Locating-hop domination in graphs

Abstract

A subset S of V (G), where G is a simple undirected graph, is a hop dominating set if for each v ∈ V (G) \ S, there exists w ∈ S such that dG(v, w) = 2 and it is a locating hop set if NG(v, 2) ∩ S 6= NG(v, 2) ∩ S for any two distinct vertices u, v ∈ V (G) \ S. A set S ⊆ V (G) is a locating-hop dominating set if it is both a locating-hop and a hop dominating set of G. The minimum cardinality of a locating-hop dominating set of G, denoted by γlh(G), is called the locating-hop domination number of G. In this paper, we investigate some properties of this newly defined parameter. In particular, we characterize the locating-hop dominating sets in graphs under some binary operations.