Outer-connected Hop Dominating Sets in Graphs

Abstract

Let G be an undirected graph with vertex and edge sets V (G) and E(G), respectively. A hop dominating set S ⊆ V (G) is called an outer-connected hop dominating set if S = V (G) or the subgraph ⟨V (G) \ S⟩ induced by V (G) \ S is connected. The minimum size of an outer-connected hop dominating set is the outer-connected hop domination number γfch(G). A dominating set
of size γfch(G) of G is called a γfch-set. In this paper, we investigate the concept and study it for graphs resulting from some binary operations. Specifically, we characterize the outer-connected hop dominating sets in the join, corona and lexicographic products of graphs, and determine bounds of the outer-connected hop domination number of each of these graphs.