A dominating induced matching, also called an efficient edge domination,
of a graph $G=(V,E)$ with $n=|V|$ vertices and $m=|E|$ edges is a subset
$F \subseteq E$ of edges in the graph such that no two edges in $F$
share a common endpoint and each edge in $E\setminus F$ is incident with
exactly one edge in $F$. It is NP-hard to decide whether a graph admits
a dominating induced matching or not. In this paper, we design a $1.1467
^nn^{O(1)}$-time exact algorithm for this problem, improving all
previous results. This problem can be redefined as a partition problem
that is to partition the vertex set of a graph into two parts $I$ and
$F$, where $I$ induces an independent set (a 0-regular graph) and $F$
induces a perfect matching (a 1-regular graph). After giving several
structural properties of the problem, we show that the problem always
contains some ``good vertices,'' branching on which by including them to
either $I$ or $F$ we can effectively reduce the graph. This leads to a
fast exact algorithm to this problem.