$2$-Layer $k$-Planar Graphs: Density, Crossing Lemma, Relationships, and Pathwidth

TL;DR

This paper explores the properties of 2-layer k-planar graphs, establishing density bounds, a crossing lemma, and relationships with other graph classes, advancing understanding of their structure and limitations.

Contribution

The paper provides the first tight density bounds for 2-layer k-planar graphs for k=2 to 5, and introduces a crossing lemma and pathwidth bounds for these graphs.

Findings

Established tight density bounds for k=2 to 5

Derived a crossing lemma for 2-layer k-planar graphs

Proved pathwidth at most k+1 for these graphs

Abstract

The $2$-layer drawing model is a well-established paradigm to visualize bipartite graphs. Several beyond-planar graph classes have been studied under this model. Surprisingly, however, the fundamental class of $k$-planar graphs has been considered only for $k=1$ in this context. We provide several contributions that address this gap in the literature. First, we show tight density bounds for the classes of $2$-layer $k$-planar graphs with $k\in\{2,3,4,5\}$. Based on these results, we provide a Crossing Lemma for $2$-layer $k$-planar graphs, which then implies a general density bound for $2$-layer $k$-planar graphs. We prove this bound to be almost optimal with a corresponding lower bound construction. Finally, we study relationships between $k$-planarity and $h$-quasiplanarity in the $2$-layer model and show that $2$-layer $k$-planar graphs have pathwidth at most $k+1$.