Finding Efficient Domination for $P_8$-Free Bipartite Graphs in Polynomial Time

A vertex set $D$ in a finite undirected graph $G$ is an {\em efficient dominating set} (\emph{e.d.s.}\ 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.s.\ in $G$, is known to be \NP-complete for $P7$-free graphs, and even for very restricted $H$-free bipartite graph classes such as for $K{1,4}$-free bipartite graphs as well as for $C4$-free bipartite graphs while it is solvable in polynomial time for $P7$-free bipartite graphs as well as for $S{2,2,4}$-free bipartite graphs. Here we show that ED can be solved in polynomial time for $P8$-free bipartite graphs.