  Given a graph $G=(V,E)$, $S\subseteq V$ is a dominating set if every $v\in
V\setminus S$ is adjacent to an element of $S$. The Minimum Dominating Set
problem asks for a dominating set with minimum cardinality. It is well known
that its decision version is $NP$-complete even when $G$ is a claw-free graph.
We give a complexity dichotomy for the Minimum Dominating Set problem for the
class of $(claw, H)$-free graphs when $H$ has at most six vertices. In an
intermediate step we show that the Minimum Dominating Set problem is
$NP$-complete for cubic graphs.
