Monotone Arc Diagrams with few Biarcs

TL;DR

The paper addresses monotone topological 2-page book embeddings of planar graphs by introducing biarcs as the spine-crossing mechanism and proving new upper bounds on their number. It develops a constructive, canonical-ordering–based algorithm that incrementally builds arc diagrams while maintaining credits to bound biarcs, achieving a global bound of $\left\lfloor \dfrac{4}{5}n \right\rfloor - 2$ on monotone biarcs for general planar graphs, with all edges crossing the spine at most once and in the same direction. It additionally obtains $\left\lfloor \dfrac{3}{4}(n-3) \right\rfloor$ biarcs for planar $3$-trees and $\left\lfloor \dfrac{n-8}{3} \right\rfloor$ biarcs for Kleetopes (tight for the latter), via specialized constructions and dual-tree analyses. The results extend previous bounds for plane arc diagrams, clarify the role of monotonicity, and provide a framework for further tightening bounds or identifying monotonicity penalties in broader graph classes.

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.

Figures (32)

• Figure 1: Arc diagrams of the octahedron: (a) proper, (b) general, and (c) monotone.
• Figure 2: Overview of notation used throughout the paper.
• Figure 4: Inserting a vertex $v_i$ into mountains, using $5-d_i$ credits (\ref{['lem:defaultApproach2']}).
• Figure 5: $\mathcal{T}(2,\frown)$
• Figure 6: $\mathcal{T}(3,\frown^2)$

Theorems & Definitions (14)

• Theorem 1
• Theorem 2
• Theorem 3
• Corollary 4
• Lemma 5
• Lemma 6
• Lemma 7
• Lemma 8
• Lemma 8
• Theorem 8