Hop Independent Hop Domination in Graphs

Abstract

Let G be an undirected graph with vertex and edge sets V (G) and E(G), respectively. A set S ⊆ V (G) is called a hop independent hop dominating set of G if S is both hop independent and hop dominating set of G. The minimum cardinality of hop independent hop dominating set of G, denoted by γhih(G), is called the hop independent hop domination number of G. In this paper, we show that the hop independent hop domination number of a graph G lies between the hop domination number and the hop independence number of graph G. We characterize these types of sets in the shadow graph, join, corona, and lexicographic product of two graphs. Moreover, either exact values or bounds of the hop independent hop domination numbers of these graphs are given.