Reducing Graph Parameters by Contractions and Deletions

TL;DR

This paper investigates the complexity of reducing graph parameters like independence number and chromatic number through fixed operations such as vertex deletion, edge contraction, and edge deletion, providing new NP-hardness results and polynomial algorithms.

Contribution

It establishes NP-hardness for several graph parameter reduction problems and offers polynomial algorithms for specific graph classes, completing the complexity classification for these problems.

Findings

NP-hardness for independence number reduction via vertex deletion and edge contraction.

Polynomial-time algorithm for bipartite graphs with edge contraction and fixed d.

NP-hardness in (C_3+P_1)-free graphs for clique number reduction.

Abstract

We consider the following problem: for a given graph $G$ and two integers $k$ and $d$, can we apply a fixed graph operation at most $k$ times in order to reduce a given graph parameter $\pi$ by at least $d$? We show that this problem is NP-hard when the parameter is the independence number and the graph operation is vertex deletion or edge contraction, even for fixed $d=1$ and when restricted to chordal graphs. We give a polynomial time algorithm for bipartite graphs when the operation is edge contraction, the parameter is the independence number and $d$ is fixed. Further, we complete the complexity dichotomy on $H$-free graphs when the parameter is the clique number and the operation is edge contraction by showing that this problem is NP-hard in $(C_3+P_1)$-free graphs even for fixed $d=1$. When the operation is edge deletion and the parameter is the chromatic number, we determine the computational complexity of the associated problem on cographs and complete multipartite graphs. Our results answer several open questions stated in [Diner et al., Theoretical Computer Science, 746, p. 49-72 (2012)].