On the Planar Split Thickness of Graphs

TL;DR

This paper studies the minimal number of vertex splits needed to convert various classes of graphs into planar graphs, providing complexity results and approximation algorithms relevant to graph drawing applications.

Contribution

It introduces the concept of planar split thickness, analyzes it for specific graph classes, and establishes NP-hardness and approximation results for recognizing and computing this parameter.

Findings

NP-hardness of recognizing 2-splittable planar graphs

Constant-factor approximation for planar split thickness

Linear-time verification for bounded treewidth graphs

Abstract

Motivated by applications in graph drawing and information visualization, we examine the planar split thickness of a graph, that is, the smallest $k$ such that the graph is $k$-splittable into a planar graph. A $k$-split operation substitutes a vertex $v$ by at most $k$ new vertices such that each neighbor of $v$ is connected to at least one of the new vertices. We first examine the planar split thickness of complete graphs, complete bipartite graphs, multipartite graphs, bounded degree graphs, and genus-1 graphs. We then prove that it is NP-hard to recognize graphs that are $2$-splittable into a planar graph, and show that one can approximate the planar split thickness of a graph within a constant factor. If the treewidth is bounded, then we can even verify $k$-splittability in linear time, for a constant $k$.