Splitting Vertices in 2-Layer Graph Drawings

TL;DR

This paper investigates vertex splitting techniques in 2-layer bipartite graph drawings to reduce edge crossings, providing algorithms for optimization problems and testing on biological data.

Contribution

It introduces new algorithms for vertex splitting to minimize crossings in bipartite graph drawings, including NP-complete variants and polynomial solutions.

Findings

Algorithms effectively reduce crossings in bipartite graphs.

NP-completeness established for certain splitting variants.

Polynomial algorithms developed for specific optimization problems.

Abstract

Bipartite graphs model the relationships between two disjoint sets of entities in several applications and are naturally drawn as 2-layer graph drawings. In such drawings, the two sets of entities (vertices) are placed on two parallel lines (layers), and their relationships (edges) are represented by segments connecting vertices. Methods for constructing 2-layer drawings often try to minimize the number of edge crossings. We use vertex splitting to reduce the number of crossings, by replacing selected vertices on one layer by two (or more) copies and suitably distributing their incident edges among these copies. We study several optimization problems related to vertex splitting, either minimizing the number of crossings or removing all crossings with fewest splits. While we prove that some variants are \NP-complete, we obtain polynomial-time algorithms for others. We run our algorithms on a benchmark set of bipartite graphs representing the relationships between human anatomical structures and cell types.