professor, Edtor-In-Chef-IJCMI, Department of Computer Scence & Engneerng, KL Unersty, Guntur, Andhra Pradesh, Inda. Abstract In Ths Paper we examne upper lmts on the estmaton of (G) the base cardnalty of a ertex K-Path Coer n G and ge a few estmaton and precse estmatons of (G). We addtonally demonstrate that n m (G) ( 1)( ) for eery graph G wth n ertces and m edges. The base cardnalty of ertex K-Path coer concepts wdely used n secure communcaton n wreless sensor networs (WSNs). Keywords: Graph, Base cardnalty of a ertex K-Path Coer, WSN. Introducton and motaton In ths paper, A subset S of ertces of a graph G s called a K-path ertex coer f eery path of order n G contans at least one ertex from S. Denote by (G) the mnmum cardnalty of a K-path ertex coer n G. It s shown that the problem of

2 1404 P. Durga Bhaan et al determnng (G) s NP-hard for each, whle for trees the problem can be soled n lnear tme. We nestgate upper bounds on the alue of (G) and prode seeral estmatons and exact alues of (G). We also proe that 3 (G) (n + m)/6, for eery graph G wth n ertces and m edges we consder fnte graphs wthout loops and multple edges and use standard graph theory notatons [3]. In partcular, by the order of a path P we understand the number of ertces on P whle the length of a path s the number of edges of P. We ntroduce a new graph narant that generalzes the ntensely studed concept of ertex coer. Our research s motated by the followng problem [] related to secure communcaton n wreless sensor networs (WSNs). The topology of WSN can be represented by a graph, n whch ertces represent sensor deces and edges represent communcaton channels between pars of sensor deces. Tradtonal securty technques cannot be appled drectly to WSN, because sensor deces are lmted n ther computaton, energy, communcaton capabltes. Moreoer, they are often deployed naccessble areas, where they can be captured by an attacer. In general, a standard sensor dece s not consdered as tamper resstant and t s undesrable to mae all deces of a sensor networ tamper-proof due to ncreasng cost. Therefore, the desgn of WSN securty protocols has become a challenge n securty research. We focus on the Canas scheme [1, 3] whch should prode data ntegrty n a sensor networ. The -generalzed Canas scheme [5] guarantees data ntegrty under the assumpton that at least one node whch s not captured exsts on each path of the length 1 n the communcaton graph. The scheme combnes the propertes of cryptographc prmtes and the networ topology. The model dstngushes between two nds of sensor deces protected and unprotected. The attacer s unable to copy secrets from a protected dece. Ths property can be realzed by mang the protected dece tamper-resstant or placng the protected dece at a safe locaton, where capture s problematc. On the other hand, an unprotected dece can be captured by the attacer, who can also copy secrets from the dece and gan control oer t. Durng the deployment and ntalzaton of a sensor networ, t should be ensured, that at least one protected node exsts on each path of the length 1 n the communcaton graph [15]. The problem to mnmze the cost of the networ by mnmzng the number of protected ertces s formulated n [15]. Formally, let G be a graph and let be a poste nteger. A subset of ertces S V(G) s called a t-path ertex coer f eery path of order n G contans at least one ertex from S. We denote by (G) the mnmum cardnalty of a t-path ertex coer n G. Clearly, -path ertex coer corresponds to the well-nown ertex coer (a subset of ertces such that each edge of the graph s ncdent to at least one ertex of the set). Therefore (G) s equal to the sze of the mnmum ertex coer of a graph G. Moreoer, the alue of 3 (G) corresponds to the so-called dssocaton number of a graph [8, 14] defned as follows. A subset of ertces n a graph G s called a dssocaton set f t nduces a sub graph wth maxmum degree 1. The number of ertces n a maxmum cardnalty

3 An Inestgaton on Some theorems on K-Path Vertex Coer 1405 dssocaton set n G s called the dssocaton number of G and s denoted by dss (G). Clearly, 3 (G) = V(G) dss(g). A related colorng s the t-path chromatc number, whch s the mnmum number of colors that are necessary for colorng the ertces of G n such a way that each color class forms a p -free set [1, 1] and t s easy to see that ( G) 1 ( G) V ( G) ( G) Snce the mnmum ertex coer problem s NP-hard [5], t s not surprsng that so s the problem of determnng for each 3. We prode detals n Secton actually; we reduce the mnmum ertex coer problem to the mnmum t-path ertex coer problem. Howeer, snce the queston whether there s a t-path ertex coer of sze at most t can be expressed n the monadc second order logc, by famous Courcelle s theorem [8], the mnmum t-path ertex coer problem can be soled n lnear tme on graphs wth bounded tree wdth, e. g. trees, seres parallel graphs, outer planar graphs, etc. In Secton 3, we determne the exact alue of for trees and present a lnear tme algorthm whch returns an optmal soluton for trees. Then Secton 4s deoted to outer planar graphs. We present a tght upper bound for (G). In Secton 5, we prode seeral estmatons for the sze of mnmum -path ertex coer on degree of ts ertces. Fnally, we proe that 3 (G) (n + m)/6, for eery graph G wth n ertces and m edges. Theorem 1. For eery graph G wthout solated ertces: 1 1 ( G) V ( G) 1 d( U) u ( G) Proof. Let us arbtrarly order the ertces of G, and let us start wth S beng the empty set. One by one let us add ertex to S unless at least two of ts negh bounds are already there. The probablty that wll eentually land n S, s, (1 d ( ) because t s the probablty that n random orderng of ertces of G, precedes each of ts negh bounds except for one. From ths one can deduce that the expected sze of the set S s at least Snce S s a 1degenerated graph, S s a forest. ( G) (1 d ( )) 1 Usng Theorem we get ( s) V ( s). Fnally, to construct a t-path ertex coer

4 1406 P. Durga Bhaan et al for G we put nto soluton V ( G) \ S and all the ertces formng the mnmum t-path ertex coer of S. Theorem. For eery poste ratonal number a >1, a, bg, and the smallest b poste nteger, such that a b + there exsts a graph G wth aerage degree d(g)= a b and (G) n m ( 1)( ) Proof. Denote by Hn a complete graph on n ertces wthout edges of one perfect matchng (assume n s een). Clearly V (Hn) = n and n( n 1) n n( n ) EH ( n) Clearly dss ( H n) ) =, because any arbtrary three ertces of Hn form a path of order three. Therefore 3 ( ( H n) n, for n. We construct the graph G as the dsjont unon of x components H y components H+4. We let x = (b a + b) ( + ) and y = (a b)( + 1). Frst, we erfy the aerage degree of the graph G. There are x( + ) ertces of degree and y( + 4) ertces of degree + ; hence we get d (G) x( )( ) y( 4)( ) x( ) y( 4) (b a b)( )( )( ) ( a b)( 1)( 4)( ) (b a b)( )( ) ( a b)( 1)( 4) (b a b)( ) ( a b) ( ) (b a b) ( a b) 4 b ( a b)( ) ( a b)( ) ( a b) b = a b.

5 An Inestgaton on Some theorems on K-Path Vertex Coer 1407 Regular graphs In the man theorem of ths secton we shall use the followng result of Erdős and Galla [ 5]. Theorem. 1 ([ 5]). If G s a graph on n ertces that does not contan a path of n order, then t cannot hae more than ( ) edges. Moreoer, the bound s acheed when the graph conssts of dsjont clques on 1 ertces. Theorem.. Let and d 1 be poste ntegers. Then, for any d-regular graph G, the followng holds: d ( G) d VG ( ) Proof. Let S VG be a ertex K-path coer and T= VG \ S. Let E s, ET be the set of edges wth both end ertces s S and T, respectely. Let EST be the set of edges wth one end ertex n S and the second ertex n T. Then obously EG ( ) 1 d V ( G) ES EST ET. Snce G s d-regular, ds E S E ST. therefore S 1. smlarly d EST E T = dt. snce the graph nduced on the set ET does not contan a path of T ( ) order. accordng to theorem 3. 1 we hae ET. Combnng all the preous formulas, we mmedately hae S 1 1 d EST ( d T ET ) ( d T T ) T d d d Then S ( T ) 1 T 1 d d S S d d and S d d VG ( ). E ST Path ertex coer for trees We begn our nestgaton wth the class of trees, that present an mportant underlyng communcaton topology for WSN. Courcelle s theorem [8] guarantees the exstence of a lnear tme algorthm, bascally usng dynamc programmng. In ths secton, we descrbe such an algorthm n detal, and we use t then to dere a sharp upper bound (T ) ( T) / for an arbtrary tree T.

6 1408 P. Durga Bhaan et al In order to smplfy our consderaton, consder that the nput tree s rooted at a ertex u. By properly rooted sub tree we denote a sub tree T, nduced by a ertex and ts descendants (wth respect to u as the root) and satsfes the followng propertes: 1. T contans a path on ertces;. T \ does not contan a path on ertces. The algorthm PVCP Tree systematcally searches for a properly rooted treet, puts nto a soluton and remoes T from the nput tree T. Functon PVCP Tree (T, ) Input: A tree T on n ertces and a poste nteger ; Output: A -path ertex coer S of T ; Form an arbtrary ertex u T, mae T rooted n u; S := ; whle T contans a properly rooted sub tree T do S := S T := T \ T ; return S; Theorem Let T be a tree and be a poste nteger. The algorthm PVCP Tree (T, ) returns an T ( ) T ( ) optmal -path ertex coer of T of sze at most. Therefore, ( T) Proof. Frst, we shall argue that PVCP Tree returns an optmal soluton. We proe ths by nducton on the number of ertces n the tree T. If T does not contan any path on ertces, the empty set s the optmal soluton. Suppose T contans a path on ertces. Let T be a properly rooted sub tree of T. Snce any -path coer of T contans a ertex of follows by nducton. T t follows that ( T) ( T \ T ) 1 and hence the result V T ( ) To proe that the returnng set S contans at most ertces, we argue that each loop of the algorthm nserts one ertex nto S and remoes from T one properly rooted sub tree hang at least ertces. _ Concernng the tme complexty of the algorthm, t s straghtforward to mplement the algorthm PVCP Tree such that the returnng set S n computed n lnear tme. Outerplanar graphs In ths secton we study the 3-path ertex coer of outerplanar graphs. n Theorem 3. Let G be an outerplanar graph of order n. Then 3 ( G)

7 An Inestgaton on Some theorems on K-Path Vertex Coer 1409 Proof. We proe the statement by nducton on the number of ertces of G. Let H be a maxmal outerplanar graph such that G s ts sub graph satsfyng V(H) = V(G). (Recall that H s maxmal outerplanar f H s outerplanar, but addng any edge destroys that property. ) It s easy to see that H s -connected and all ts nner faces are trangles. Obsere that eery t-path ertex coer n H s a -path ertex coer n G, snce G s a sub graph of H. Therefore, t suffces to fnd a 3-path ertex coer n H n 3 n of sze at most Obously, f H conssts of a sngle trangle, then 3( H) 1 Assume H has at least four ertces. Snce H s a -connected outerplanar graph, the closed tral boundng the outer face contans each ertex of H precsely once, hence H s Hamltonan. Let 1,,... n be the cyclc orderng of ertces of H along the Hamltonan cycle. Colour a ertex whte, f the degree of n H s, otherwse colour t blac. Snce all the nner faces of H are trangles, there are no two consecute whte ertces, unless H conssts of a sngle trangle, whch s excluded. Hence, the whte ertces nduce an ndependent set. The edge 1s called good, f t has a whte end ertex, otherwse t s bad. If all the edges ncdent wth the outer faces are good, t mples that n s een, and half of the ertces are whte. Then the set of all blac ertces s a 3-path ertex coer of sze n snce the whte ertces form an ndependent set. Assume there s a bad edge,. e. there s at least one par of consecute blac ertces, say, 1We clam that there s an edge of H such that, 1and form a trangular face adjacent to the outer face, all the three ertces, 1, and are blac and all the edges 1, 3,..., 1, (ndces taen modulo n) are good For the sae of contradcton, suppose that ths s not the case. Let e =, 1 be a bad edge. There s a unque trangular face of H ncdent both wth and 1 ; let be the thrd ertex ncdent wth the face. It s easy to see that must be blac. The ertces, 1 and, 1 cut the Hamltonan cycle nto the edge, 1 and two paths; let (e) be the smaller of ther lengths. Let e be the bad edge wth (e) mnmal; let 0 be the correspondng blac ertex such that together wth 0 and 0 1 they form a trangular face of H. Wthout loss of generalty assume that P = 0 1,..., 0 s the path of length (e). Obsere that P contans at least one whte ertex, thus, t contans good edges. If all the edges 01 0,..., are good, we are done. Otherwse, we fnd a bad edge e 1 = 11 of P. Agan, there s a unque trangular face of H ncdent both wth 1 and 11 ; let 1 be the thrd (blac) ertex ncdent wth the face. Snce 0 1, 0 s a -cut of H separatng the ertces 0, from the rest of the graph, we hae 1 0 1, 0 But then ( e1) ( e0), a contradcton wth the choce of e 0

8 1410 P. Durga Bhaan et al Therefore, there s an edge e = of H such that, 1, and form a trangular face adjacent to the outer face, all the three ertces, 1 and are blac, and all the edges 1,..., 1 are good. It means that on the p,..., 1 blac and whte ertces alternate, thus, (e) = s for some nteger S Let W be the set of whte ertces on P, let B be the set of blac ertces on P, ncludng 1 and. Obously W = Sand B = s + 1; ertces from W nduce an ndependent set n H and 1 s adjacent to precsely one of them. Let S be a 3-path VG ( ') ertex coer n the graph H ' H \, 1,..., of sze at most = ns gen by nducton. Then S s' ( B \ 1 ) s a 3-path ertex n s n coer n H of sze at most 1 S The bound n n Theorem 3 s the best possble, snce t s easy to fnd outerplanar n graphs wth. 3 Consder an arbtrary -connected outerplanar graph G. Let 1,,..., n be the orderng of ts ertces along the Hamltonan cycle (the boundary of the outer face). For each edge, 1ncdent wth the outer face ( n1 1), add a new ertex u and two new edges u and, 1 Let the resultng graph be H. It s easy to see that H s a -connected outerplanar graph on nertces. We clam that ( H) n 3. Let S be a 3-path ertex coer n H. Dde the ertces of H nto n pars of the form u Obsere that f u S and, S thenu 1 S and \ 1 S (otherwse there s a path on 3 ertces n H not coered by S). Therefore, the aerage number of ertces n S s ether at least 1 (f at least one ertex of a par s n S), or (f no ertex s. n S, then n the preous par both are). Altogether, S n 4 Upper bounds on degree of ertces In ths secton we prode seeral upper bounds on path ertex coer based on degrees of a graph. Frst, recall theorem of Caro [6] and We [9], whch states 1 ( G) V ( G) u ( G) 1 du ( ) for eery graph G. Snce the only graphs for whch ths s best possble are the dsjont unons of clques, addtonal structural assumptons allow mproements (see e. g. [15 18, 3]). The problem of dssocaton number of graph was also studed n [14], where Görng et al. proed

9 An Inestgaton on Some theorems on K-Path Vertex Coer 1411 n m Theorem Let G be a graph on n ertces and m edges. Then 3( G) 6 Theorem 5.. Let a, b are ntegers such that b a b There s a graph on n ertces and m edges such that m a n m and 3( G) n b 6 Proof. Let x = 3(b a) and y = (a b). We construct the graph G hang two types of components: x components c 4 (a cycle on 4 ertces) wth 4 edges, 3( C4). y components H 6 ( 6 wth a perfect matchng remoed) wth 1 edges, 3( C6) 4 It s easy to see that n = 4x + 6y and m = 4x + 1y. Then. n m 1x 4y x 4 y 3( G) 6 6 Moreoer, m x 6y 6( b a) 1( a b) 6a a n x 3y 6( b a) 6( a b) 6b b References [1] J. Ayama, H. Era, S. V. Geraco, M. Watanabe, Path chromatc numbers of graphs, J. Graph Theory 13 (1989) [] M. O. Albertson, D. M. Berman, The acyclc chromatc number, n: Proceedngs of the eenth Southeastern Conference on Combnatorcs, Graph Theory, and Computng, n: Congr. Numer., ol. 17, 1976, pp Utltas Math.. [3] N. Alon, Transersal numbers of unform hypergraphs, Graphs Combn. 6 (1990) 1 4. [4] S. Bau, L. W. Benee, R. C. Vandell, Decyclng snaes, Congr. Numer. 134 (1998) [5] R. Bolac, K. Cameron, V. V. Lozn, On computng the dssocaton number and the nduced matchng number of bpartte graphs, Ars Combn. 7 (004) [6] Y. Caro, New results on the ndependence number, Techncal Report, Tel- A Unersty, [7] T. Cormen, C. E. Leserson, R. L. Rest, Introducton to Algorthms, The MIT Press, 1990.

10 141 P. Durga Bhaan et al [8] B. Courcelle, The monadc second-order logc of graphs. I. Recognzable sets of fnte graphs, Inform. and Comput. 85 (1990) [9] L. J. Cowen, C. E. Jesurum, Colorng wth defects, DIMACS Techncal Report 95-5, [10] V. E. Alesee, R. Bolac, D. V. Korobtsyn, V. V. Lozn, NP-hard graph problems and boundary classes of graphs, Theoret. Comput. Sc. 389 (1 ) (007) [1] R. Bolac, K. Cameron, V. V. Lozn, On computng the dssocaton number and the nduced matchng number of bpartte graphs, Ars Combn. 7 (004) [13] B. Brešar, F. Kardoš, J. Katrenč, G. Semanšn, Mnmum -path ertex coer, Dscrete Appl. Math. 159 (1) (011) [14] K. Cameron, P. Hell, Independent pacngs n structured graphs, Math. Program. 105 ( 3) (006) [15] P. Erdős, T. Galla, On maxmal paths and crcuts of graphs, Acta Math. Acad. Sc. Hungar. 10 (1959) Authors : Dr. P. DURGA BHAVANI s an assocate professor, scholar born n Inda. she receed her M. Sc from Andhra unersty, Andhra Pradesh n mathematcs, and M. Tech from jntu Hyderabad n computer scence and engneerng, Ph. D from unersty of Allahabad. Dr. Durga Bhaan early research wor was manly on graph theory related to neural networs. She s lfetme member n ISTE. Dr. K. VIJAY KUMAR s an assocate professor, scholar born n Inda. he receed hs M. Sc from Madura Kamaraj unersty and Ph. D from Unersty of Allhabad. Dr. jay Kumar early research wor was manly on Graph theory related to sem group theory. Dr. S. Satyanarayana s Dstngushed scentst, Professor, nentor, author, and busness leader Born n Inda. He receed hs M. Tech(CSE) & PhD(Computer Scence & Engneerng) from Reputed Unerstes. Dr. S. Satyanarayana early scentfc wor was manly Graph Theory to Software Engneerng, Cloud Computng, Cryptography, SAS, Busness Intellgence and Artfcal Neural Networs. He Publshed More than 35 Research Papers n Internatonal Journals & 6 Boos n Internatonal Publcatons. He s Fellow of Assocaton of Research Socety n Computng (FARSC)-USA.

The Erdős Pósa property for vertex- and edge-disjoint odd cycles in graphs on orientable surfaces

Dscrete Mathematcs 307 (2007) 764 768 www.elsever.com/locate/dsc Note The Erdős Pósa property for vertex- and edge-dsjont odd cycles n graphs on orentable surfaces Ken-Ich Kawarabayash a, Atsuhro Nakamoto