On Computing Vertex Connectivity of 1-Plane Graphs

TL;DR

This paper explores the vertex connectivity of 1-plane graphs, extending known algorithms from planar graphs to certain subclasses of 1-plane graphs with efficient linear-time solutions.

Contribution

It introduces structural properties of minimal vertex cuts in specific classes of 1-plane graphs and develops linear-time algorithms for computing their vertex connectivity.

Findings

For 1-plane graphs with K4 endpoint crossings, vertex cut structure matches that of planar graphs.

For other 1-plane graphs, minimal vertex cuts relate to cycles of bounded diameter in an auxiliary graph.

Linear-time algorithms are achieved for these subclasses of 1-plane graphs.

Abstract

The vertex connectivity of a graph $G$ is the size of the smallest set of vertices $S$ such that $G \setminus S$ is disconnected. For the class of planar graphs, the problem of vertex connectivity is well-studied, both from structural and algorithmic perspectives. Let $G$ be a plane embedded graph, and $\Lambda(G)$ be an auxiliary graph obtained by inserting a face vertex inside each face and connecting it to all vertices of $G$ incident with the face. If $S$ is a minimal vertex cut of $G$, then there exists a cycle of length $2|S|$ whose vertices alternate between vertices of $S$ and face vertices. This structure facilitates the designing of a linear-time algorithm to find minimum vertex cuts of planar graphs. In this paper, we attempt a similar approach for the class of 1-plane graphs -- these are graphs with a drawing on the plane where each edge is crossed at most once. We consider different classes of 1-plane graphs based on the subgraphs induced by the endpoints of crossings. For 1-plane graphs where the endpoints of every crossing induce the complete graph $K_4$, we show that the structure of minimum vertex cuts is identical to that in plane graphs, as mentioned above. For 1-plane graphs where the endpoints of every crossing induce at least three edges (i.e., one edge apart from the crossing pair of edges), we show that for any minimal vertex cut $S$, there exists a cycle of diameter $O(|S|)$ in $\Lambda(G)$ such that all vertices of $S$ are in the neighbourhood of the cycle. This structure enables us to design a linear time algorithm to compute the vertex connectivity of all such 1-plane graphs.