on γ-free, γ-totally-free and γ-fixed sets in graphs

Let $g=(v,e)$ be a connected graph. a subset $s$ of $v$ is called a $gamma$-free set if there exists a $gamma$-set $d$ of $g$ such that $s cap d= emptyset$. if further the induced subgraph $h=g[v-s]$ is connected, then $s$ is called a  $cc$-$gamma$-free set of $g$. we use this concept to identify connected induced subgraphs $h$ of a given graph $g$ such that $gamma(h) leq gamma(g)$. we also introduce the concept of $gamma$-totally-free and $gamma$-fixed sets and present several basic results on the corresponding parameters.