Thickness and Antithickness of Graphs

TL;DR

This paper investigates the duality between crossings and non-crossings in graph drawings by analyzing the concepts of thickness and antithickness, revealing their relationship and extremal properties across different models.

Contribution

It introduces and studies the notions of thickness and antithickness, establishing their relationship and extremal bounds in various graph drawing contexts.

Findings

Thickness and antithickness are dual measures of graph complexity.

Bounds relating thickness and antithickness are established.

Extremal cases for specific graph classes are characterized.

Abstract

This paper studies questions about duality between crossings and non-crossings in graph drawings via the notions of thickness and antithickness. The "thickness" of a graph $G$ is the minimum integer $k$ such that in some drawing of $G$, the edges can be partitioned into $k$ noncrossing subgraphs. The "antithickness" of a graph $G$ is the minimum integer $k$ such that in some drawing of $G$, the edges can be partitioned into $k$ thrackles, where a "thrackle" is a set of edges, each pair of which intersect exactly once. (Here edges with a common endvertex $v$ are considered to intersect at $v$.) So thickness is a measure of how close a graph is to being planar, whereas antithickness is a measure of how close a graph is to being a thrackle. This paper explores the relationship between the thickness and antithickness of a graph, under various graph drawing models, with an emphasis on extremal questions.