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 have 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。$G$的定向是指将其中所有无向边转换为有向边（即每条无向边$\{u,v\}$被赋予方向$(u,v)$或$(v,u)$）。如何寻找混合图的强连通定向问题已有充分研究且可在线性时间内解决。本文提出以下混合图定向问题：给定混合图$G$，计算其最大顶点子集族$C_1,C_2,\ldots,C_k$，满足性质：从$G$中删除任意一条边$e$（有向或无向）后，存在$G\setminus{e}$的定向$R_i$，使得$C_i$中所有顶点在$R_i$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问题，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，若其包含一个无反向平行边（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）是最大无孪生强连通子图。而$G$的2-边无孪生强连通分量（2eTSCC）是最大顶点子集$C$，使得对于任意边$e$，任意两顶点$u,v\in C$均属于$G\setminus e$的同一无孪生强连通分量。这些概念在道路与通信网络设计、建筑结构稳定性等领域具有多种应用价值。