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

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.Comment: 38 pages, 13 figure