Abstract Let G = ( V , E ) be a finite undirected graph. An edge set E ′ ⊆ E is a dominating induced matching (d.i.m.) in G if every edge in E is intersected by exactly one edge of E ′ . The Dominating Induced Matching (DIM) problem asks for the existence of a d.i.m. in G ; this problem is also known as the 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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在多项式时间内在无 S1,1,5 的图中寻找主导诱导匹配

摘要 令 G = ( V , E ) 为有限无向图。如果 E 中的每条边恰好与 E ' 的一条边相交，则边集 E ' ⊆ E 是 G 中的主导诱导匹配（dim）。支配诱导匹配 (DIM) 问题要求 G 中存在暗点；这个问题也被称为 Efficient Edge Domination 问题；这是线图的有效支配问题。DIM 问题是 NP 完全的，即使对于非常有限的图类，例如最大度数为 3 的平面二部图，但对于无 P 7 的图可以在线性时间内求解，对于无 S 1 , 2 , 4 的图可以在多项式时间内求解为以及 S 2 , 2 , 2 自由图和 S 2 , 2 , 3 自由图。在本文中，结合两种不同的方法，我们在多项式时间内求解无 S 1 , 1 , 5 的图。