CN-edge domination in graphs

Let $G=(V,E)$ be a graph. A subset $D$ of $V$ is called common neighbourhood dominating set (CN-dominating set) if for every $v\in V-D$ there exists a vertex $u\in D$ such that $uv\in E(G)$ and $|\Gamma(u,v)|\geq1$, where $|\Gamma(u,v)|$ is the number of common neighbourhood between the vertices $u$ and $v$. The minimum cardinality of such CN-dominating set denoted by $\gamma_{cn}(G)$ and is called common neighbourhood domination number (CN-edge domination) of $G$. In this paper we introduce the concept of common neighbourhood edge domination (CN-edge domination) and common neighbourhood edge domatic number (CN-edge domatic number) in a graph, exact values for some standard graphs, bounds and some interesting results are established.