Equality of strong domination and chromatic strong domination in graphs

A subset $D$ of a vertex set $V$ of a graph $G$ is a chromatic strong dominating set if $D$ is a strong dominating set and $chi()=chi(G)$. The minimum cardinality of chromatic strong dominating set is called chromatic strong domination number of $G$ and is denoted by $gamma_{s}^c(G)$. In this paper we study relationship between strong domination and chromatic strong domination of a graph.