Locating-Dominating Sets in Graphs

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1 Applied Mathematical Sciences, Vol. 8, 2014, no. 88, HIKARI Ltd, Locating-Dominating Sets in Graphs Sergio R. Canoy, Jr. 1, Gina A. Malacas Department of Mathematics and Statistics Mindanao State University-Iligan Institute of Technology Tibanga Highway, Iligan City, Philippines Dennis Tarepe Department of Mathematical Sciences College of Arts and Sciences Mindanao University of Science and Technology Cagayan de Oro City, Philippines Copyright c 2014 Sergio R. Canoy, Jr., Gina A. Malacas and Dennis Tarepe. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper we characterize the locating dominating sets in the corona and composition of graphs. We also determine the locatingdomination numbers of these graphs. Mathematics Subject Classification: 05C69 Keywords: domination, locating set, strictly locating set, corona, composition 1 Introduction Let G =(V (G),E(G)) be a connected graph and v V (G). The neighborhood of v is the set N G (v) =N(v) ={u V (G) :uv E(G)}. IfX V (G), then the open neighborhood of X is the set N G (X) =N(X) = v X N G (v). The 1 This research is funded by the National Research Council of the Philippines-DOST

2 4382 Sergio R. Canoy, Jr., Gina A. Malacas and Dennis Tarepe closed neighborhood of X is N G [X] =N[X] =X N(X). A connected graph G of order n 3ispoint distinguishing if for any two distinct vertices u and v of G, N G [u] N G [v]. It is totally point determining if for any two distinct vertices u and v of G, N G (u) N G (v) and N G [u] N G [v]. A subset X of V (G) isadominating set of G if for every v (V (G)\X), there exists x X such that xv E(G), i.e., N[X] =V (G). The domination number γ(g) ofg is the smallest cardinality of a dominating set of G. A subset S of V (G) isalocating set in a connected graph G if every two vertices u and v of V (G)\S, N G (u) S N G (v) S. Itisastrictly locating set if it is locating and N G (u) S S for all u V (G)\S. The minimum cardinality of a locating set in G, denoted by ln(g), is called the locating number of G. The minimum cardinality of a strictly locating set in G, denoted by sln(g), is called the strict locating number of G. A locating (resp. strictly locating) subset S of V (G) which is also dominating is called a locating-dominating (resp. strictly locating-dominating) set in a (connected) graph G. The minimum cardinality of a locating-dominating (resp. strictly locating-dominating) set in G, denoted by γ L (G) (resp. (γ SL (G)), is called the L-domination (resp. SL-domination) number of G. In a given network or graph, a locating set can be viewed as a set of monitors which can actually determine the exact location of an intruder (e.g. a burglar, a fire, etc.). By requiring such as set to be dominating implies that every node where there is no monitor in it is connected to at least one monitor devise. Hence, determination of the locating domination number of a graph is equivalent to finding the least number of monitors that can do the certain task in a given graph or network. Domination in graphs and other types of domination are found in the book by Haynes et al. [4]. The concepts of locating set, locating dominating set and the associated parameters are studied in [1], [5], and [3]. On the other hand, the concepts of point distinguishing and totally point determining are defined and studied in [6] and [2]. 2 Preliminary Results and Characterizations From the definitions, the following relationships are immediate: Remark 2.1 For any connected graph G, ln(g) γ L (G) γ SL (G). The next two results in [1] give specific relationships between these parameters. Theorem 2.2 (1) Let G be a connected graph such that ln(g) <γ L (G). Then 1+ln(G) =γ L (G).

3 Locating-dominating sets in graphs 4383 Theorem 2.3 Let G be a connected graph such that γ L (G) <γ SL (G). Then 1+γ L (G) =γ SL (G). 3 Locating Dominating Sets in the Corona of Graphs Let G and H be graphs of order m and n, respectively. The corona of two graphs G and H is the graph G H obtained by taking one copy of G and m copies of H, and then joining the ith vertex of G to every vertex of the ith copy of H. For every v V (G), denote by H v the copy of H whose vertices are attached one by one to the vertex v. Denote by v + H v the subgraph of the corona G H corresponding the join {v} + H v. Theorem 3.1 Let G and H be non-trivial connected graphs. Then S V (G H) is a locating dominating set in G H if and only if S = A B C D, where A V (G), B= {B v : v A and B v is a locating set in H v }, C= {E v : v/ A, N G (v) A and E v is a locating- dominating set in H v } and D= {D v : v / A, N G (v) A = and D v is a strictly locating- dominating set in H v }. Proof : Suppose S is a locating-dominating set in G H. Let A = V (G) S and let v A. Put B v = V (H v ) S. Since S is a dominating set, B v. Let x, y V (H v ) \ B v with x y. Then x, y / S. Since S is a locating set in G H, (N H v(x) B v ) {v} = N G H (x) S N G H (y) S =(N H v(y) B v ) {v}. Thus, N H v(x) B v N H v(y) B v. Hence, B v is a locating set in H v. Next, let v/ A. Consider the following cases: Case1: Suppose N G (v) A. Let E v = V (H v ) S and let x, y V (H v ) \ E v with x y. Then x, y / S. Since S is a locating set and v/ S, N H v(x) E v = N G H (x) S N G H (y) S = N H v(y) E v. Thus, E v is a locating set in H v. Furthermore, because S is a dominating set and v/ S, E v must be a dominating set in H v. Case2: Suppose N G (v) A =. Let D v = V (H v ) S. As in Case 1, D v is a locating set in H v. Suppose that there exists x V (H v ) such that N H v(x) D v = N G H (x) S = D v. Since v/ S (v / A) and N G (v) A =, N G H (x) S = N G H (v) S = D v.

4 4384 Sergio R. Canoy, Jr., Gina A. Malacas and Dennis Tarepe This implies that S is not a locating set in G H, contrary to our assumption. Thus, D v is a strictly locating set in H v. Since v/ S and S is a dominating set, it follows that D v is a dominating set in H v. Therefore D v is a strictly locating-dominating set in H v. Now let B= {B v : v A and B v is a locating set in H v }, C= {E v : v/ A, N G (v) A and E v is a locating- dominating set in H v } and D= {Dv : v/ A, N G (v) A = and D v is a strictly locating- dominating set in H v }. Then S = A B C D. For the converse, suppose that S = A B C D, where A, B, C, and D are the sets possessing the properties as described. Let x V (G H) \ S and let v V (G) such that x V (v + H v ). If v S, then xv E(G H). If v/ S, then x/ S v = S V (H v ), where S v (E v or D v ) is a dominating set of H v. Hence, there exists y V (H v ) S such that xy E(G H). Therefore S is a dominating set of G H. Next, let a, b V (G H) \ S with a b. Let u, v V (G) such that a V (u + H u ) and b V (v + H v ). Suppose first that u = v. Consider the following cases: Case1: Suppose v S. Then, by assumption, a, b V (H v ) \ B v, where B v is a locating set in H v. Hence, N H v(a) B v N H v(b) B v. Thus, N G H (a) S =(N H v(a) B v ) {v} (N H v(b) B v ) {v} = N G H (b) S. Case2: Suppose v/ S. Ifa, b V (H v ), then a, b / S v = V (H v ) S where S v ( E v or D v )is a locating set in H v by assumption. Hence, N G H (a) S = N H v(a) S v N H v(b) S v = N G H (b) S. Suppose a = v and b V (H v ). If N G (v) S, sayz N G (v) S, then z [N G H (a) S] \ [N G H (b) S]. Thus, N G H (a) S N G H (b) S. If N G (v) S =, then S v = V (H v ) S = D v is a strictly locating set by assumption. This implies that N G H (a) S = D v N H v(b) S v = N G H (b) S. Suppose now that u v. Consider the following cases: Case1: Suppose u, v S. Then a u and b v. Since u N G H (a) and v N G H (b), it follows that N G H (a) S N G H (b) S. Case2: Suppose u / S or v / S. Without loss of generality, assume that u / S. Then S u = V (H u ) S (equal to E u or D u )is a dominating set by assumption. Pick any c S u such that ac E(G H). Then c N G H (a) \ N G H (b). Therefore N G H (a) S N G H (b) S. Therefore S is a locating set. Accordingly, S is a locating-dominating set in G H.

5 Locating-dominating sets in graphs 4385 Theorem 3.2 Let G and H be non-trivial connected graphs with V (G) = n. Then nγ L (H) γ L (G H) nγ SL (H). Proof : Let S be a minimum locating-dominating set in G. Then S = A B C D where A, B, C, and D are the sets described in Theorem 3.1. By Remark 2.1 and Theorem 2.2, it follows that γ L (G H) = S = A + B + C + D A + A ln(h)+(n A )γ L (H) = A (1 + ln(h))+(n A )γ L (H) A γ L (H)+(n A )γ L (H) = nγ L (H). Now let A be a minimum strictly locating-dominating set in H. For each v V (G), pick A v V (H v ) with A v = A. Then S = A v is a locating-dominating set in G H by Theorem 3.1. Hence, This proves the desired result. γ L (G H) S = nγ SL (H). v V (G) Corollary 3.3 Let G and H be non-trivial connected graphs with V (G) = n. Ifγ L (H) = γ SL (H), then γ LT (G H) =nγ SL (H). 4 Locating Dominating Sets in the Composition of Graphs Theorem 4.1 Let G and H be non-trivial connected graphs with Δ(H) V (H) 2. Then C = x S ({x} T x ), where S V (G) and T x V (H) for each x S, is a locating-dominating set in G[H] if and only if (i) S = V (G); (ii) T x is a locating set in H for every x V (G); (iii) T x or T y is strictly locating in H whenever x and y are adjacent vertices of G with N G [x] =N G [y]; and (iv) T x or T y is (locating) dominating in H whenever x and y are distinct non-adjacent vertices of G with N G (x) =N G (y).

6 4386 Sergio R. Canoy, Jr., Gina A. Malacas and Dennis Tarepe Proof : Suppose C is a locating-dominating set in G[H]. Suppose there exists x V (G)\S. Pick a, b V (H), where a b. Then (x, a), (x, b) / C and (x, a) (x, b). Since {(x, c) :c V (H)} C =, it follows that N G[H] ((x, a)) C = N G[H] ((x, b)) C. This implies that C is not a locating set in G[H], contrary to our assumption. Therefore, S = V (G). Now let x V (G) and suppose that T x is not locating in H. Then there exists distinct vertices p and q in V (H)\T x such that N H (p) T x = N H (q) T x. Let D x = N H (p) T x. Then ({x} D x ) C. Since N G[H] ((x, p)) C = {{y} T y : y N G (x)} ({x} D x )=N G[H] ((x, q)) C, it follows that C is a not a locating set in G[H]. Again, this gives a contradiction. Therefore, T x is a locating set in H. To prove (iii), let x and y be adjacent vertices of G with N G [x] =N G [y]. Suppose that T x and T y are not strictly locating in H. Then there exist c V (H)\T x and d V (H)\T y such that N H (c) T x = T x and N H (d) T y = T y. It follows that ({x} T x ) ({y} T y ) N G[H] ((x, c)) N G[H] ((y, d)). Since N G [x] =N G [y], it follows that N G[H] ((x, c)) C = N G[H] ((y, d)) C, i.e., C is not a locating set in G[H]. This contradicts our assumption. Therefore, T x or T y is strictly locating in H. To prove (iv), let x and y be distinct non-adjacent vertices of G with N G (x) =N G (y). Suppose that T x is not a dominating set in H. Then there exists a V (H)\T x such that ab / E(H) for all b T x. It follows that (x, a) / C and N G[H] ((x, a)) = {{z} T z : z N G (x)}. Let c V (H)\T y. Then (y, c) / C. Since N G (x) =N G (y), it follows that {{z} T z : z N G (x)} N G[H] ((y, c)). Since C is a locating set in G[H], N G[H] (x, a) N G[H] (y, c). This implies that there exists (y, d) {y} T y such that (y, d)(y, c) E(G[H]). This implies that d T y and cd E(H). Therefore, T y is a dominating set in H. This shows that (iv) holds. For the converse, suppose that conditions (i), (ii), (iii), and (iv) hold. By (i) and the fact that G is connected, it follows that C is a dominating set in G[H]. Now let (x, a), (y, b) V (G[H])\C with (x, a) (y, b). Consider the following cases: Case1. Suppose x = y. Then a b and a, b / T x = T y. By (ii), T x is a locating set in H; hence, N H (a) T x = A B = N H (b) T y. Suppose that ca\b. Then (x, c) N G[H] ((x, a))\n G[H] ((x, b)). Consequently, N G[H] ((x, a)) C N G[H] ((y, b)) C. Case2. Suppose x y. Consider the following subcases: Subcase1. Suppose xy / E(G). If N G (x) N G (y), then N G[H] ((x, a)) C N G[H] ((y, b)) C. Suppose N G (x) =N G (y). By (iv), T x or T y is a (locating) dominating set. Assume, without loss of generality, that T x is a dominating set in H. Since a/ T x, there

7 Locating-dominating sets in graphs 4387 exists c T x such that ac E(H). It follows that (x, c) C and (x, a)(x, c) E(G[H]). Since (x, c)(y, b) / E(G[H]), it follows that N G[H] ((x, a)) C N G[H] ((y, b)) C. Subcase2. Suppose xy E(G). If N G [x] N G [y], then N G[H] ((x, a)) C N G[H] ((y, b)) C. Suppose N G [x] = N G [y]. By (iii), assume without loss of generality that T x is a strictly locating set in H. Then there exists (x, d) C such that (x, d)(x, a) / E(G[H]). Since (x, d)(y, b) E(G[H]), it follows that N G[H] ((x, a)) C N G[H] ((y, b)) C. Accordingly, C is a locating-dominating set in G[H]. The following is a direct consequence of Theorem 4.1. Corollary 4.2 Let G be a connected totally point determining graph and let H be a non-trivial connected graph. Then C = x S ({x} T x ) is a minimum locating-dominating set in G[H] if and only if S = V (G) and T x is a minimum locating set in H for every x V (G). The next result is immediate from Corollary 4.2. Corollary 4.3 Let G be a connected totally point determining graph and let H be a non-trivial connected graph. Then γ L (G[H]) = V (G).ln(H). Proof : Let C = x S ({x} T x ) be a minimum locating-dominating set in G[H]. Then S = V (G) and T x is a minimum locating set in H for every x V (G), by Corollary 4.2. Therefore γ L (G[H]) = C = V (G) ln(h). References [1] S.R. Canoy, Jr, and G. A. Malacas, Determining the Intruder s Location in a Given Network: Locating-Dominating Sets in a Graph, NRCP Research Journal, 13(2013), No. 1, 1-8. [2] D. Geoffrey, Nuclei for totally point determining graphs. Discrete Math. 21(1978), [3] J. Gimbel, B. D. van Gorden, M. Nicolescu, C. Umstead, and N. Vaiana, Location with dominating sets, Congr. Numer. 151 (2001) [4] T. W. Haynes, S. T. Hedetniemi, and P. J. Slater, Fundamentals of Domination in Graphs. Marcel Dekker, New York, [5] T.W. Haynes, M.A. Henning, and J. Howard, Locating and total dominating sets in trees, Discrete Applied Mathematics, 154(2006), Issue 8,

8 4388 Sergio R. Canoy, Jr., Gina A. Malacas and Dennis Tarepe [6] D.P. Summer, Point Determination in graphs. Discrete Math, 5(1973), Received: June 6, 2014