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