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 $k+1$.

中文翻译：

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$2$-层 $k$-平面图：密度、交叉引理、关系和路径宽度

$2$-layer 绘图模型是一个完善的范式，用于可视化二部图。在这个模型下已经研究了几个超平面图类。然而，令人惊讶的是，在这种情况下，仅在 $k=1$ 的情况下才考虑了 $k$-平面图的基本类别。我们提供了一些贡献来解决文献中的这一差距。首先，我们用 $k\in\{2,3,4,5\}$ 显示 $2$-layer $k$-planar graphs 类的紧密密度边界。基于这些结果，我们为 $2$-层 $k$-平面图提供了一个交叉引理，这意味着 $2$-层 $k$-平面图的一般密度界限。我们证明这个界限几乎是最优的，具有相应的下界构造。最后，