Weighted Efficient Domination for $(P_5+kP_2)$-Free Graphs in Polynomial Time

Let $G$ be a finite undirected graph. A vertex {\em dominates} itself and all its neighbors in $G$. A vertex set $D$ is an {\em efficient dominating set} (\emph{e.d.}\ for short) of $G$ if every vertex of $G$ is dominated by exactly one vertex of $D$. The \emph{Efficient Domination} (ED) problem, which asks for the existence of an e.d.\ in $G$, is known to be \NP-complete even for very restricted graph classes such as for claw-free graphs, for chordal graphs and for $2P3$-free graphs (and thus, for $P7$-free graphs). We call a graph $F$ a {\em linear forest} if $F$ is cycle- and claw-free, i.e., its components are paths. Thus, the ED problem remains \NP-complete for $F$-free graphs, whenever $F$ is not a linear forest. Let WED denote the vertex-weighted version of the ED problem asking for an e.d. of minimum weight if one exists. In this paper, we show that WED is solvable in polynomial time for $(P5+kP2)$-free graphs for every fixed $k$, which solves an open problem, and, using modular decomposition, we improve known time bounds for WED on $(P4+P2)$-free graphs, $(P6,S{1,2,2})$-free graphs, and on $(2P3,S{1,2,2})$-free graphs and simplify proofs. For $F$-free graphs, the only remaining open case is WED on $P_6$-free graphs.