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 chordal bipartite graphs as well as for $P_7$-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 $C_4$-free bipartite graphs while it is solvable in polynomial time for $P_8$-free bipartite graphs as well as for $S_{1,3,3}$-free bipartite graphs and for $S_{1,1,5}$-free bipartite graphs. Here we show that ED can be solved in polynomial time for $(S_{1,2,5},S_{3,3,3})$-free chordal bipartite graphs.
翻译:在限定的无方向图形中设定的顶点$D$G$是 $7美元的高效支配设置 (\ emph{ e.d.s. ⁇ @s. surrect) $G$,如果每张G$的顶点完全由1美元的顶点支配的话,顶点为$D美元。 顶点为$G$。 以$G$计的顶点设置为$G$,已知是 $NP- 完成的,用于chordal 双面图和$P$7的非正方块图,甚至对于非常有限的H$免费双面图类,如$1,4美元无双面图,以及$4美元无双面图,而对于$8美元免费双面图和$1,3美元无双面图。