An FPT Algorithm for Bipartite Vertex Splitting

TL;DR

This paper presents a fixed parameter tractable algorithm for the bipartite vertex splitting problem, which aims to minimize crossings in bipartite graph drawings by splitting vertices.

Contribution

The paper introduces an FPT algorithm for deciding if a bipartite graph can be made planar through splitting at most k vertices, advancing graph drawing techniques.

Findings

The problem is fixed parameter tractable in k.

Efficient algorithms can determine minimal vertex splits for planarity.

Improves understanding of crossing minimization in bipartite graphs.

Abstract

Bipartite graphs model the relationship between two disjoint sets of objects. They have a wide range of applications and are often visualized as a 2-layered drawing, where each set of objects is visualized as a set of vertices (points) on one of the two parallel horizontal lines and the relationships are represented by edges (simple curves) between the two lines connecting the corresponding vertices. One of the common objectives in such drawings is to minimize the number of crossings this, however, is computationally expensive and may still result in drawings with so many crossings that they affect the readability of the drawing. We consider a recent approach to remove crossings in such visualizations by splitting vertices, where the goal is to find the minimum number of vertices to be split to obtain a planar drawing. We show that determining whether a planar drawing exists after splitting at most $k$ vertices is fixed parameter tractable in $k$.