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

Abstract: 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 at most $k$ graph operations from some fixed set $S$? As parameters we take the chromatic number $\chi$, clique number $\omega$ and independence number $\alpha$, and as operations we choose the edge contraction ec and vertex deletion 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 of subclasses of perfect graphs. We use these results to determine the complexity of the problem for $S={\mbox{ec}}$ and $S={\mbox{vd}}$ and $\pi\in {\chi,\omega,\alpha}$ restricted to $H$-free graphs.