  A vertex set $D$ in a finite undirected graph $G$ is an efficient dominating
set (e.d.s. for short) of $G$ if every vertex of $G$ is dominated by exactly
one vertex of $D$. The Efficient Domination (ED) problem, which asks for the
existence of an e.d.s.\ in $G$, is known to be \NP-complete even for very
restricted graph classes such as for $2P_3$-free chordal graphs while it is
solvable in polynomial time for $P_6$-free chordal graphs (and even for
$P_6$-free graphs). A standard reduction from the \NP-complete Exact Cover
problem shows that ED is \NP-complete for a very special subclass of chordal
graphs generalizing split graphs. The reduction implies that ED is \NP-complete
e.g.\ for double-gem-free chordal graphs while it is solvable in linear time
for gem-free chordal graphs (by various reasons such as bounded clique-width,
distance-hereditary graphs, chordal square etc.), and ED is \NP-complete for
butterfly-free chordal graphs while it is solvable in linear time for
$2P_2$-free graphs.
  We show that (weighted) ED can be solved in polynomial time for $H$-free
chordal graphs when $H$ is net, extended gem, or $S_{1,2,3}$.
