The local metric dimension of edge corona and corona product of cycle graph and path graph

Abstract

Let G be a nontrivial connected graph with vertex set V(G). For an ordered set W = {w1, w2, …, wn} of n distinct vertices in G, the representation of a vertex v ∈ V(G) with respect to W is an ordered value of distance between v and every vertex of W . The set W is a local metric set of G if the representations of every pair of adjacent vertices with respect to W are different. The local metric set with minimum cardinality is called local metric basis and its cardinality is the local metric dimension of G and denoted by diml(G). The edge corona product of cycle graph and path graph denoted by Cm ʘ Pn, this graph is obtained from a cycle graph Cm and m copies of path graph Pn, and then joining two end-vertices of ith edge of Cm to every vertex in the ith copy of Pn, where 1 ≤ i ≤ m. The corona product of cycle graph and path graph denoted by Cm ◊ Pn, this graph is obtained from a cycle graph Cm and m copies of path graph Pn, and then joining by an edge each vertex from the ith copy of Pn with ith vertex of Cm. In this paper, we determine the local metric dimension of edge corona and corona product of cycle graph and path graph for positive integer m ≥ 3 and n ≥ 1. We obtain the local metric dimension of edge corona product of cycle and path graphs is diml(Cm ◊ Pn)=2 for n = 1 and for n ≥ 2. The local metric dimension of corona product of cycle and path graphs is diml(Cm ◊ Pn)=1 for n = 1 and even positive integer m ≥ 3, diml(Cm ʘ Pn)=2 for n = 1 and odd positive integer m ≥ 3, and for n ≥ 2.