The complexity of blocking (semi)total dominating sets with edge contractions

TL;DR

This paper investigates the computational complexity of reducing a graph's (semi)total domination number through a fixed number of edge contractions, establishing NP-hardness results and complexity classifications for specific cases.

Contribution

It proves NP-hardness for deciding the necessity of exactly two or three edge contractions and provides complexity dichotomies for these cases on certain graph classes.

Findings

Deciding if two edge contractions suffice is NP-hard.

Deciding if three edge contractions suffice is NP-hard.

Complexity dichotomies are established for monogenic graph classes.

Abstract

We consider the problem of reducing the (semi)total domination number of graph by one by contracting edges. It is known that this can always be done with at most three edge contractions and that deciding whether one edge contraction suffices is an $\mathsf{NP}$-hard problem. We show that for every fixed $k \in \{2,3\}$, deciding whether exactly $k$ edge contractions are necessary is $\mathsf{NP}$-hard and further provide for $k=2$ complete complexity dichotomies on monogenic graph classes.