  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.
