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Pathwidth of 2-Layer $k$-Planar Graphs

Yuto Okada

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, 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 $(k+3)/2$.

Pathwidth of 2-Layer $k$-Planar Graphs

Abstract

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

Paper Structure

This paper contains 9 sections, 6 theorems, 2 figures.

Table of Contents

  1. Introduction
  2. Our results.
  3. Related results.
  4. Preliminaries
  5. 2-layer $k$-planar graphs.
  6. Pathwidth.
  7. Node searching number.
  8. Lower Bound
  9. Conclusions

Key Result

Lemma 1

For every graph $G$, $\mathrm{ns}(G) = \mathrm{pw}(G) + 1$ holds.

Figures (2)

  • Figure 1: An illustration of $G_k$, which has $k + 2$ rows and $3k + 6$ columns.
  • Figure 2: An illustration of $W_k$. The same pattern appears every $4k$ columns.

Theorems & Definitions (6)

  • Lemma 1: DBLP:journals/tcs/Bodlaender98DBLP:journals/dm/KirousisP85
  • Lemma 2: DBLP:journals/tcs/KirousisP86DBLP:journals/jacm/LaPaugh93
  • Theorem 3
  • Lemma 4
  • Lemma 6
  • Lemma 7