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, s1), d(v, s2), . . . , 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 = {s1, s2, . . . , sk} of vertex set G. If rm(v|W) 6= 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 ∼= K1, G H is equal to the graph produced by adding one pendant edge to every vertex of G. If H ∼= mK1 where mK1 is union of trivial graph K1, 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