TIGHT LOWER BOUND FOR LOCATING-TOTAL DOMINATION NUMBER. VIT University Chennai, INDIA

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1 International Journal of Pure and Applied Mathematics Volume 101 No , ISSN: (printed version); ISSN: (on-line version) url: PAijpam.eu TIGHT LOWER BOUND FOR LOCATING-TOTAL DOMINATION NUMBER R. Jayagopal 1, R. Sundara Rajan 2, Indra Rajasingh 3 1,2,3 School of Advanced Sciences VIT University Chennai, INDIA Abstract: Locating-dominating set in a connected graph G is a dominating set S of G such that for every pair of vertices u and v in V(G)\S, N(u) S N(v) S. Further, if S is a total dominating set, then S is called a locatingtotal dominating set. The locating-domination number γ L (G) is the minimum cardinality of a locating-dominating set of G and the locating-total domination number γ L t (G) is the minimum cardinality of a locating-total dominating set of G. In this paper we prove that if G is a necklace or a windmill graph, then γ L (G) = γ L t (G). AMS Subject Classification: 05C69 Key Words: dominating set, total dominating set, locating-dominating set, locating-total dominating set, necklace, windmill graphs 1. Introduction A vertex in a graph G dominates itself and its neighbors. A set of vertices S in a graph G is a dominating set, if each vertex of G is dominated by some vertex of S. A dominating set S is called a total dominating set if each vertex v of G is dominated by some vertex u v of S. The domination and total domination problem is to determine the minimum cardinality of dominating Received: March 12, 2015 c 2015 Academic Publications, Ltd. url:

2 662 R. Jayagopal, R.S. Rajan, I. Rajasingh and total dominating set in G respectively. Domination problems has been widely used in areas like locating radar stations problem, coding theory, modelling biological networks and nuclear power plants (see [1, 2, 3, 4], etc.). Total domination plays a role in the problem of placing monitoring devices in a system in such a way that every site in the system, including the monitors, is adjacent to a monitor site so that, if a monitor goes down, then an adjacent monitor can still protect the system. Installing minimum number of expensive sensors in the system which will transmit a signal at the detection of faults and uniquely determining the location of the faults motivates the concept of locating sets and locating-total dominating sets [5]. Determining if an arbitrary graph has a dominating set of a given size is a well-known N P-complete problems [6]. Locating-dominating set (LDS) in a connected graph G is a dominating set S of G such that for every pair of vertices u and v in V(G) S, N(u) S N(v) S. The minimum cardinality of a locating-dominating set of G is the locating-domination number γ L (G) [5]. Determining if an arbitrary graph has a locating-dominating set of a given size is also a well-known N P-complete problems [7]. Locating-total dominating set (LT DS) in a connected graph G is a total dominating set S of G such that for every pair of vertices u and v in V(G) \ S, N(u) S N(v) S. The minimum cardinality of a locating total-dominating set of G is the locating-total domination number γt L (G) [5]. The locating-total domination problem has been discussed for critical graphs [8], trees [9], cubic graphs and grid graphs [10], corona and composition of graphs [11], claw-free cubic graphs [12], edge-critical graphs [13] and so on. In this paper we show that the locating-domination and locating-total domination number are equal for some special classes of graphs like necklace and windmill graphs. The rest of the paper is organized as follows. Section 2 gives background of the paper. Section 3 establishes the main results. 2. Basic Concepts In this section, we give the preliminaries that are required for this study. All graphs considered in this paper are simple and connected. For S V(G), let G[S] denote the subgraph of G induced by S. Definition 2.1. Let K m and K ti be complete graphs on m and t i vertices respectively, 1 i m. The graph obtained by identifying any one vertex of

3 TIGHT LOWER BOUND FOR LOCATING-TOTAL K ti with the i th vertex of K m, 1 i m, is called a K-necklace and is denoted by KN(K m ;K t1,k t2,...,k tm ). See Figure 1(a). Definition 2.2. [14] The windmill graph Wd(n,m), n,m 2, is constructed by joining m copies of the complete graph K n at a shared vertex. See Figure 1(b). For our convenience we denote the windmill graph Wd(n,m) as WM(K t1, K t2,...,k tm ), where K ti is a complete graphs on t i vertices, t i 2, 1 i m, m 2. Definition 2.3. Let P m be a path on m vertices and K ti be a complete graphs on t i vertices, 1 i m. The graph obtained by identifying any one vertex of K ti with the i th vertex of P m, 1 i m, is called a P-necklace and is denoted by PN(P m ;K t1,k t2,...,k tm ). See Figure 2(a). Definition 2.4. Let C m beacycle on m vertices and let K ti beacomplete graphs on t i vertices, 1 i m. The graph obtained by identifying any one vertex of K ti with the i th vertex of C m, 1 i m, is called a C-necklace and is denoted by CN(C m ;K t1,k t2,...,k tm ). See Figure 2(b). Figure1: (a)kn(k 4 ;K t1,k t2,k t3,k t4 )and(b)wm(k t1,k t2,k t3,k t4 ).

4 664 R. Jayagopal, R.S. Rajan, I. Rajasingh 3. Main Results In this section we prove that the locating-domination and locating-total domination number are equal for some special classes of graphs like necklace and windmill graphs. Lemma 3.1. Let G be a graph on n-vertices with cut vertices v 1,v 2,...,v m, m 1. Suppose for every cut vertex v i of G, there exists components H i1,h i2,...,h iki,k i 1, such that G[V(H ij ) {v i }] is a complete graph on r ij vertices, r ij 3,1 j k i,1 i m. Then any locatingdominating set of G contains at least r ij 2 arbitrarily chosen vertices of H ij, i,j,1 j k i,1 i m. In other words γ L (G) m ki i=1 j=1 (r i 2). j Proof. SupposeS isalocating-dominating setwhich contains at most r ij 3 vertices of H ij for some i, j, 1 j k i and 1 i m. Then there are at least two vertices u and v of H ij which are not in S satisfying N(u) S = N(v) S, a contradiction. Theorem 3.2. Let G be the K-necklace KN(K m ;K t1,k t2,...,k tm ), m 1,t i 4,1 i m on n = m i=1 t i vertices. Then γl (G) = γt L (G) = n 2m+1. Proof. Let v 1,v 2,...,v m be the vertices of the subgraph K m of G such that each complete graph K ti, shares exactly one vertex v i of K m, 1 i m. It is easy to see that v i is a cut vertex of G, 1 i m. Let H i be a component of G\v i such that G[H i {v i }] = K ti, 1 i m. By Lemma 3.1, any locatingdominating set of G contains at least t i 2 vertices of H i, 1 i m. Let S denote the union of the sets of arbitrarily chosen t i 2 vertices from each H i = Kti 1, 1 i m. Now, S = m i=1 (t i 2) = n 2m. Let v t i be the vertex in H i not in S, 1 i m. Then N(v i ) S = N(v ti ) S, i,1 i m. ThisimpliesthatS isnotalocating-dominating set. Henceγ L (G) n 2m+1. Let S = S {v 1 }. We claim that S is a locating-dominating set of G. Clearly, S is a dominating set of G. We have only to show that for u,v V(G) \ S, N(u) S N(v) S. Now V(G) \ S = {v 2,v 3,...,v m,v t1,v t2,...,v tm }. If u = v i and v = v j, 2 i,j m, then N(v i ) S = {v 1 } (V(K ti 1) \v ti ) {v 1 } (V(K tj 1) \ v tj ) = N(v j ) S. If u = v i, 2 i m and v = v t1, then N(v i ) S = {v 1 } (V(K ti 1)\v ti ) {v 1 } (V(K t1 1)\v t1 ) = N(v t1 ) S. If

5 TIGHT LOWER BOUND FOR LOCATING-TOTAL u = v i and v = v tj, 2 i,j m, then N(v i ) S = {v 1 } (V(K ti 1) \v ti ) V(K tj 1) \ v tj = N(v tj ) S. If u = v ti and v = v tj, 2 i,j m, then N(v ti ) S = V(K ti 1)\v ti V(K tj 1)\v tj = N(v tj ) S. Thus S is a locatingdominating set of G. Since S = n 2m+1, we conclude that S is a minimum locating-dominating set. As S contains no isolated vertex, S is a minimum locating-total dominating set and γt L (G) = n 2m+1. Theorem 3.3. Let G be a windmill graph WM(K t1,k t2,...,k tm ), m 2,t i 4,1 i m on n = m i=1 t i (m 1) vertices. Then γl (G) = γt L (G) = n m 1. Proof. Let w be the cut-vertex of G. For each i, 1 i m, H i be the component of G\w such that G[H i {w}] = K ti. By Lemma 3.1, any locatingdominating set of G contains at least t i 2 vertices of H i. Hence γ L (G) m i=1 (t i 2) = n m 1. Let S be the union of the sets of arbitrarily chosen t i 2 vertices from each H i = Kti 1, 1 i m. Let v ti be the vertex in H i not in S, 1 i m. We claim that S is a minimum locatingdominating set of G. We have S = V(G)\{w,v t1,v t2,...,v tm } where v ti H i. It is clear that S is a dominating set of G. We have only to show that for u,v V(G) \S, N(u) S N(v) S. Now V(G)\S = {w,v t1,v t2,...,v tm }. If u = w and v = v tj, 1 j m, then N(w) S = m i=1 (V(K t i 1) \ v ti ) V(K tj 1) \ v tj = N(v tj ) S. If u = v ti and v = v tj, 1 i,j m then N(v ti ) S = V(K ti 1 ) \ v t i V(K tj 1) \ v tj = N(v tj ) S. Hence S is a locating-dominating set of G. Since S = n m 1, we conclude that S is a minimum locating-dominating set. As S contains no isolated vertex, S is a minimum locating-total dominating set and γt L (G) = n m 1. Theorem 3.4. Let G be a P-necklace graph PN(P m ;K t1,k t2,...,k tm ), m 3,t i 4,1 i monnvertices. Thenγ L (G) = γt L (G) = n 2m+ m/3. Proof. Let v 1,v 2,...,v m be the vertices of the subgraph P m of G such that each complete graph K ti, shares exactly one vertex v i of P m, 1 i m. It is easy to see that v i is a cut vertex of G, 1 i m. Let H i be the component of G\v i such that G[H i {v i }] = K ti, 1 i m. By Lemma 3.1, any locatingdominating set of G contains at least t i 2 vertices of H i, 1 i m. Let S denote the union of the sets of arbitrarily chosen t i 2 vertices from each H i = Kti 1, 1 i m. Now, S = m i=1 (t i 2) = n 2m. Let v t i be the

6 666 R. Jayagopal, R.S. Rajan, I. Rajasingh Figure 2: (a) Vertices of H i, 1 i 9 and v 2,v 5,v 8 are in S of PN(P 9 ;K t1,k t2...,k t9 ) and (b) CN(C m ;K t1,k t2,...,k tm ). vertex in H i not in S, 1 i m. Then N(v i ) S = N(v ti ) S, i,1 i m. This implies that S is not a locating-dominating set of G. Now, we claim that every three consecutive vertices on P m should contain a member of S. Suppose not, then there exist vertices v i 1,v i and v i+1 on P m not belonging to S, for some i in G. Then N(v i ) S = N(v ti ) S, a contradiction. Thus we need at least one vertex for every three consecutive vertices on P m. Hence S n 2m+ m/3. We proceed to prove that the lower boundis tight. We consider the cases where m 0,1,2 (mod 3). When m 0 (mod 3) select vertices v 3k 1,1 k m/3 in S. When m 1 (mod 3) select vertices v 3k 1,1 k m/3 in S. When m 2 (mod 3) select vertices v 3k 1,1 k m/3 in S. Let S = S S. We claim that S is a locating-dominating set of G. Clearly, S is a dominating set of G. We have only to show that for u,v V(G)\S, N(u) S N(v) S. If u V(H i )andv = v i thenn(u) S = V(H i )\uandn(v) S = (V(H i )\u) {w} where w S is a vertex adjacent to v on P m. Thus N(u) S N(v) S. Similarly in all other cases of u and v in V(G)\S, N(u) S N(v) S. Hence S is a locating-dominating set of G. Since S = n 2m+ m/3, we conclude that S is a minimum locating-dominating set. As S contains no isolated vertex, S is a minimumlocating-total dominatingset andγ L t (G) = n 2m+ m/3. See Figure 2 (a). If in Theorem 3.4, P m is replaced by C m, then we have the following result.

7 TIGHT LOWER BOUND FOR LOCATING-TOTAL Theorem 3.5. Let G be a C-necklace graph CN(C m ;K t1,k t2,...,k tm ), t i 4,1 i m,m 3 on n = m i=1 t i vertices. Then γl (G) = γt L (G) = n 2m+ m/3. 4. Conclusion In this paper, we have obtained the locating-domination number γ L (G) and the locating-total domination number γt L (G) for some special classes of graphs like necklace and windmill graphs and proved that they are equal for these graphs. References [1] E. J. Cockayne and S. T. Hedetniemi, Towards a theory of domination in graphs, Networks, 7(1977), [2] C. Berge, Graphs and hypergraphs, North Holland Publisher, Amsterdam, Netherlands, [3] T. Haynes, D. Knisley, E. Seier and Y. Zou, A quantitative analysis of secondary RNA structure using domination based parameters on trees, BMC Bioinformatics, 7(2006), [4] J. G. Kalbeisch, R. G. Stanton and J. D. Horton, On covering sets and error-correcting codes, Journal of Combinatorial Theory, 11A(1971), [5] T. W. Haynes, M. A. Henning and J. Howard, Locating and total dominating sets in trees, Discrete Applied Mathematics, 154(2006), [6] M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, W. H. Freeman and company publisher, San Francisco, US, [7] I. Charon, O. Hudry and A. Lobstein, Minimizing the size of an identifying or locating-dominating code in a graph is NP-hard, Theoretical Computer Science, 290(2003), [8] M. Chellali, Locating-total domination critical graphs, Australasian Journal of Combinatorics, 45(2009),

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