On the middle CN-dominating graphs

Abstract

Let $G = (V,E)$ be a graph and $A(G)$ is the collection of all minimal CN-dominating sets of $G$. The middle CN-dominating graph of $G$ is the graph denoted by $M_{cnd}(G)$ with vertex set the disjoint union of $V \cup A(G)$ and $(u, v)$ is an edge if and only if $u \cap v \neq\phi$ whenever $u, v \in A(G)$ or $u \in v$ whenever $u \in v$ and $v \in A(G)$. In this paper, characterizations are given for graphs whose middle CN-dominating graph is connected and $K_{p}\subseteq M_{cnd}(G)$. Other properties of middle CN-dominating graphs are also obtained.