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Treewidth of Outer $k$-Planar Graphs

Rafał Pyzik

TL;DR

The paper analyzes how geometric constraints from outer $k$-planarity and outer min-$k$-planarity influence treewidth and separation number. It provides a sharp lower bound of $1.5k+0.5$ for odd $k$ and improves upper bounds to $3\cdot\lfloor k/2\rfloor+4$ for treewidth and $2\cdot\lfloor k/2\rfloor+4$ for the separation number, using bramble arguments, planarization, and dual-tree decompositions. Explicit constructions, such as $G_k$ and $F_k$, establish tightness up to parity, while the bounds have algorithmic implications via structural graph parameters. Overall, the results substantially sharpen our understanding of how convex drawings with bounded edge crossings constrain global graph structure.

Abstract

Treewidth is an important structural graph parameter that quantifies how closely a graph resembles a tree-like structure. It has applications in many algorithmic and combinatorial problems. In this paper, we study the treewidth of outer $k$-planar graphs, that is, graphs admitting a convex drawing (a straight-line drawing where all vertices lie on a circle) in which every edge crosses at most $k$ other edges. We also consider the more general class of outer min-$k$-planar graphs, which are graphs admitting a convex drawing where for every crossing of two edges at least one of these edges is crossed at most $k$ times. Firman, Gutowski, Kryven, Okada and Wolff [GD 2024] proved that every outer $k$-planar graph has treewidth at most $1.5k+2$ and provided a lower bound of $k+2$ for even $k$. We establish a lower bound of $1.5k+0.5$ for every odd $k$. Additionally, they showed that every outer min-$k$-planar graph has treewidth at most $3k+1$. We improve this upper bound to $3 \cdot \lfloor k/2 \rfloor+4$. Our approach also allows us to upper bound the separation number, a parameter closely related to treewidth, of outer min-$k$-planar graphs by $2 \cdot \lfloor k/2 \rfloor+4$. This improves upon the previous bound of $2k+1$ and achieves a bound with an optimal multiplicative constant.

Treewidth of Outer $k$-Planar Graphs

TL;DR

The paper analyzes how geometric constraints from outer -planarity and outer min--planarity influence treewidth and separation number. It provides a sharp lower bound of for odd and improves upper bounds to for treewidth and for the separation number, using bramble arguments, planarization, and dual-tree decompositions. Explicit constructions, such as and , establish tightness up to parity, while the bounds have algorithmic implications via structural graph parameters. Overall, the results substantially sharpen our understanding of how convex drawings with bounded edge crossings constrain global graph structure.

Abstract

Treewidth is an important structural graph parameter that quantifies how closely a graph resembles a tree-like structure. It has applications in many algorithmic and combinatorial problems. In this paper, we study the treewidth of outer -planar graphs, that is, graphs admitting a convex drawing (a straight-line drawing where all vertices lie on a circle) in which every edge crosses at most other edges. We also consider the more general class of outer min--planar graphs, which are graphs admitting a convex drawing where for every crossing of two edges at least one of these edges is crossed at most times. Firman, Gutowski, Kryven, Okada and Wolff [GD 2024] proved that every outer -planar graph has treewidth at most and provided a lower bound of for even . We establish a lower bound of for every odd . Additionally, they showed that every outer min--planar graph has treewidth at most . We improve this upper bound to . Our approach also allows us to upper bound the separation number, a parameter closely related to treewidth, of outer min--planar graphs by . This improves upon the previous bound of and achieves a bound with an optimal multiplicative constant.

Paper Structure

This paper contains 5 sections, 13 theorems, 3 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lower bound on the treewidth of outer $k$-planar graphs
  4. Upper bound on the treewidth of outer min-$k$-planar graphs
  5. The separation number of outer min-$k$-planar graphs

Key Result

Theorem 1

For every graph $G$, $\operatorname{tw}(G) = \operatorname{bn}(G)-1$.

Figures (3)

  • Figure 1: The transformation described in \ref{['obs:expand_okpl']}.
  • Figure 2: The graph $G_k$, for $k=3$, with a subgraph of $\mathcal{B}_1$ colored green and a subgraph of $\mathcal{B}_2$ colored orange.
  • Figure 3: The graph $F_k$, which is a modification of the graph $G_k$, for $k=3$.

Theorems & Definitions (13)

  • Theorem 1: Seymour and Thomas, seymour1993graph
  • Theorem 4: Kammer and Tholey, kammer2009lower
  • Corollary 5
  • Theorem 6
  • Theorem 7
  • Lemma 8: Folklore
  • Lemma 9: Folklore
  • Lemma 10
  • Lemma 11
  • Theorem 12
  • ...and 3 more