Monotone Arc Diagrams with few Biarcs

TL;DR

This paper presents new bounds on monotone topological 2-page book embeddings of planar graphs, minimizing spine crossings and ensuring consistent crossing directions, with improved bounds for specific graph classes.

Contribution

It introduces improved upper bounds on the number of spine crossings in monotone embeddings for general planar graphs, planar 3-trees, and Kleetopes, with tight bounds for Kleetopes.

Findings

Every planar graph has a monotone 2-page embedding with at most (4n-10)/5 spine crossings.

Planar 3-trees can be embedded with at most (3n-9)/4 spine crossings.

Kleetopes have a tight bound of at most (n-8)/3 spine crossings.

Abstract

We show that every planar graph has a monotone topological 2-page book embedding where at most (4n-10)/5 (of potentially 3n-6) edges cross the spine, and every edge crosses the spine at most once; such an edge is called a biarc. We can also guarantee that all edges that cross the spine cross it in the same direction (e.g., from bottom to top). For planar 3-trees we can further improve the bound to (3n-9)/4, and for so-called Kleetopes we obtain a bound of at most (n-8)/3 edges that cross the spine. The bound for Kleetopes is tight, even if the drawing is not required to be monotone. A Kleetope is a plane triangulation that is derived from another plane triangulation T by inserting a new vertex v_f into each face f of T and then connecting v_f to the three vertices of f.