Using Edge Contractions and Vertex Deletions to Reduce the Independence Number and the Clique Number

TL;DR

This paper investigates the computational complexity of reducing graph parameters like independence and clique numbers using vertex deletions and edge contractions, establishing NP-hardness results and providing efficient algorithms for specific graph classes.

Contribution

It proves NP-hardness of parameter reduction via vertex deletions and edge contractions in various graph classes and offers a polynomial-time algorithm for bipartite graphs.

Findings

NP-hardness for independence number reduction in chordal graphs

NP-hardness for clique number reduction in certain H-free graphs

Polynomial-time algorithm for bipartite graphs with edge contraction

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 also 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. Our results answer several open questions stated in [Diner et al., Theoretical Computer Science, 746, p. 49-72 (2012)].