  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 edge
contraction? 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 when
restricted to subcubic claw-free graphs and $2P_3$-free graphs. As a
consequence, we are able to establish a complexity dichotomy for the problem on
$H$-free graphs when $H$ is connected.
