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arXiv · 2008.09329

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

Patrizio Angelini·Giordano Da Lozzo·Henry Förster·Thomas Schneck
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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 $k+1$.

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Patrizio Angelini, Giordano Da Lozzo, Henry Förster, Thomas Schneck. 2020-08-21. $2$-Layer $k$-Planar Graphs: Density, Crossing Lemma, Relationships, and Pathwidth. https://arxiv.org/abs/2008.09329

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