Convex Hop Domination in Graphs

Abstract

Let G be an undirected connected graph with vertex and edge sets V (G) and E(G), respectively. A set C ⊆ V (G) is called convex hop dominating if for every two vertices x, y ∈ C, the vertex set of every x-y geodesic is contained in C and for every v ∈ V (G) \ C, there exists w ∈ C such that dG(v, w) = 2. The minimum cardinality of convex hop dominating set of G, denoted by γconh(G), is called the convex hop domination number of G. In this paper, we show that every two positive integers a and b, where 2 ≤ a ≤ b, are realizable as the connected hop domination number and convex hop domination number, respectively, of a connected graph. We also characterize the convex hop dominating sets in some graphs and determine their convex hop domination numbers.