On Locating-Dominating Set of Regular Graphs

Abstract

Let G be a simple, connected, and finite graph. For every vertex v∈VG, we denote by NGv the set of neighbours of v in G. The locating-dominating number of a graph G is defined as the minimum cardinality of W ⊆ VG such that every two distinct vertices u,v∈VG\W satisfies ∅≠NGu∩W≠NGv∩W≠∅. A graph G is called k-regular graph if every vertex of G is adjacent to k other vertices of G. In this paper, we determine the locating-dominating number of k-regular graph of order n, where k=n−2 or k=n−3.