Computing Vertex and Edge Connectivity of Graphs Embedded with Crossings

TL;DR

This work tackles the problem of computing vertex and edge connectivity for graphs embedded in the plane with crossings by introducing the ribbon radius μ(G) and a radial planarization framework Λ(G). A central structural result shows that for any minimal cut S, the relevant region Λ(S, μ) has bounded diameter, enabling the construction of co-separating triples and efficient dynamic programming on bounded-width decompositions to decide κ(G) and λ(G) in time 2^{O(μκ)}|V(G^×)| and 2^{O(μλ)}|V(G^×)| respectively. The authors develop a full algorithmic pipeline—from BFS-based layer partitioning to tree decompositions and MSOL-free dynamic programming—that extends linear-time connectivity testing from planar and special near-planar graphs to broad classes with small ribbon radius, including k-plane graphs with bounded ×-crossings, d-map/d-framed graphs, and bounded-crossing variants. They also provide near-term applications to various near-planar graph families, along with constructions showing limitations when μ is large. The results offer a unifying, scalable approach to connectivity in embedded graphs and establish a foundation for future extensions to weighted settings and other surfaces.

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.

Figures (6)

• Figure 1: A graph drawing where all crossings are $\times$-crossings
• Figure 2: A figure for \ref{['theorem: vertex connectivity']} depicting the structure of a co-separating triple of $(\Lambda(G),G)$. The graph is partitioned into different regions. The regions containing vertices of $S$ (shown with black dots) represent $\Lambda(s,\mu)$ for $s \in S$. The union of all such regions forms the graph $\Lambda(S,\mu)$. The remaining regions are holes of $\Lambda(S,\mu)$. These are bounded by solid edges representing the boundary $\mathcal{Z}(S,\mu)$. All vertices of $\Lambda(G)$ inside a hole belong entirely to $A$, or to $B$.
• Figure 3: Constructing $\Lambda_i$ for $i=4$
• Figure 4: An edge $e = (u,v)$; the paths of $e^\times$ in $\Lambda_i$ are bold. We add edges from all vertices of $e^\times \cap \Lambda_i$ to the corresponding transition-points.
• Figure 5: An illustration showing the graph constructed in the proof of \ref{['thm: counterex edge conn']} for the values $k=4$, $p = 3$, and $r= 3$. Graph $R$ is dashed (red) while $B$ is solid (blue). The graph is drawn on a flat cylinder (the left and right boundaries are identified).

Theorems & Definitions (25)

• Definition 1: Radial Planarization
• Definition 2: Co-separating Triple and Nucleus
• Theorem 5
• Proposition 6
• Corollary 7
• Theorem 8
• Corollary 11
• Corollary 12
• Proposition 13
• Proposition 14