  A graph is called 1-plane if it has an embedding in the plane where each edge
is crossed at most once by another edge.A crossing of a 1-plane graph is called
an $\times$-crossing if there are no other edges connecting the endpoints of
the crossing (apart from the crossing pair of edges). In this paper, we show
how to compute the vertex connectivity of a 1-plane graph $G$ without
$\times$-crossings in linear time. To do so, we show that for any two vertices
$u,v$ in a minimum separating set $S$, the distance between $u$ and $v$ in an
auxiliary graph $\Lambda(G)$ (obtained by planarizing $G$ and then inserting
into each face a new vertex adjacent to all vertices of the face) is small. It
hence suffices to search for a minimum separating set in various subgraphs
$\Lambda_i$ of $\Lambda(G)$ with small diameter. Since $\Lambda(G)$ is planar,
the subgraphs $\Lambda_i$ have small treewidth. Each minimum separating set $S$
then gives rise to a partition of $\Lambda_i$ into three vertex sets with
special properties; such a partition can be found via Courcelle's theorem in
linear time.
