On the local multiset dimension of graph with homogenous pendant edges

Abstract

Let G be a connected graph with E as edge set and V as vertex set. rm (v|W) = {d(v, s 1), d(v, s 2),…, d(v, sk )} is the multiset representation of a vertex v of G with respect to W where d(v, si ) is a distance between of the vertex v and the vertices in W for k—ordered set W = {s 1, s 2,…,sk } of vertex set G. If rm (v|W) = rm (u|W) for every pair u, v of adjacent vertices of G, we called it as local resolving set of G. The minimum cardinality of local resolving set W is called local multiset dimension. It is denoted by μl (G). Hi ≅ H, for all i ∈ V(G). If H ≅ K 1, G ⊙ H is equal to the graph produced by adding one pendant edge to every vertex of G. If H ≅ mK 1 where mK 1 is union of trivial graph K 1, G ⊙ H is equal to the graph produced by adding one m pendant edge to every vertex of G. In this paper, we analyze the exact value of local multiset dimension on some graphs with homogeneous pendant edges.