Metric dimension of helm graph and double cones graph

Abstract

Let G be a graph with vertex set V(G) = {v1, v2,., vn} and edge set E(G) = {e1, e2,., en}. G in connected if there exist a path connecting every vertex in G. The distace between two vertices u and v, denoted by d(u, v), is the lenght of the shortest path from u to v in G. Let W = {w1, w2,., wk} be a subset of vertices in a connected graph G. For v ∈ V(G), a representation of v of u wih respect o W is defined as the k-tuple r(v|W) = (d(v, w1), d(v, w2),., d(v, wk)). The set of W is called a resolving set of G if every two distinct vertices u, v ∈ V(G) satisfy r(u|W) ≠ r(v|W). The resolving set of G with minimum cardinality is called a minimum resolving set or basis of G and the cardinality of minimum resolving set is called metric dimension, denoted by Dim(G). In this paper, we obtained the metric dimension of helm graph Hn and double cones graph DCn.