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In this paper, we consider the following problem: given a connected graph$G$, can we reduce the domination number of $G$ by one by using only one edgecontraction? We show that the problem is $\mathsf{NP}$-hard when restricted to$\{P_6,P_4+P_2\}$-free graphs and that it is $\mathsf{coNP}$-hard whenrestricted to subcubic claw-free graphs and $2P_3$-free graphs. As aconsequence, we are able to establish a complexity dichotomy for the problem on$H$-free graphs when $H$ is connected.

Blocking dominating sets for $H$-free graphs via edge contractions