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.

翻译：一个图被称为1-平面图，如果它能在平面上嵌入，使得每条边最多被另一条边交叉一次。1-平面图中的交叉称为×-交叉，若没有其他边连接该交叉的端点（除交叉边对之外）。本文展示了如何在无×-交叉的1-平面图$G$中线性时间计算其顶点连通性。为此，我们证明：对于最小分离集$S$中的任意两个顶点$u,v$，它们在辅助图$\Lambda(G)$（通过对$G$平面化，然后在每个面内插入一个与面内所有顶点相邻的新顶点得到）中的距离很小。因此，只需在$\Lambda(G)$的若干具有小直径的子图$\Lambda_i$中搜索最小分离集即可。由于$\Lambda(G)$是平面图，子图$\Lambda_i$具有较小的树宽。每个最小分离集$S$进而将$\Lambda_i$划分为三个具有特殊性质的顶点集；此类划分可通过Courcelle定理在线性时间内找到。