  In this note, we consider the following problem: given a connected graph $G$,
can we reduce the domination number of $G$ by using only one edge contraction?
We show that the problem is polynomial-time solvable on $P_3+kP_2$-free graphs
for any $k \geq 0$ which combined with results of [1,2] leads to a complexity
dichotomy of the problem on $H$-free graphs.
