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For given graph $H$, the independence number $\alpha(H)$ of $H$, is the sizeof the maximum independent set of $V(H)$. Finding the maximum independent setin a graph is a NP-hard problem. Another version of the independence number isdefined as the size of the maximum induced forest of $H$, and called the forestnumber of $H$, and denoted by $f(H)$. Finding $f(H)$ is also a NP-hard problem.Suppose that $H=(V(H),E(H))$ be a graph, and $\G$ be a family of graphs, agraph $H$ has a $\G$-free $k$-coloring if there exists a decomposition of$V(H)$ into sets $V_i$, $i-1,2,\ldots,k$, so that $G\nsubseteq H[V_i]$ for each$i$, and $G\in\G$. $S\subseteq V(H)$ is $G$-free, where the subgraph of $H$induced by $S$, be $G$-free, i.e. it contains no copy of $G$. Finding a maximumsubset of $H$, so that $H[S]$ be a $G$-free graph is a very hard problem aswell. In this paper, we study the generalized version of the independencenumber of a graph. Also giving some bounds about the size of the maximum$G$-free subset of graphs is another purpose of this article.

Some bounds on the size of Maximum $G$-free sets in graph