The Bundled Crossing Number

TL;DR

This paper explores the computational complexity of the bundled crossing number in graph drawings, relating it to graph genus and providing approximation algorithms for minimizing bundled crossings in various graph layout scenarios.

Contribution

It establishes the relationship between bundled crossing number and graph genus, and introduces approximation algorithms for minimizing bundled crossings in different graph layout settings.

Findings

Bundled crossing number equals the orientable genus when multiple crossings are allowed.

Approximation algorithms achieve constant factors for circular layouts with fixed vertex order.

For general layouts, the algorithms provide specific approximation ratios depending on edge density.

Abstract

We study the algorithmic aspect of edge bundling. A bundled crossing in a drawing of a graph is a group of crossings between two sets of parallel edges. The bundled crossing number is the minimum number of bundled crossings that group all crossings in a drawing of the graph. We show that the bundled crossing number is closely related to the orientable genus of the graph. If multiple crossings and self-intersections of edges are allowed, the two values are identical; otherwise, the bundled crossing number can be higher than the genus. We then investigate the problem of minimizing the number of bundled crossings. For circular graph layouts with a fixed order of vertices, we present a constant-factor approximation algorithm. When the circular order is not prescribed, we get a $\frac{6c}{c-2}$ approximation for a graph with $n$ vertices having at least $cn$ edges for $c>2$. For general graph layouts, we develop an algorithm with an approximation factor of $\frac{6c}{c-3}$ for graphs with at least $cn$ edges for $c > 3$.