Computing Vertex and Edge Connectivity of Graphs Embedded with Crossings

TL;DR

This paper develops a unified linear-time algorithm for computing vertex and edge connectivity in a broad class of embedded graphs with crossings, extending previous results from planar and 1-plane graphs.

Contribution

It generalizes the face-distance approach to a wider class of graphs, enabling efficient connectivity computation for graphs with bounded crossings.

Findings

Linear-time algorithms for connectivity in various embedded graph classes.

Extension of face-distance methods beyond planar and 1-plane graphs.

Applicable to optimal 2-planar, 3-planar, and other crossing-bounded graphs.

Abstract

Vertex connectivity and edge connectivity are fundamental concepts in graph theory that have been widely studied from both structural and algorithmic perspectives. The focus of this paper is on computing these two parameters for graphs embedded on the plane with crossings. For planar graphs -- which can be embedded on the plane without any crossings -- it has long been known that vertex and edge connectivity can be computed in linear time. Recently, the algorithm for vertex connectivity was extended from planar graphs to 1-plane graphs (where each edge is crossed at most once) without $\times$-crossings -- these are crossings whose endpoints induce a matching. The key insight, for both these classes of graphs, is that any two vertices/edges of a minimum vertex/edge cut have small face-distance (distance measured by number of faces) in the embedding. In this paper, we attempt at a comprehensive generalization of this idea to a wider class of graphs embedded on the plane. Our method works for all those embedded graphs where every pair of crossing edges is connected by a path whose vertices and edges have a small face-distance from the crossing point. 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. For all these graph classes, we get a linear-time algorithm for computing vertex and edge connectivity.