Edge injective domination of graphs

Abstract

Let G = (V,E) be a simple graph. A subset X of E is called edge injective dominating set (edge Inj-dominating set) if for every edge f ∈ E - X there exists an edge g ∈ X such that |Γ(f,g)| ≥ 1, where |Γ(f,g)| is the number of common edge neighbors between the edges f and g. The minimum cardinality of such edge dominating set denoted by γ'_in(G) and is called edge injective domination number (edge Inj-domination number) of G. In this paper, we introduce the edge injective domination number, injective independence edge number (Inj-independence edge number) β'_in(G) and edge injective domatic number (edge Inj-domatic number) d'_in(G) of a graph G. Exact values for some standard graphs, bounds and some interesting results are established.