关于混合图的2-强连通定向及相关问题
On 2-strong connectivity orientations of mixed graphs and related problems
AI总结：
本文研究混合图的2-强连通定向问题，定义了移除任意边后仍可通过定向保持强连通的极大顶点集，通过归约为有向图2eTSCC计算实现线性时间求解，可应用于网络设计与建筑结构稳定性领域。
摘要：
混合图$G$是同时包含无向边和有向边的图。$G$的一个定向通过为$G$的所有无向边指定方向来形成，即将每条无向边$\u200b{u,v}$转化为有向边$(u,v)$或$(v,u)$。寻找混合图的一个定向使其成为强连通图的问题已得到充分研究，并且可以在线性时间内求解。本文中，我们提出混合图中的如下定向问题：给定一个混合图$G$，我们希望计算其极大顶点集$C_1,C_2,\u2026,C_k$，满足如下性质：从$G$中移除任意一条边$e$（有向或无向）后，$G\u200b{e}$存在一个定向$R_i$，使得$C_i$中的所有顶点在$R_i$中是强连通的。我们讨论了这些集合的性质，并证明可以通过将该问题归约为计算有向图的2-边无双孪生边强连通分量（2eTSCC），在线性时间内求解该问题。一个有向图$G=(V,E)$是双孪生边强连通的，若它包含一个不含任何一对反向（或称双生）边的强连通生成子图。有向图$G$的双孪生边强连通分量（TSCC）是其极大的双孪生边强连通子图。$G$的2-边双孪生边强连通分量（2eTSCC）是一个极大顶点子集$C$，满足对任意边$e$，任意两个顶点$u,v\u200bC$都位于$G\u200b{e}$的同一个双孪生边强连通分量中。这些概念的研究动机来自多种不同的应用场景，例如道路与通信网络的设计，以及建筑物的结构稳定性。
英文摘要：
A mixed graph $G$ is a graph that consists of both undirected and directed edges. An orientation of $G$ is formed by orienting all the undirected edges of $G$, i.e., converting each undirected edge $\{u,v\}$ into a directed edge that is either $(u,v)$ or $(v,u)$. The problem of finding an orientation of a mixed graph that makes it strongly connected is well understood and can be solved in linear time. Here we introduce the following orientation problem in mixed graphs. Given a mixed graph $G$, we wish to compute its maximal sets of vertices $C_1,C_2,\ldots,C_k$ with the property that by removing any edge $e$ from $G$ (directed or undirected), there is an orientation $R_i$ of $G\setminus{e}$ such that all vertices in $C_i$ are strongly connected in $R_i$. We discuss properties of those sets, and we show how to solve this problem in linear time by reducing it to the computation of the $2$-edge twinless strongly connected components of a directed graph. 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. A $2$-edge twinless strongly connected component (2eTSCC) of $G$ is a maximal subset of vertices $C$ such that any two vertices $u, v \in C$ are in the same twinless strongly connected component of $G \setminus e$, for any edge $e$. These concepts are motivated by several diverse applications, such as the design of road and telecommunication networks, and the structural stability of buildings.
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