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We study the following problem: for given integers $d$, $k$ and graph $G$,can we reduce some fixed graph parameter $\pi$ of $G$ by at least $d$ via atmost $k$ graph operations from some fixed set $S$? As parameters we take thechromatic number $\chi$, clique number $\omega$ and independence number$\alpha$, and as operations we choose the edge contraction ec and vertexdeletion vd. We determine the complexity of this problem for $S=\{\mbox{ec}\}$and $S=\{\mbox{vd}\}$ and $\pi\in \{\chi,\omega,\alpha\}$ for a number ofsubclasses of perfect graphs. We use these results to determine the complexityof the problem for $S=\{\mbox{ec}\}$ and $S=\{\mbox{vd}\}$ and $\pi\in\{\chi,\omega,\alpha\}$ restricted to $H$-free graphs.

Contraction and Deletion Blockers for Perfect Graphs and $H$-free Graphs