Neighborhood connected perfect domination in graphs

Let $G = (V, E)$ be a connected graph. A set $S$ of vertices in $G$ is a perfect dominating set if every vertex $v$ in $V-S$ is adjacent to exactly one vertex in $S$. A perfect dominating set $S$ is said to be a neighborhood connected perfect dominating set (ncpd-set) if the induced subgraph $$ is connected. The minimum cardinality of a ncpd-set of $G$ is called the neighborhood connected perfect domination number of $G$ and is denoted by $gamma_{ncp}(G)$. In this paper we initiate a study of this parameter.