On the Metric Dimension of Corona Product of a Graph with K1

Abstract

For an ordered set W = {w1,w2,…,wk} of vertices and a vertex v in a connected graph G, the k-vector (Formula Presented) is called the metric representation of v with respect to W, where d(x, y) is the distance between the vertices x and y. A set W is called a resolving set for G if distinct vertices of G have distinct metric representations with respect to W. The minimum cardinality of a resolving set for G is its metric dimension dim(G), and a resolving set of minimum cardinality is a basis of G. The corona product, (Formula Presented) of graphs G and H is obtained by taking one copy of G and n(G) copies of H, and by joining each vertex of the ith copy of H to the ith vertex of G. In this paper, we obtain bounds for dim(Formula Presented), characterize all graphs G with dim(Formula Presented), and prove that dim(Formula Presented) if and only if G is the complete graph Kn or the star graph K1,n−1. © Kyungpook Mathematical Journal