Blocking total dominating sets via edge contractions

TL;DR

This paper investigates how a single edge contraction can reduce the total domination number in various graph classes, providing a complete complexity classification for $H$-free graphs.

Contribution

It offers a comprehensive complexity analysis of the 1-Edge Contraction($ ext{γ}_t$) problem across different graph classes, culminating in a full dichotomy for $H$-free graphs.

Findings

Complexity results for specific graph classes

Complete dichotomy for $H$-free graphs

Identification of cases where contraction reduces total domination number

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($\gamma_t$)). 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 dichotomy for $H$-free graphs.