Pathwidth of 2-Layer k-Planar Graphs

A bipartite graph G = (X ∪ 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 ≥ 0. This improves their lower bound of (k + 3) / 2.