The edge hop domination number of a graph

Abstract

Let G = (V,E) be a graph. A set S ⊆ E(G) is called an edge hop dominating set if S=E(G) or for every g∈E(G)\S, there exists h∈S such that d(g,h) = 1. The minimum cardinality of an edge hop domination set of G is called the edge hop domination number of G is denoted by γ_eh(G). The edge hop domination number of some standard graphs are determined. It is proved that for any two connected graphs H and K of orders n₁ and n₂ respectively, γ_eh(H+K)=3. Also it is proved that for any two connected graphs of sizes m₁≥3 and m₂≥3 respectively, γ_eh(H◦K)≤m1.