Local Metric Dimension of Certain Classes of Circulant Networks

Abstract

Let G(V,E) be a graph with a set of vertices V and a set of edges E. Then, a minimum subset Wl of V is said to be a local metric basis of G if for any two adjacent vertices u,v∈V∖Wl there exists a vertex w∈Wl such that d(u,w) ≠ d(v,w). The cardinality of a local metric basis is referred to as the local metric dimension of the graph G denoted by βl(G). In this paper, we investigate the local metric dimensions of certain circulant-related architectures such as Harary graphs Hk,n with even k or n, Toeplitz networks, and ILLIAC networks.