Blocking total dominating sets via edge contractions

E Galby,F Mann,B Ries

doi:10.1016/j.tcs.2021.03.028

Abstract

In this paper, we study the problem of deciding whether the total domination number of a given graph G can be reduced using exactly one edge contraction (called 1-Edge Contraction(γt)). We focus on several graph classes and determine the computational complexity of this problem. By putting together these results, we manage to obtain a complete complexity dichotomy for H-free graphs.

Highlights

In this paper, we consider the problem of reducing the total domination number of a graph by contracting a single edge
We ask for a specific graph parameter π to decrease: given a graph G, a set O of one or more graph operations and an integer k ≥ 1, the question is whether G can be transformed into a graph G by using at most k operations from O such that π (G ) ≤ π (G) − d for some threshold d ≥ 0
We provide a complete dichotomy for 1-Edge Contraction(γt ) in H -free graphs

Summary

Introduction

We consider the problem of reducing the total domination number of a graph by contracting a single edge. One can always reduce by at least 1 the total domination number of a connected graph G by using at most 3 edge contractions ([14, Theorem 4.3]) They prove the following theorem, which is a crucial result for our work. The authors in [10] considered the domination number, i.e. they considered the problem above but with γ (G) instead of γt (G) denoted by 1-Edge Contraction(γ ) In particular they showed that if H is not an induced subgraph of P3 + p P2 + t K1, for p ≥ 1 and t ≥ 0 1-Edge Contraction(γ ) is polynomial-time solvable on H -free graphs if and only if H is an induced subgraph of P5 + t K1, for t ≥ 0.

Preliminaries

Hardness results

Algorithms

Proof of Theorem 3

Conclusion