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On the Strong Metric Dimension of Antiprism Graph, King Graph, and Km ⊙ Kn Graph

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On the Strong Metric Dimension of Antiprism Graph, King Graph, and Km ⊙ Kn Graph On the strong metric dimension of antiprism graph, king graph, and Km ⊙ Kn graph Yuyun Mintarsih and Tri Atmojo Kusmayadi Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Sebelas Maret, Surakarta, Indonesia E-mail: [email protected], [email protected] Abstract. Let G be a connected graph with a set of vertices V (G) and a set of edges E(G). The interval I[u; v] between u and v to be the collection of all vertices that belong to some shortest u-v path. A vertex s 2 V (G) is said to be strongly resolved for vertices u, v 2 V (G) if v 2 I[u; s] or u 2 I[v; s]. A vertex set S ⊆ V (G) is a strong resolving set for G if every two distinct vertices of G are strongly resolved by some vertices of S. The strong metric dimension of G, denoted by sdim(G), is defined as the smallest cardinality of a strong resolving set. In this paper, we determine the strong metric dimension of an antiprism An graph, a king Km;n graph, and a Km ⊙ Kn graph. We obtain the strong metric dimension of an antiprim graph An are n for n odd and n + 1 for n even. The strong metric dimension of King graph Km;n is m + n − 1. The strong metric dimension of Km ⊙ Kn graph are n for m = 1, n ≥ 1 and mn − 1 for m ≥ 2, n ≥ 1. 1. Introduction The strong metric dimension was introduced by Seb¨oand Tannier [6] in 2004. Let G be a connected graph with a set of vertices V (G) and a set of edges E(G). Oelermann and Peters- Fransen [5] defined the interval I[u; v] between u and v to be the collection of all vertices that belong to some shortest u − v path. A vertex s 2 S is said to strongly resolve two vertices u and v if u 2 I[v; s] or v 2 I[u; v]. A vertex set S of G is a strong resolving set for G if every two distinct vertices of G are strongly resolved by some vertices of S. The strong metric basis of G is a strong resolving set with minimal cardinality. The strong metric dimension of a graph G is defined as the cardinality of strong metric basis denoted by sdim(G). Some authors have investigated the strong metric dimension to some graph classes. Seb¨oand Tannier [6] observed the strong metric dimension of complete graph Kn, cycle graph Cn, and tree. Kratica et al. [2] observed the strong metric dimension of hamming graph Hn;k. At the same year, Kratica et al [3] determined the strong metric dimension of convex polytope Dn and Tn. Yi [8] determined that sdim(G) = 1 if only if G is path graph and sdim(G) = n−1 if only if G is complete graph. Kusmayadi et al. [4] determined the strong metric dimension of sunflower graph, t-fold wheel graph, helm graph, and friendship graph. In this paper, we determine the strong metric dimension of an antiprism graph, a king graph, and a Km ⊙ Kn graph. 2. Main Results 2.1. Strong Metric Dimension Let G be a connected graph with a set of vertices V (G), a set of edges E(G), and S=fs1; s2; : : : ; skg 2 V (G). Oelermann and Peters-Fransen [5] defined the interval I[u; v] between u and v to be the collection of all vertices that belong to some shortest u − v path. A vertex s 2 S is said to strongly resolve two vertices u and v if u 2 I[v; s] or v 2 I[u; s]. A vertex set S of G is a strong resolving set for G if every two distinct vertices of G are strongly resolved by some vertices of S. The strong metric basis of G is a strong resolving set with minimal cardinality. The strong metric dimension of a graph G is defined as the cardinality of strong metric basis denoted by sdim(G). We often make use of the following lemma and properties about strong metric dimension given by Kratica et al. [3]. Lemma 2.1 Let u, v 2 V(G), u =6 v, (i) d(w,v) ≤ d(u,v) for each w such that u w 2 E(G), and (ii) d(u,w) ≤ d(u,v) for each w such that v w 2 E(G). Then there does not exist vertex a 2 V(G), a =6 u,v that strongly resolves vertices u and v. Property 2.1 If S ⊂ V(G) is strong resolving set of graph G, then for every two vertices u, v 2 V(G) satisfying conditions 1 and 2 of Lemma 2:1, obtained u 2 S or v 2 S. Property 2.2 If S ⊂ V(G) is strong resolving set of graph G, then for every two vertices u, v 2 V(G) satisfying d(u, v) = diam(G), obtained u 2 S or v 2 S. 2.2. The Strong Metric Dimension of antiprism graph Baˇca [1] defined the antiprism graph An for n ≥ 3 is a 4-regular graph with 2n vertices and 4n edges. It consists of outer and inner Cn, while the two cycles connected by edges viui and viu1+i(mod n) for i = 1, 2, 3, . , n. The antiprism graph An can be depicted as in Figure 1. Figure 1. Antiprism graph An Lemma 2.2 For every integer n ≥ 3 and n odd, if S is a strong resolving set of antiprism graph An then j S j ≥ n. Proof. We know that S is a strong resolving set of antiprism graph An. Suppose that S contains at most n - 1 vertices, then j S j < n. Let V1, V2 ⊂ V (An), with V1 = fu1; u2; : : : ; ung and V2 = fv1; v2; : : : ; vng. Now, we define S1 = V1 \ S and S2 = V2 \ S. Without loss of generality, we may take j S1 j = p, p > 0 and j S2 j = q, q ≥ 0. Clearly p + q ≥ n, if not then there are two distinct vertices va and vb where va 2 V1 n S1 and vb 2 V2 n S2 such that for every s 2 S, we obtain va 2= I[vb; s] and vb 2= I[va; s]. This contradicts with the supposition that S is a strong resolving set. Thus, j S j ≥ n. ut Lemma 2.3 For every integer n ≥ 3 and n odd, a set S = fu1; u2; : : : ; ung is a strong resolving set of antiprism graph An. Proof. For every integer i; j 2 [1, n] with 1 ≤ i < j ≤ n, a vertex ui 2 S which strongly resolves vi dan vj so that vj 2 I[vi; uj]. Thus, S = fu1; u2; : : : ; ung is a strong resolving set of antiprism graph An. ut Lemma 2.4 For every integer n ≥ 3 and n even, if S is a strong resolving set of antiprism graph An then j S j ≥ n+1. Proof. We know that S is a strong resolving set of antiprism graph An. Suppose that S contains at most n vertices, then j S j < n + 1. Let V1, V2 ⊂ V (An), with V1 = fu1; u2; : : : ; ung and V2 = fv1; v2; : : : ; vng. Now, we define S1 = V1 \ S and S2 = V2 \ S. Without loss of generality, we may take j S1 j = p, p ≥ 0 and j S2 j = q, q ≥ 0. Clearly p + q ≥ n + 1, if not then there are two distinct vertices va and vb where va 2 V1 n S1 and vb 2 V2 n S2 such that for every s 2 S, we obtain va 2= I[vb, s] and vb 2= I[va, s]. This contradicts with the supposition that S is a strong resolving set. Thus, j S j ≥ n+1. ut Lemma 2.5 For every integer n ≥ 3 and n even, a set S = fu1; u2; : : : ; u n ; u n ; v1; v2; : : : ; v n g 2 2 +1 2 is a strong resolving set of antiprism graph An. Proof. We prove that for every two distinct vertices u; v 2 V (An)nS, u =6 v there exists a vertex s 2 S which strongly resolves u and v. There are three possible pairs of vertices. n n 6 (i) A pair of vertices (ui; uj) with i; j = 2 + 2; 2 + 3; : : : ; n, i = j. 2 n For every integer i; j [ 2 + 2; n] with i < j, we obtain the shortest ui - u1 path: ui; ui+1; : : : ; uj; : : : ; un; u1. Thus, uj 2 I[ui; u1]. n n 6 (ii) A pair of vertices (vi; vj) with i; j = 2 + 1; 2 + 2; : : : ; n, i = j. 2 n For every integer i; j [ 2 + 1; n] with i < j, we obtain the shortest vi - v1 path: vi; vi+1; : : : ; vj; : : : ; vn; v1. Thus, vj 2 I[vi; v1]. n n n n (iii) A pair of vertices (ui; vj) with i = 2 + 2; 2 + 3; : : : ; n dan j = 2 + 1; 2 + 2; : : : ; n. 2 n 2 n ≤ For every integer i [ 2 + 2; n] and j [ 2 + 1; n] with i j, we obtain the shortest ui - v1 2 2 n path: ui; vi; : : : ; vj, :::, vn, v1. Thus, vj I[ui; v1]. Then, for every integer i [ 2 +2; n] and n j 2 [ + 1; n] with i > j, we obtain the shortest ui - v n path: ui; vi−1; : : : ; vj; : : : ; v n ; v n .
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