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

## TL;DR

This paper investigates the computational complexity of reducing graph parameters like chromatic, clique, and independence numbers using edge contractions and vertex deletions within perfect and H-free graphs.

## Contribution

It characterizes the complexity of parameter reduction problems for specific graph classes and operations, extending understanding of graph modification problems.

## Key findings

- Complexity results for contraction and deletion operations on perfect graphs.
- Extension of complexity analysis to H-free graphs.
- Identification of cases where parameter reduction is computationally feasible or hard.

## 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.