On the partition dimension of edge corona product of path and cycle

Abstract

Let v be a vertex of a connected graph G(V, E). Let S be a subset of V (G). For an ordered partition of V (G), the representation of a vertex with respect to is the k-vectors , where d(v, Sk) represents the distance between the vertex v and the set Sk, defined by . The partition of V (G) is a resolving partition set if the k-vektors are distinct. The minimum resolving partition is a partition dimension of G, denoted by pd(G). The edge corona of G1 and G2 is defined as the graph obtained by taking one copy of G1 and copies of G2 and then joining two end-vertices of the i-th edge of G1 to every vertex in the i-th copy of G2. In this paper, we will study the partition dimension of the edge corona of cycle and path, namely and for n ≤ 2 and m ≤ 3.