Computing the vertex connectivity of a locally maximal 1-plane graph in linear time

TL;DR

This paper extends the linear-time algorithm for computing vertex connectivity from planar graphs to locally maximal 1-plane graphs, which are nearly planar with specific crossing and embedding properties.

Contribution

It introduces a linear-time method for determining vertex connectivity in locally maximal 1-plane graphs, broadening the class of graphs with efficient connectivity algorithms.

Findings

Vertex connectivity can be computed in linear time for locally maximal 1-plane graphs.

The method generalizes planar graph algorithms to a broader class of nearly planar graphs.

The approach leverages the specific embedding and crossing properties of these graphs.

Abstract

It is known that the vertex connectivity of a planar graph can be computed in linear time. We extend this result to the class of locally maximal 1-plane graphs: graphs that have an embedding with at most one crossing per edge such that the endpoints of each pair of crossing edges induce the complete graph $K_4$