On the local metric dimension of a lollipop graph, a web graph, and a friendship graph

Abstract

Let G be a simple connected graph with the vertex set V(G) and the edge set E(G). The distance between two vertices u and v, denoted by d(u, v), is the length of a shortest u − v path. If W = {w1, w2, w3, …, wn} is a finite set of vertices of G and v ∈ V(G), then the representation of v with respect to W is an ordered n-tuple r(v|W) = (d(v, w1), d(v, w2), d(v, w3), …, d(v, wn)). The set W is called a local metric generator for G if every two adjacent vertices of G has distinct representations. A minimum local metric generator is called a local metric basis for G and its cardinality is called the local metric dimension of G. A lollipop graph Lm,n for m ≥ 3 and n ≥ 2 is the graph obtained by joining a complete graph Km to a path graph Pn with a bridge. A web graph Wn for n ≥ 3 is a generalized prism graph Yn+1,3 with the edge of the outer cycle removed. A friendship graph fn for n ≥ 2 is a graph constructed by joining n copies of the cycle graph C3 with a common vertex. In this paper, we determine the local metric dimension of a lollipop graph, a web graph, and a friendship graph.