A Note on Multiset Dimension and Local Multiset Dimension of Graphs

Abstract

All graphs in this paper are nontrivial and connected simple graphs. For a set W = {s1,s2,...,sk} of verticesof G, the multiset representation of a vertex v of G with respect to W is r(v|W) = {d(v,s1),d(v,s2),...,d(v,sk)} whered(v,si) is the distance between of v and si. If the representation r(v|W)̸= r(u|W) for every pair of vertices u,v of a graph G, the W is called the resolving set of G, and the cardinality of a minimum resolving set is called the multiset dimension, denoted by md(G). A set W is a local resolving set of G if r(v|W) ̸= r(u|W) for every pair of adjacent vertices u,v of a graph G. The cardinality of a minimum local resolving set W is called local multiset dimension, denoted by µl(G). In our paper, we discuss the relationship between the multiset dimension and local multiset dimension of graphs and establish bounds of local multiset dimension for some families of graph.