Metric Identification of Vertices in Polygonal Cacti

Abstract

The distance between two vertices u and v in a connected graph G is the number of edges lying in a shortest path (geodesic) between them. A vertex x of G performs the metric identification for a pair (u, v) of vertices in G if and only if the equality between the distances of u and v with x implies that u = v (That is, the distance between u and x is different from the distance between v and x). The minimum number of vertices performing the metric identification for every pair of vertices in G defines the metric dimension of G. In this paper, we perform the metric identification of vertices in two types of polygonal cacti: chain polygonal cactus and star polygonal cactus.