A Variant of Hop Domination in Graphs

Let G be a connected graph with vertex and edge sets V (G) and E(G), respectively. A set S subset of V (G) is a hop dominating set of G if for each v is an element of V (G) \ S, there exists w is an element of S such that dG(v,w) = 2. A set S subset of V (G) is a super hop dominating set if ehpn(G)(v, V (G) \ S) not equal Phi for each v is an element of V (G) \ S, where ehpn(G)(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 gamma(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.