A variant of hop domination in graphs

Abstract

Let G be a connected graph with vertex and edge sets V (G) and E(G), respectively. A set S ⊆ V (G) is a hop dominating set of G if for each v ∈ V (G) \ S, there exists w ∈ S such that dG(v, w) = 2. A set S ⊆ V (G) is a super hop dominating set if ehpnG(v, V (G) \ S) ̸= ∅ for each v ∈ V (G) \ S, where ehpnG (v, V (G) \ S) is the set containing all the external hop private neighbors of v with respect to V (G) \ S. The minimum cardinality of a super hop dominating set of G, denoted by γ s h (G), is called the super hop domination number of G. In this paper, we investigate the concept and study it for graphs resulting from some binary operations. Specifically, we characterize the super hop dominating sets in the join, and lexicographic products of graphs, and determine bounds of the super hop domination number of each of these graphs.