$s$ - $t$路径、小径和步道的安全
Safety in $s$-$t$ Paths, Trails and Walks
摘要：
给定有向图$G$和一对节点$s$和$t$，$G$的\emph{$s$-$t$ bridge}是一条边，该边的移除破坏了$G$的所有$s$ - $t$路径（因此出现在所有$s$ - $t$路径中）。计算$G$的所有$s$ - $t$桥是一个基本的图问题，可以在线性时间内求解.在本文中，我们认为这个问题的自然概括，从生物信息学的“安全”的概念。如果$W$是$\mathcal{W}$中所有遍历的子遍历，则我们说遍历$W$是关于$s$ - $t$遍历的集合$\mathcal{W}$的\emph{safe}。我们首先考虑当$\mathcal{W}$由以下各项组成时的最大安全行走：所有$s$ - $t$路径、所有$s$ - $t$踪迹或$G$的所有$s$ - $t$行走.我们发现，前两个问题是立即线性时间概括找到所有$s$ - $t$桥梁，而第三个问题是更多的参与。特别是，我们表明，存在一个紧凑的表示可计算的线性时间，允许输出所有最大的安全行走的时间线性的长度。我们进一步概括这些问题，通过假设，安全性只定义相对于一个子集的\emph{visible}边缘。在这里，我们证明了一个二分法之间的$s$ - $t$路径和$s$ - $t$线索的情况下，和$s$ - $t$散步的情况下：前两个是NP-难的，而后者是可解的，具有相同的复杂性，当所有的边缘是可见的。我们还表明，同样的复杂性结果适用于类似的概括\emph{$s$-$t$ articulation points}（节点出现在所有$s$ - $t$路径）。因此，我们获得了最好的结果自然的“安全”-概括这两个基本的图形问题。此外，我们的算法是简单的，不采用任何复杂的数据结构，使他们在实践中使用的理想。
英文摘要：
Given a directed graph $G$ and a pair of nodes $s$ and $t$, an \emph{$s$-$t$ bridge} of $G$ is an edge whose removal breaks all $s$-$t$ paths of $G$ (and thus appears in all $s$-$t$ paths). Computing all $s$-$t$ bridges of $G$ is a basic graph problem, solvable in linear time. In this paper, we consider a natural generalisation of this problem, with the notion of "safety" from bioinformatics. We say that a walk $W$ is \emph{safe} with respect to a set $\mathcal{W}$ of $s$-$t$ walks, if $W$ is a subwalk of all walks in $\mathcal{W}$. We start by considering the maximal safe walks when $\mathcal{W}$ consists of: all $s$-$t$ paths, all $s$-$t$ trails, or all $s$-$t$ walks of $G$. We show that the first two problems are immediate linear-time generalisations of finding all $s$-$t$ bridges, while the third problem is more involved. In particular, we show that there exists a compact representation computable in linear time, that allows outputting all maximal safe walks in time linear in their length. We further generalise these problems, by assuming that safety is defined only with respect to a subset of \emph{visible} edges. Here we prove a dichotomy between the $s$-$t$ paths and $s$-$t$ trails cases, and the $s$-$t$ walks case: the former two are NP-hard, while the latter is solvable with the same complexity as when all edges are visible. We also show that the same complexity results hold for the analogous generalisations of \emph{$s$-$t$ articulation points} (nodes appearing in all $s$-$t$ paths). We thus obtain the best possible results for natural "safety"-generalisations of these two fundamental graph problems. Moreover, our algorithms are simple and do not employ any complex data structures, making them ideal for use in practice.
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