The complement bi-metric dimension of graphs

Abstract

Given G=(V, E) be a connected graph with vertex set V(G), edge set E(G). For S={s1, s2, s3, … , sk} ⊆ V(G) and each vertex u∈V(G), we associate a pair of k-dimensional vectors (a, b), with a=(d(u, s1), d(u, s2), … , d(u, sk)) and b=(δ(u, s1), δ(u, s2), … , δ(u, sk)), where d(u, sk) and δ(u, sk) respectively denote lengths of a shortest and longest paths between u and sk. If for every two vertices u, v∈V(G) with u≠v resulting in r(u|S)≠r(v|S), then S is a bi-resolving set in G. Bi-metric dimension of G denoted by βb(G) is bi-resolving set S whose cardinality is minimum. The purpose of this study is to develop a new concept of a type of bimetric dimension of a graph G called complement bi-metric dimension of G, βb¯(G), and give exact value of βb¯(G), where G is a graph Pn, Kn, Cn, and Sn, as well as further analyze.