ON THE STRONG METRIC DIMENSION OF BROKEN FAN GRAPH, STARBARBELL GRAPH, AND C m k P n GRAPH Ratih Yunia Mayasari, Tri Atmojo Kusmayadi, Santoso Budi Wiyono Department of Mathematics Faculty of Mathematics and Natural Sciences Universitas Sebelas Maret Abstract. Let G be a connected graph with vertex set V (G) and edge set E(G). For every pair of vertices u, v V (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 V (G) strongly resolves two vertices u and v if u belongs to a shortest v s path or v belongs to a shortest u s path. A vertex set S of G is a strong resolving set of G if every two distinct vertices of G are strongly resolved by some vertex of S. The strong metric basis of G is a strong resolving set with minimal cardinality. The strong metric dimension sdim(g) of a graph G is defined as the cardinality of strong metric basis. In this paper we determine the strong metric dimension of a broken fan graph, starbarbell graph, and C m k P n graph. Keywords : strong metric dimension, strongly resolved set, broken fan graph, starbarbell graph, C m k P n graph 1. Introduction The concept of strong metric dimension was presented by Sebö and Tannier [9] in 2004. Oellermann and Peters-Fransen [8] defined for two vertices u and v in a connected graph G, the interval I[u, v] between u and v to be collection of all vertices that belong to some shortest path. A vertex s strongly resolves two vertices u and v if v I[u, s] or u I[v, s A set S of vertices in a connected graph G is a strong resolving set for G if every two vertices of G are strongly resolved by some vertex of S. The smallest cardinality of a strong resolving set of G is called its strong metric dimension and is denoted by sdim(g). Some researchers have investigated the strong metric dimension to some graph classes. In 2004 Sebö and Tannier [9] observed the strong metric dimension of complete graph K n, cycle graph C n, and tree. In 2012, Kuziak et al. [6] observed the strong metric dimension of corona product graph. In the same year, Kratica et al. [3] determined the metric dimension of hamming graph H n,k. Kratica et al. [4] determined the metric dimension of convex polytope D n and T n in 2012 too. In 2013 Yi [10] determined the metric dimension of P n. Kusmayadi et al. [5] determined the strong metric dimension of some related wheel graph such as sunflower graph, t-fold wheel graph, helm graph, and friendship graph. In this paper, we determine 1
the strong metric dimension of a broken fan graph, starbarbell graph, and C m k P n graph. 2. Strong Metric Dimension Let G be a connected graph with vertex set V (G), edge set E(G), and S = {s 1, s 2,..., s k } V (G). Oellermann and Peters-Fransen [8] 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 S strongly resolves two vertices u and v if u I[v, s] or v I[u, s A vertex set S of G is a strong resolving set of G if every two distinct vertices of G are strongly resolved by some vertex 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. [4 Lemma 2.1. Let u, v V(G), u v, (1) d(w,v) d(u,v) for each w such that uw E(G), and (2) d(u,w) d(u,v) for each w such that vw E(G). Then there does not exist vertex a V(G), a 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 V(G) satisfying conditions 1 and 2 of Lemma 2.1, obtained u S or v S. Property 2.2. If S V(G) is strong resolving set of graph G, then for every two vertices u, v V(G) satisfying d(u,v) = diam(g), obtained u S or v S. 3. The Strong Metric Dimension of a Broken Fan Graph Gallian [2] defined the broken fan graph BF (a, b) is a graph with V (BF (a, b)) = {c} {v 1, v 2,..., v a } {u 1, u 2,..., u b } and E(BF (a, b)) = {(c, v i ) i = 1, 2,..., a} {(c, u i ) i = 1, 2,..., b} E(P a ) E(P b ) (a 2 and b 2). Lemma 3.1. For every integer a 2 and b 3, if S is a strong resolving set of broken fan graph BF (a, b) then S a + b 2. Proof. We prove for every two distinct vertices (v i, u b ) and (u j, u b ). For every i = {1, 2,..., a} so d(v i, u b ) = 2 = diam(bf (a, b)) then by using Property 2.2, v i S 2 2018
or u b S and for every j {1, 2,..., b 2} so d(u j, u b ) = 2 = diam(bf (a, b)) then using Property 2.2, u j S or u b S. Therefore, S contains at least one vertex from distinct set X ib = {v i, u b } for i {1, 2,..., a} and one vertex from distinct set Y jb = {u j, u b } for j {1, 2,..., b 2}. The minimum number of vertices from distinct set X ib is a and the minimum number of vertices from distinct set Y jb is b 2. Therefore, S a + b 2. Lemma 3.2. For every integer a 2 and b 3, a set S = {v 1, v 2,..., v a, u 1, u 2,..., u b 2 } is a strong resolving set of broken fan graph BF (a, b). Proof. We prove that every two distinct vertices u, v (BF (a, b)) \ S, u v there exists a vertex s S which strongly resolves u and v. There are two pairs of vertices from V (BF (a, b)) \ S. (1) A pair of vertices (c, u j ). For every integer i = {1, 2,..., a} and j {b 1, b}, d(v i, u j ) = 2 = diam(bf (a, b)), we obtain the shortest v i u j path : v i, c, u j. Thus, c I[v i, u j (2) A pair of vertices (u b 1, u b ) For j = b 2, d(u j, u b ) = 2 = diam(bf (a, b)), we obtain the shortest u b 2 u b path: u b 2, u b 1, u b. Thus, u b 1 I[u j, u b For every possible pairs of vertices, there exists a vertex s S which strongly resolves every two distinct vertices BF (a, b) \ S. Thus, S is a strong resolving set of BF (a, b). Theorem 3.1. Let BF (a, b) be the broken fan graph, then { 3, a = 2 and n = 2; sdim(bf (a, b)) = a + b 2, a 2 and b 3. Proof. There are two cases to determine the strong metric dimension of broken fan graph. (1) Case 1 (For a = 2 and b = 2). By using Theorem from Kusmayadi et al. [5] that sdim(f n ) = 2n 1, so that sdim(bf (2, 2)) = 3 because of BF (2, 2) = f 2. Hence, sdim(bf (2, 2)) = 3. (2) Case 2 (For a 2 and b 3). By using Lemma 3.2 a set S = {v 1, v 2,..., v a, u 1, u 2,..., u b 2 } is strong resolving set of broken fan graph BF (a, b) with a 2 and b 3. According to Lemma 3.1, S a + b 2, S is strong metric basis of broken fan graph BF (a, b). Hence, sdim(bf (a, b)) = a + b 2. 3 2018
4. The Strong Metric Dimension of Starbarbell Graph Starbarbell graph SB m1,m 2,...,m n is a graph obtained from a star graph S n and n complete graph K mi by merging one vertex from each K mi and the i th -leaf of S n, where m i 3, 1 i n, and n 2. The vertices set of starbarbell graph is c, v 1,1, v 1,2,..., v 1,m1, v 2,1, v 2,2,..., v 2,m2,..., v n,1, v n,2,..., v n,mn. Lemma 4.1. For every integer m i 3 and n 2, if S is a strong resolving set of starbarbell graph SB m1,m 2,...,m n then S n i=1 (m i 1) 1. Proof. Let us consider a pair of vertices (v i,j, v k,l ) for every i, k = 1, 2,..., n with v i,j v k,l and j, l = 2,..., m i satisfying both of the conditions of Lemma 2.1. According to Property 2.1, we obtain v i,j S or v k,l S. It means that S contains one vertex from distinct sets X i,j,k,l = {v i,j, v k,l }. The minimum number of vertices from distinct sets X i,j,k,l is n i=1 (m i 1) 1. Therefore, S n i=1 (m i 1) 1. Lemma 4.2. For every integer m i 3 and n 2, a set S = {v 1,2, v 1,3,..., v 1,m1, v 2,2, v 2,3,..., v 2,m2,..., v n,2, v n,3,..., v n,mn 1} is a strong resolving set of starbarbell graph SB m1,m 2,...,m n. Proof. We prove that for every two distinct vertices u, v V (SB m1,m 2,...,m n )\S, u v, there exists a vertex s S which strongly resolves u and v. There are four possible pairs of vertices. (1) A pair of vertices (u, v n,mn ). For every integer i {1, 2,..., n 1} and j {2, 3,..., m i }, d(v i,j, v n,mn )= 4 = diam(sb m1,m 2,...,m n ), we obtain the shortest v n,mn v i,j path: v n,mn, v n,1, u, v i,1, v i,j. Thus, u I[v n,mn, v i,j (2) A pair of vertices (u, v i,1 ). For every integer i {1, 2,..., n 1} and j {2, 3,..., m i }, d(u, v i,j )= 2, we obtain the shortest u v i,j path: u, v i,1, v i,j. Thus, v i,1 I[u, v i,j For i = n, l {2, 3,..., m i 1}, d(u, v n,l ) = 2, we obtain the shortest u v n,l path: u, v n,1, v n,l. Thus, v n,1 I[u, v n,l (3) A pair of vertices (v i,1, v n,mn ). For every integer i {1, 2,..., n 1} and j {2, 3,..., m i }, d(v i,j, v n,mn ) = 4 = diam(sb m1,m 2,...,m n ), we obtain the shortest v i,j v n,mn path: v i,j, v i,1, u, v n,1, v n,mn. Thus, v i,1 I[v i,j, v n,mn For i = n, k {1, 2,..., n 1}, and l {2, 3,..., m i }, d(v k,l, v n,mn ) = 4 = diam(sb m1,m 2,...,m n ), we obtain the shortest v k,l v n,mn path: v k,l, v k,1, u, v n,1, v n,mn. Thus, v n,1 I[v k,l, v n,mn 4 2018
(4) A pair of vertices (v i,1, v k,1 ). For every integer i, k {1, 2,..., n 1}, i k, and l {2, 3,..., m i }, d(v i,1, v k,l )= 3, we obtain the shortest v i,1 v k,l path: v i,1, u, v k,1, v k,l. Thus, v k,1 I[v i,1, v k,l For k = n and l {2, 3,..., m i 1}, d(v i,1, v k,l ) = 3, we obtain the shortest v i,1 v k,l path: v i,1, u, v k,1, v k,l. Thus, v k,1 I[v i,1, v k,l From every possible pairs of vertices, there exists a vertex s S which strongly resolves every two distinct vertices SB m1,m 2,...,m n \ S. Thus S is a strong resolving set of starbarbell graph SB m1,m 2,...,m n. Theorem 4.1. Let SB m1,m 2,...,m n be the starbarbell graph with m 3 and n 2. Then sdim(sb m1,m 2,...,m n ) = n i=1 (m i 1) 1. Proof. By using Lemma 4.2, we have a set S = {v 1,2, v 1,3,..., v 1,m1, v 2,2, v 2,3,..., v 2,m2,..., v n,2, v n,3,..., v n,(mn 1)} is a strong resolving set of SB m1,m 2,...,m n graph with m 3 and n 2. According to Lemma 4.1, S n i=1 (m i 1) 1, S is a strong metric basis of SB m1,m 2,...,m n. Hence, sdim(sb m1,m 2,...,m n ) = n i=1 (m i 1) 1. 5. The Strong Metric Dimension of C m k P n Graph By using the definiton from Frucht and Harary [1], the corona product C m P n graph is graph obtained from C m and P n by taking one copy of C m and n copies of P n and joining by an edge each vertex from i th -copy of P n with the i th -vertex of C m. Then, by using the definiton from Marbun and Salman [7], the k-multilevel corona product C m k P n graph is graph obtained from the corona product C m k 1 P n and P n graph and it can be written as C m k P n = (C m k 1 P n ) P n. The C 3 2 P 2 can be depicted as in Figure 1. v1,2,2 v1,2,1 v1,1,2 v1,1,1 v1,2 v1,1 v1,0,2 v1 v1,0,1 v2,0,1 v2,0,2 v2,1 v2,1,1 v2 v2,1,2 v2,2,1 v2,2 v3,0,2 v3 v3,0,1 v3,1 v3,2 v3,2,2 v3,2,1 v3,1,2 v2,2,2 v3,1,1 Figure 1. C 3 2 P 2 graph 5 2018
Lemma 5.1. For every integer m 3, n = 2, and k 1, if S is a strong resolving set of C m k P n graph then S (mn(n + 1) k 1 ) 1. Proof. Let us consider a pair of vertices (v a1,a 2,...,a y ) with y, z = k + 1, 1 a 1, b 1 m, 0 a i, b i 2, and 2 i k + 1 satisfying both of the conditions of Lemma 2.1. According to Property 2.1, we obtain v a1,a 2,...,a y S or v b1,b 2,...,b z S. It means that S contains one vertex from distinct sets X yz = {v a1,a 2,...,a y }. The minimum number of vertices from distinct sets X yz is (mn(n + 1) k+1 ) 1. Therefore, S (mn(n + 1) k+1 ) 1. Lemma 5.2. For every integer m 3, n = 2, and k 1, a set S = {v 1,0,...,1, v 1,0,...,2,..., v 2,0,...,1, v 2,0,...,2,..., v m,2,...,1 } is a strong resolving set of C m k P n graph. Proof. We prove that every two distinct vertices u, v (C m k P n ) \ S, there exists a vertex s S which strongly resolves u and v. There are two pairs of vertices from V (C m k P n ) \ S. (1) A pair of vertices (v a1,a 2,...,a y ). For every integer 1 y, z k, 1 a 1, b 1 m, 0 a i, b i 2, and 2 i k, we obtain the shortest v a1,a 2,...,a y v b1,b 2,...,b z,b z+1,...,b k+1 path: v a1,a 2,...,a y, v a1,a 2,...,a y 1,..., v a1,..., v b1,...,b z+1,...,,b z+1,...,b k+1. So that I[v a1,a 2,...,a y,b z+1,...,b k+1 (2) A pair of vertices (v a1,a 2,...,a y ). For every integer 1 y k, 1 a 1 m 1, 0 a i 2, and 2 i k, z = k + 1, b 1 = m, b j = 2, and 2 j k + 1, we obtain the shortest v a1,a 2,...,a y,a y+1,...,a k+1 v b1,b 2,...,b z path: v a1,a 2,...,a y,a y+1,...,a k+1,..., v a1,a 2,...,a y,..., v a1,..., v b1,.... So that, v a1,a 2,...,a y I[v a1,a 2,...,a y,a y+1,...,a k+1 For every possible pairs of vertices, there exists a vertex s S which strongly resolves every two distinct vertices (C m k P n ) \ S. Thus S is a strong resolving set of C m k P n. Lemma 5.3. For every integer m 3, n 3, and k 1, if S is a strong resolving set of C m k P n graph then S (mn(n + 1) k 1 ) 2. Proof. We know that S is strong resolving set of C m k P n graph. Suppose that S contains at most (mn(n + 1) k 1 ) 3 vertices, then S < (mn(n + 1) k 1 ) 2. Let V 1 is set of vertices v a1,a 2,...,a y with y = k + 1, 1 a 1 m, 0 a i 2, and 2 i k and V 2 is set of vertices v b1,b 2,...,b z with 1 z k, 1 b 1 m, 0 b i 2, and 2 i k. Now, we define S 1 = V 1 S and S 2 = V 2 S. Without loss of generality, we may take S 1 = p, p > 0 and S 2 = q, q 0. Clearly p + q (mn(n + 1) k 1 ) 3, there are two distinct vertices v a and v b where v a V 1 \ S and v b V 2 \ S such that 6 2018
for every s S we obtain v a / I[v b, s] and v b / I[v a, s This contradicts with the supposition that S is a strong resolving set. Thus S (mn(n + 1) k 1 ) 2. Lemma 5.4. For every integer m 3, n 3, and k 1, a set S = {v 1,0,...,1, v 1,0,...,2,..., v 1,0,...,n,..., v 2,0,...,1, v 2,0,...,2,..., v 2,0,...,n,..., v m,n,...,1, v m,n,...,2,..., v m,n,...,n 2 } is a strong resolving set of C m k P n graph. Proof. We prove that every two distinct vertices u, v (C m k P n ) \ S, there exists a vertex s S which strongly resolves u and v. There are three pairs of vertices from V (C m k P n ) \ S. (1) A pair of vertices (v a1,a 2,...,a y ). For every integer 1 y, z k, 1 a 1, b 1 m, 0 a i, b i n, and 2 i k, we obtain the shortest v a1,a 2,...,a y v b1,b 2,...,b z,b z+1,...,b k+1 path: v a1,a 2,...,a y, v a1,a 2,...,a y 1,..., v a1,..., v b1,...,b z+1,..., v b1,b 2,...,b z,b z+1,...,b k+1. So that I[v a1,a 2,...,a y,b z+1,...,b k+1 (2) A pair of vertices (v a1,a 2,...,a y ) For every integer 1 y k, 1 a 1 m 1, 0 a i n, 2 i k, z = k + 1, b 1 = m, b j = n, 2 j k, b z = {n 1, n} we obtain the shortest v a1,a 2,...,a y,a y+1,...,a k+1 v b1,b 2,...,b z path: v a1,a 2,...,a y,a y+1,...,a k+1,..., v a1,a 2,...,a y,..., v a1,..., v b1,.... So that, v a1,a 2,...,a y I[v a1,a 2,...,a y,a y+1,...,a k+1 (3) A pair of vertices (v a1,a 2,...,a y, v a1,a 2,...,a z ) For every integer x, y, z = k + 1, a 1 = m, a i = n, 2 i k, a y = n 1, a z = n a x = n 2 we obtain the shortest v a1,a 2,...,a x v a1,a 2,...,a z path: v a1,a 2,...,a x, v a1,a 2,...,a y, v a1,a 2,...,a z. So that, v a1,a 2,...,a y I[v a1,a 2,...,a x, v a1,a 2,...,a z For every possible pairs of vertices, there exists a vertex s S which strongly resolves every two distinct vertices (C m k P n ) \ S. Thus S is a strong resolving set of C m k P n. Theorem 5.1. Let C m k P n be the corona product of cycle graph and path graph, then { sdim(c m k (mn(n + 1) k 1 ) 1, m 3, k 1, dan n = 2; P n ) = (mn(n + 1) k 1 ) 2, m 3, k 1, dan n 3. Proof. By Lemma 5.1 and Lemma 5.2, we have sdim(c m k P n ) = (mn(n+1) k 1 ) 1 for m 3, n = 2, and k 1. By Lemma 5.3 and Lemma 5.4, we have sdim(c m k P n ) = (mn(n + 1) k 1 ) 2 for m 3, n 3, and k 1. 7 2018
6. Conclusion According to the discussion above it can be concluded that the strong metric dimension of a broken fan graph BF (a, b), a starbarbell graph SB m1,m 2,...,m n, and a C m k P n graph are as stated in Theorem 3.1, Theorem 4.1, and Theorem 5.1, respectively. References [1] Frucht, and F. Harary, On The Corona of Two Graphs, Aequationes Math. Vol 4 (1970), 322-325. [2] Gallian, J. A., A Dynamic Survey of Graph Labeling, The Electronic Journal of Combinatorics #DS6, 2016. [3] Kratica, J., V. Kovačević-Vujčić, and M. Čangalović, Minimal Doubly Resolving Sets and The Strong Metric Dimension of Hamming Graph, Applicable Analysis and Discrete Mathematics 6 (2012), 63-71. [4] Kratica, J., V. Kovačević-Vujčić, and M. Čangalović, Minimal Doubly Resolving Sets and The Strong Metric Dimension of Some Convex Polytope, Applied Mathematics and Computation 218 (2012), 9790-9801. [5] Kusmayadi, T. A., S. Kuntari, D. Rahmadi, and F. A. Lathifah, On the Strong Metric Dimension of Some Related Wheel Graphs, Far East Journal of Mathematical Sciences (FJMS) 99 (2016), no. 9, 1322-1334. [6] Kuziak, D., I. G. Yero, J. A. Rodríguez-Velázquez, On The Strong Metric Dimension of Corona Product Graphs and Join Graph, Discrete Applied Mathematics 161 (2013), 1022-1027. [7] Marbun, H.T, and M. Salman, Wheel Supermagic Labelings for a Wheel k Multilevel Corona with a Cycle, AKCE International Journal Graphs Combinatorics 2 (2013), 183-191. [8] Oellermann, O. and J. Peters-Fransen, The Strong Metric Dimension of Graph and Digraph, Discrete Applied Mathematics 155 (2007), 356-364. [9] Sebö, A. and E. Tannier, On Metric Generators of Graph, Mathematics and Operations Research 29(2) (2004), 383-393. [10] Yi, E., On Strong Metric Dimension Graph and Their Complements, Acta Mathematica Sinica 29(8) (2013), 1479-1492. 8 2018