The doubly metric dimension of corona product graphs

Abstract

The doubly metric dimension of a connected graph G is the minimum cardinality of doubly resolving sets in it. It is well known that deciding the doubly metric dimension of G is NP-complete. The corona product G ? H of two vertex-disjoint graphs G and H is defined as the graph obtained from G and H by taking one copy of G and |V(G)| copies of H, then joining the ith vertex of G to every vertex in the ith copy of H. In this paper some formulae on the doubly metric dimension of corona product G?H of graphs G and H are established in terms of the order of G with the adjacency dimension of H and the doubly metric dimension of K1 ? H, respectively. We determine both sharp upper and lower bounds on doubly metric dimension of corona product graphs with disconnected and connected coronas involved, respectively, and characterize the corresponding extremal graphs. We also characterize all graphs G of diameter two with doubly metric dimension two. Furthermore, the exact values are obtained for the doubly metric dimensions of corona product graphs, being the corona either a path or a cycle.