Locating Hop Sets in a Graph

Abstract

Let G be a connected graph with vertex set V (G) and edge set E(G). The open hop neighborhood of vertex v ∈ V (G) is the set NG(v, 2) = {w ∈ V (G) : dG(v, w) = 2}, where dG(v, w) denotes the distance between v and w. A non-empty set S ⊆ V (G) is a locating hop set of G if NG(u, 2) ∩ S ̸= NG(v, 2) ∩ S for every pair of distinct vertices u, v ∈ V (G) \ S. The smallest cardinality of a locating hop set of G, denoted by lhn(G) is called the locating hop number of G. This study focuses mainly on the concept of locating hop set in graphs. Characterizations of locating hop sets in the join and corona of two graphs are given and bounds for the corresponding locating hop numbers of these graphs are determined.