  The regularity of an edge ideal of a finite simple graph $G$ is at least the
induced matching number of $G$ and is at most the minimum matching number of
$G$. If $G$ possesses a dominating inuduced matching, i.e., an induced matching
which forms a maximal matching, then the induced matching number of $G$ is
equal to the minimum matching number of $G$. In the present paper, from
viewpoints of both combinatorics and commutative algebra, finite simple graphs
with dominating induced matchings will be mainly studied.
