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.

翻译：混合图 $G$ 是由无向边和有向边共同构成的图。对 $G$ 的定向指的是将其所有无向边定向，即将每条无向边 $\{u,v\}$ 转换为有向边 $(u,v)$ 或 $(v,u)$。寻找使混合图强连通的定向问题已得到充分研究，并可在线性时间内求解。本文引入如下混合图定向问题：给定混合图 $G$，我们希望计算其最大顶点集 $C_1,C_2,\ldots,C_k$，这些集合满足如下性质：从 $G$ 中移除任意边 $e$（有向或无向）后，存在 $G\setminus{e}$ 的一个定向 $R_i$，使得 $C_i$ 中的所有顶点在 $R_i$ 中强连通。我们讨论了这些集合的性质，并通过将其归约为有向图的 $2$ 边无孪生强连通分量计算，证明了该问题可在线性时间内求解。有向图 $G=(V,E)$ 称为无孪生强连通图，若它包含一个不含任何反向平行（即孪生）边的强连通生成子图。有向图 $G$ 的无孪生强连通分量（TSCC）是它最大的无孪生强连通子图。$G$ 的 $2$ 边无孪生强连通分量（2eTSCC）是最大顶点子集 $C$，满足对于任意边 $e$，$C$ 中任意两顶点 $u, v$ 均属于 $G \setminus e$ 的同一无孪生强连通分量。这些概念的提出源自多种实际应用，如道路与通信网络设计以及建筑物结构稳定性分析。