  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.
