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arXiv 1706.09301cs.DM
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无$S_{1,2,4}$图的支配诱导匹配

Dominating Induced Matchings in $S_{1,2,4}$-Free Graphs

Andreas Brandstädt, Raffaele Mosca

备注：
arXiv admin note: substantial text overlap with arXiv:1507.06541, arXiv:1706.04894
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摘要：

设$G=(V,E)$是一个无圈无重边的有限无向图.边的子集$M \subseteq E$是{ \em支配诱导匹配}（{ \em d.i.m.}）在$G$中，如果$E$中的每条边都恰好被$M$的一条边包围。特别地，这意味着$M$是一个诱导匹配，并且不在$M$中的每条边与$M$中的一条边恰好共享一个顶点。显然，不是每个图都有d.i.m.。\emph{Dominating Induced Matching}（\emph{DIM}）问题要求存在d.i.m.\在$G$中;这个问题也被称为\emph{Efficient Edge Domination}问题;它是线图的{ \em有效支配}问题。DIM问题是PDMATHAMV-完全的一般情况下，即使是非常有限的图类，如平面二分图与最大程度3。然而，DIM对于无爪（即，$S_{1,1,1}$ -free）图、$S_{1,2,3}$ -free图以及$S_{2,2,2}$ -free图、$P_7$ -free图在线性时间内以及$P_8$ -free图在多项式时间内（$P_k$是$S_{i,j,\ell}$的特殊情况）.在Hertz，Lozin，Ries，Zamaraev和de Werra的一篇论文中，证明了对于每个固定的$i,j,k$，无$S_{i,j,k}$图，DIM在多项式时间内是可解的。在本文中，结合两种不同的方法，我们解决了它在多项式时间的$S_{1,2,4}$ -free图，推广了$S_{1,2,3}$ -free以及$P_7$ -free的情况。

英文摘要：

Let $G=(V,E)$ be a finite undirected graph without loops and multiple edges. A subset $M \subseteq E$ of edges is a {\em dominating induced matching} ({\em d.i.m.}) in $G$ if every edge in $E$ is intersected by exactly one edge of $M$. In particular, this means that $M$ is an induced matching, and every edge not in $M$ shares exactly one vertex with an edge in $M$. Clearly, not every graph has a d.i.m. 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 {\em Efficient Domination} problem for line graphs. The DIM problem is \NP-complete in general, and even for very restricted graph classes such as planar bipartite graphs with maximum degree 3. However, DIM is solvable in polynomial time for claw-free (i.e., $S_{1,1,1}$-free) graphs, for $S_{1,2,3}$-free graphs as well as for $S_{2,2,2}$-free graphs, in linear time for $P_7$-free graphs, and in polynomial time for $P_8$-free graphs ($P_k$ is a special case of $S_{i,j,\ell}$). In a paper by Hertz, Lozin, Ries, Zamaraev and de Werra, it was conjectured that DIM is solvable in polynomial time for $S_{i,j,k}$-free graphs for every fixed $i,j,k$. In this paper, combining two distinct approaches, we solve it in polynomial time for $S_{1,2,4}$-free graphs which generalizes the $S_{1,2,3}$-free as well as the $P_7$-free case.

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