On Efficient Domination for Some Classes of $H$-Free Chordal Graphs

Abstract

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 $S_{1,2,3}$.

References in corpus (3)

• Polynomial-time Algorithms for Weighted Efficient Domination Problems in AT-free Graphs and Dually Chordal Graphs
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• Weighted Efficient Domination for $P_6$-Free Graphs in Polynomial Time

Cited by in corpus (1)

• On Chordal-$k$-Generalized Split Graphs