A bipartite graph $G = (X \cup Y, E)$ is a 2-layer $k$-planar graph if it admits a drawing on the plane such that the vertices in $X$ and $Y$ are placed on two parallel lines respectively, edges are drawn as straight-line segments, and every edge involves at most $k$ crossings. Angelini, Da Lozzo, Förster, and Schneck [GD 2020; Comput. J., 2024] showed that every 2-layer $k$-planar graph has pathwidth at most $k + 1$. In this paper, we show that this bound is sharp by giving a 2-layer $k$-planar graph with pathwidth $k + 1$ for every $k \geq 0$. This improves their lower bound of $(k+3)/2$.

翻译：若一个二分图 $G = (X \cup Y, E)$ 存在一种平面绘制方式，使得 $X$ 与 $Y$ 中的顶点分别置于两条平行线上，边以直线段绘制，且每条边最多涉及 $k$ 次交叉，则称该图为双层 $k$-平面图。Angelini、Da Lozzo、Förster 和 Schneck [GD 2020; Comput. J., 2024] 证明了每个双层 $k$-平面图的路径宽度至多为 $k + 1$。本文通过构造出对每个 $k \geq 0$ 路径宽度均为 $k + 1$ 的双层 $k$-平面图，证明该上界是紧的。这改进了他们先前给出的 $(k+3)/2$ 的下界。