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arXiv 1402.7213cs.DMmath.CO
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无$P_k$图的一个新刻画

A new characterization of $P_k$-free graphs

Eglantine Camby, Oliver Schaudt

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13 pages, 4 figures
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摘要：

在$k$顶点上不包含导出路径的图的类，$P_k$ -free图，在算法图论中起着突出的作用。这激发了对$P_k$ -free图的特殊结构性质的研究，包括替代性的刻画。设$G$是一个连通的无$P_k$图，$k \ge 4$ .我们证明了$G$允许一个连通控制集，其导出子图要么是$P_{k-2}$ -free的，要么同构于$P_{k-2}$。令人惊讶的是，事实证明，$G$的每个最小连通支配集都具有这个性质。这就给出了$P_k$ -free图的一个新的刻画：图$G$是$P_k$ -free的当且仅当$G$的每个连通导出子图都有一个连通控制集，其导出子图要么是$P_{k-2}$ -free的，要么是与$C_k$同构的.这改进和推广了以前的几个结果; $k=7$的特殊情况解决了van 't Hof和Paulusma提出的问题[A new characterization of $P_6$ -free graphs，COCOON 2008]。本文的第二部分给出了一个有效的算法，即给定一个$G$的$n$顶点和$m$边的连通图，计算$G$的连通控制集$X$，该连通控制集具有以下性质：对于最小$k$，使得$G$是$P_k$ -free的，由$X$诱导的子图是$P_{k-2}$ -free的或同构于$P_{k-2}$ .作为应用，我们证明了对于顶点超边关联图是$P_7$ -free的超图，超图2-着色性问题是一个NP-完全问题，可以在多项式时间内得到解决。

英文摘要：

The class of graphs that do not contain an induced path on $k$ vertices, $P_k$-free graphs, plays a prominent role in algorithmic graph theory. This motivates the search for special structural properties of $P_k$-free graphs, including alternative characterizations. Let $G$ be a connected $P_k$-free graph, $k \ge 4$. We show that $G$ admits a connected dominating set whose induced subgraph is either $P_{k-2}$-free, or isomorphic to $P_{k-2}$. Surprisingly, it turns out that every minimum connected dominating set of $G$ has this property. This yields a new characterization for $P_k$-free graphs: a graph $G$ is $P_k$-free if and only if each connected induced subgraph of $G$ has a connected dominating set whose induced subgraph is either $P_{k-2}$-free, or isomorphic to $C_k$. This improves and generalizes several previous results; the particular case of $k=7$ solves a problem posed by van 't Hof and Paulusma [A new characterization of $P_6$-free graphs, COCOON 2008]. In the second part of the paper, we present an efficient algorithm that, given a connected graph $G$ on $n$ vertices and $m$ edges, computes a connected dominating set $X$ of $G$ with the following property: for the minimum $k$ such that $G$ is $P_k$-free, the subgraph induced by $X$ is $P_{k-2}$-free or isomorphic to $P_{k-2}$. As an application our results, we prove that Hypergraph 2-Colorability, an NP-complete problem in general, can be solved in polynomial time for hypergraphs whose vertex-hyperedge incidence graph is $P_7$-free.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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