2-Layer k-Planar Graphs Density, Crossing Lemma, Relationships And Pathwidth

• Published in: Computer journal Vol. 67; no. 3; pp. 1005 - 1016
• Main Authors: Angelini, Patrizio, Da Lozzo, Giordano, Förster, Henry, Schneck, Thomas
• Format: Journal Article
• Language: English
• Published: Oxford University Press 14.04.2024
• Subjects: pathwidth; quasiplanar graphs; density; crossing lemma; 2-layer graph drawing; planar graphs
• ISSN: 0010-4620; 1460-2067
• DOI: 10.1093/comjnl/bxad038

The $2$-layer drawing model is a well-established paradigm to visualize bipartite graphs where vertices of the two parts lie on two horizontal lines and edges lie between these lines. 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 $k+1$ while there are also $2$-layer $k$-planar graphs with pathwidth at least $(k+3)/2$.