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Given an integer $P>0$, a graph $G$, and a set of graphs $\\mathcal{F}$, we say that $G$ {\\em admits an $(\\mathcal{F},P)$-partition} if the vertex set of $G$ can be partitioned into $P$ subsets $X_1,..., X_P$, so that for every $i \\in \\{1,..., P\\}$, either $|X_i|=1$, or the subgraph of $G$ induced by $X_i$ is $\\{F\\}$-free for some $F \\in \\mathcal{F}$.\n  Our first result is the following. 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