Let $G=(V,E)$ be a finite undirected graph. An edge set $E' \subseteq E$ is a {\em dominating induced matching} ({\em d.i.m.}) in $G$ if every edge in $E$ is intersected by exactly one edge of $E'$. The \emph{Dominating Induced Matching} (\emph{DIM}) problem asks for the existence of a d.i.m.\ in $G$; this problem is also known as the \emph{Efficient Edge Domination} problem; it is the Efficient Domination problem for line graphs. The DIM problem is \NP-complete even for very restricted graph classes such as planar bipartite graphs with maximum degree 3 but is solvable in linear time for $P_7$-free graphs, and in polynomial time for $S_{1,2,4}$-free graphs as well as for $S_{2,2,2}$-free graphs and for $S_{2,2,3}$-free graphs. In this paper, combining two distinct approaches, we solve it in polynomial time for $S_{1,1,5}$-free graphs.

中文翻译：

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在多项式时间内在 $S_{1,1,5}$-Free 图中寻找主导诱导匹配

令 $G=(V,E)$ 是一个有限无向图。如果$E$ 中的每条边都与$E'$ 的一条边相交，则边集$E' \subseteq E$ 是$G$ 中的{\em 支配诱导匹配}（{\em dim}）。\emph{支配诱导匹配}（\emph{DIM}）问题要求在$G$ 中存在一个dim\；这个问题也被称为 \emph{Efficient Edge Domination} 问题；这是线图的有效支配问题。即使对于非常有限的图类（例如最大阶数为 3 的平面二部图），DIM 问题也是 \NP 完全的，但对于无 $P_7$ 的图可以在线性时间内求解，对于 $S_{1,2,4 可以在多项式时间内求解}$-free 图形以及 $S_{2,2,2}$-free 图形和 $S_{2,2,3}$-free 图形。在本文中，结合两种不同的方法，我们在多项式时间内求解 $S_{1,1,5}$-free 图。