Outer-Convex Hop Domination in Graphs Under Some Binary Operations

Abstract

Let G be a graph with vertex and edge sets V (G) and E(G), respectively. A set C ⊆ V (G) is called an outer-convex hop dominating if for every two vertices x, y ∈ V (G) \ C, the vertex set of every x−y geodesic is contained in V (G) \ C and for every a ∈ V (G) \ C, there exists b ∈ C such that dG(a, b) = 2. The minimum cardinality of an outer-convex hop dominating set of G, denoted by ̃γconh(G), is called the outer-convex hop domination number of G. In this paper, we generate some formulas for the parameters of some special graphs and graphs under some binary operations by characterizing first the outer-convex hop dominating sets of each of these
graphs. Moreover, we establish realization result that identifies and determines the connection of this parameter with the standard hop domination parameter. It shows that given any graph, this new parameter is always greater than or equal to the standard hop domination parameter.