计算无孪生强连接点的线性时间算法及相关问题
Linear-Time Algorithms for Computing Twinless Strong Articulation Points and Related Problems
摘要：
一个有向图$G=(V,E)$是无孪生强连通的，如果它包含一个强连通生成子图，且没有任何一对反平行（或孪生）边.有向图$G$的无孪生强连通分支（TSCC）是它的极大无孪生强连通子图。这些概念有几个不同的应用，如电信网络的设计和建筑物的结构稳定性。如果$v$的缺失增加了$G$的TSCC的数量，则顶点$v \in V$是$G$的无孪生强铰接点。在这里，我们提出了第一个线性时间算法，找到所有的双强连接点的有向图。我们证明了无孪生强连接点的计算归结为无向图中的以下问题，这可能是独立的兴趣：给定一个$2$ -顶点连通（双连通）的无向图$H$，找到属于顶点边割对的所有顶点$v$，即，对于其存在边缘$e$使得$H \setminus \{v,e\}$不被连接。我们开发了一个线性时间算法，不仅找到所有这样的顶点$v$，但也计算边$e$的数量，使$H \setminus \{v,e\}$不连接。这也意味着，对于强连通有向图$G$中的每个非孪生强铰接点$v$，我们可以计算$G \setminus v$中TSCC的数目。我们注意到，计算属于顶点-边缘割对的所有顶点的问题可以通过利用$H$的$3$ -顶点连接（三连接）分量的结构在线性时间内解决，该结构由$H$的SPQR树表示。然而，我们的方法在概念上是简单的，因此可能更适合于实际实现。
英文摘要：
A directed graph $G=(V,E)$ is twinless strongly connected if it contains a strongly connected spanning subgraph without any pair of antiparallel (or twin) edges. The twinless strongly connected components (TSCCs) of a directed graph $G$ are its maximal twinless strongly connected subgraphs. These concepts have several diverse applications, such as the design of telecommunication networks and the structural stability of buildings. A vertex $v \in V$ is a twinless strong articulation point of $G$ if the deletion of $v$ increases the number of TSCCs of $G$. Here, we present the first linear-time algorithm that finds all the twinless strong articulation points of a directed graph. We show that the computation of twinless strong articulation points reduces to the following problem in undirected graphs, which may be of independent interest: Given a $2$-vertex-connected (biconnected) undirected graph $H$, find all vertices $v$ that belong to a vertex-edge cut-pair, i.e., for which there exists an edge $e$ such that $H \setminus \{v,e\}$ is not connected. We develop a linear-time algorithm that not only finds all such vertices $v$, but also computes the number of edges $e$ such that $H \setminus \{v,e\}$ is not connected. This also implies that for each twinless strong articulation point $v$ which is not a strong articulation point in a strongly connected digraph $G$, we can compute the number of TSCCs in $G \setminus v$. We note that the problem of computing all vertices that belong to a vertex-edge cut-pair can be solved in linear-time by exploiting the structure of $3$-vertex-connected (triconnected) components of $H$, represented by an SPQR tree of $H$. Our approach, however, is conceptually simple, and thus likely to be more amenable to practical implementations.
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