切割路径及其余项结构及其应用
Cut paths and their remainder structure, with applications
摘要：
在强连通图$G = (V,E)$中，割弧（也称为强桥）是一个弧$e \in E$，它的移除使得图不再是强连通的。等价地，存在$u,v \in V$，使得所有$u$ - $v$遍历都包含$e$。割弧是一个基本的图论概念，有无数的应用，特别是在可达性问题。在本文中，我们开始研究切割路径，作为一个概括的切割弧，我们自然地定义为这些路径$P$存在$u,v \in V$，使所有的$u$ - $v$散步包含$P$作为子散步。我们首先证明了各种性质的切割路径和定义其剩余结构，我们使用它来提出一个简单的$O(m)$时间验证算法的切割路径（$|V| = n$，$|E| = m$）。其次，我们应用割路及其剩余结构来改进生物信息学中的几个可达性问题。一条路称为安全的，如果它是强连通图的每个节点覆盖闭路的子路。多安全性的定义类似，通过考虑节点覆盖集的封闭行走代替。我们表明，切割路径提供简单的$O(m)$时间算法验证，如果步行是安全的或多安全的。对于多安全性，我们提出了第一个线性时间算法，而对于安全性，我们提出了一个简单的算法，其中最先进的采用复杂的数据结构。最后，我们表明，同时计算剩余结构的所有子路的切割路径可以在线性时间。这些属性产生输出所有最大多安全行走的$O(mn)$算法，改进了在时间上运行的最新算法$O(m^2+n^3)$。本文的结果只是触及了割路研究的表面，我们相信，一个图的丰富的结构可以揭示，考虑到一个路径的角度，而不仅仅是一个弧。
英文摘要：
In a strongly connected graph $G = (V,E)$, a cut arc (also called strong bridge) is an arc $e \in E$ whose removal makes the graph no longer strongly connected. Equivalently, there exist $u,v \in V$, such that all $u$-$v$ walks contain $e$. Cut arcs are a fundamental graph-theoretic notion, with countless applications, especially in reachability problems. In this paper we initiate the study of cut paths, as a generalisation of cut arcs, which we naturally define as those paths $P$ for which there exist $u,v \in V$, such that all $u$-$v$ walks contain $P$ as subwalk. We first prove various properties of cut paths and define their remainder structures, which we use to present a simple $O(m)$-time verification algorithm for a cut path ($|V| = n$, $|E| = m$). Secondly, we apply cut paths and their remainder structures to improve several reachability problems from bioinformatics. A walk is called safe if it is a subwalk of every node-covering closed walk of a strongly connected graph. Multi-safety is defined analogously, by considering node-covering sets of closed walks instead. We show that cut paths provide simple $O(m)$-time algorithms verifying if a walk is safe or multi-safe. For multi-safety, we present the first linear time algorithm, while for safety, we present a simple algorithm where the state-of-the-art employed complex data structures. Finally we show that the simultaneous computation of remainder structures of all subwalks of a cut path can be performed in linear time. These properties yield an $O(mn)$ algorithm outputting all maximal multi-safe walks, improving over the state-of-the-art algorithm running in time $O(m^2+n^3)$. The results of this paper only scratch the surface in the study of cut paths, and we believe a rich structure of a graph can be revealed, considering the perspective of a path, instead of just an arc.
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