The problem of computing vertex and edge connectivity of a graph are classical problems in algorithmic graph theory. The focus of this paper is on computing these parameters on embedded graphs. A typical example of an embedded graph is a planar graph which can be drawn with no edge crossings. It has long been known that vertex and edge connectivity of planar embedded graphs can be computed in linear time. Very recently, Biedl and Murali extended the techniques from planar graphs to 1-plane graphs without $\times$-crossings, i.e., crossings whose endpoints induce a matching. While the tools used were novel, they were highly tailored to 1-plane graphs, and do not provide much leeway for further extension. In this paper, we develop alternate techniques that are simpler, have wider applications to near-planar graphs, and can be used to test both vertex and edge connectivity. Our technique works for all those embedded graphs where any pair of crossing edges are connected by a path that, roughly speaking, can be covered with few cells of the drawing. Important examples of such graphs include optimal 2-planar and optimal 3-planar graphs, $d$-map graphs, $d$-framed graphs, graphs with bounded crossing number, and $k$-plane graphs with bounded number of $\times$-crossings.

翻译：计算图的顶点连通度与边连通度是算法图论中的经典问题。本文聚焦于嵌入图上这些参数的计算。嵌入图的典型例子是平面图，即可以无交叉边绘制的图。众所周知，平面嵌入图的顶点与边连通度可在线性时间内计算完成。最近，Biedl与Murali将平面图的技术推广至不含×型交叉（即交叉端点构成匹配的交叉）的1-平面图。尽管所用工具新颖，但这些方法高度特化于1-平面图，难以进一步扩展。本文提出一套更简洁的替代技术，其适用于更广泛的近平面图，并能同时检测顶点与边连通度。我们的技术适用于所有满足以下条件的嵌入图：任意两条交叉边均存在一条路径相连，且粗略而言，该路径可被绘图中少数单元覆盖。此类图的重要示例包括最优2-平面图与最优3-平面图、$d$-地图图、$d$-框架图、交叉数有界图，以及具有有限×型交叉数的$k$-平面图。