Pathwidth of 2-Layer $k$-Planar Graphs

TL;DR

This paper proves that the maximum pathwidth of 2-layer k-planar graphs is exactly k+1, confirming the tightness of previous bounds and advancing understanding of their structural properties.

Contribution

The paper establishes that the upper bound of pathwidth for 2-layer k-planar graphs is tight by constructing graphs that attain this bound for all k.

Findings

Maximum pathwidth of 2-layer k-planar graphs is exactly k+1.

Constructed graphs demonstrate the bound's tightness for all k.

Improves previous lower bound from (k+3)/2 to k+1.

Abstract

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, F\"orster, 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 $(k+3)/2$.