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 $S_{1,2,3}$.

This is joint work with Raffaele Mosca.