A Parameterized Algorithm for Vertex and Edge Connectivity of Embedded Graphs

2024-06-30

Embedded Systems and Applications

AI Summary

This work addresses the vertex and edge connectivity determination problem for embedded graphs with bounded crossings—such as $k$-planar graphs and $d$-map graphs. Moving beyond classical algorithms restricted to planar graphs, we propose the first unified parameterized algorithmic framework applicable to crossing-restricted graph classes. Our approach integrates embedding-driven unit covering analysis, path-constraint modeling, and parameterized dynamic programming, thereby lifting the planarity assumption. The algorithm achieves linear or near-linear time complexity, attaining theoretical optimality across multiple near-planar graph families. It is the first method capable of exact connectivity computation for optimal $k$-planar graphs and other non-planar embeddings. Moreover, it simultaneously handles both vertex and edge connectivity testing within a single framework, unifying previously disparate treatments.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

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 $ imes$-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 $ imes$-crossings.

Problem

• Computing vertex and edge connectivity for graphs with crossings
• Extending connectivity algorithms from planar to crossing-rich embeddings
• Developing linear-time methods for graphs with bounded crossing patterns

Innovation

• Extends connectivity algorithms to graphs with crossings
• Uses small face-distance property near crossing points
• Applies linear-time method to multiple embedded graph classes