Reducing graph transversals via edge contractions

Paloma T Lima,Vinicius F Dos Santos,Ignasi Sau,Uéverton S Souza

doi:10.1016/j.jcss.2021.03.003

Abstract

For a graph invariant π, the Contraction(π) problem consists of, given a graph G and positive integers k,d, deciding whether one can contract k edges of G to obtain a graph in which π has dropped by at least d. Galby et al. [ISAAC 2019, MFCS 2019] studied the case where π is the size of a minimum dominating set. We focus on graph invariants defined as the minimum size of a vertex set that hits all the occurrences of graphs in a collection H according to a fixed containment relation. We prove co-NP-hardness results under some assumptions on the graphs in H, in particular implying that Contraction(π) is co-NP-hard for fixed k=d=1 when π is the size of a minimum feedback vertex set or an odd cycle transversal. In sharp contrast, when π is the size of a minimum vertex cover, the problem is in XP parameterized by d.

Highlights

Graph modification problems play a central role in algorithmic graph theory and have been widely studied in the last few years [5, 12, 20]
We show (Theorem 4) that 1-Contraction(τH≺, 1) is co-NPhard when H is a family of 2-connected graphs containing at least one non-complete graph and ≺ is any of the subgraph, induced subgraph, minor, or topological minor containment relations
Checking whether an monadic second order (MSO) formula φ holds on an n-vertex graph of treewidth at most tw can be done in time f (φ, tw) · n, for a computable function f

Summary

Introduction

Graph modification problems play a central role in algorithmic graph theory and have been widely studied in the last few years [5, 12, 20]. We show (Theorem 4) that 1-Contraction(τH≺, 1) is co-NPhard when H is a family of 2-connected graphs containing at least one non-complete graph and ≺ is any of the subgraph, induced subgraph, minor, or topological minor containment relations This implies that it is co-NP-hard to test whether we can reduce the feedback vertex set number or the odd cycle transversal number of a graph by performing one edge contraction. We show that the picture changes completely when the parameter in question is the vertex cover number of a graph (denoted by vc): we prove (Theorem 15) that Contraction(vc) can be solved in XP time parameterized by d on general graphs, in polynomial time for fixed d, in particular for d = 1 This result should be compared to Proposition 1, which shows that the problem is hard for d = 1 if the edge to be contracted is prescribed. The proofs of the results marked with “( )” have been moved to the full version of this article, available at https: //arxiv.org/abs/2005.01460

Preliminaries

Hardness results

The case of Vertex Cover

Conclusions and further research