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Bounding the Treewidth of Outer $k$-Planar Graphs via Triangulations

Oksana Firman, Grzegorz Gutowski, Myroslav Kryven, Yuto Okada, Alexander Wolff

TL;DR

We address bounding the treewidth of outer $k$-planar graphs via a new triangulation method of the outer cycle that limits edge piercings. The core approach yields a tighter treewidth bound of $\frac{3}{2}k+2$ (tight for $k=2$ with $\mathrm{tw}=4$) and a matching lower bound of $k+2$ for even $k$, while also improving the separation number. The constructions are constructive, enabling efficient tree decompositions and balanced separators, and extend to outer min-$k$-planar graphs with analogous gains. The results have direct implications for algorithmic applications and the cop-number, and raise open questions about the exact constant and broader applicability of the triangulation technique.

Abstract

The treewidth is a structural parameter that measures the tree-likeness of a graph. Many algorithmic and combinatorial results are expressed in terms of the treewidth. In this paper, we study the treewidth of outer $k$-planar graphs, that is, graphs that admit a straight-line drawing where all the vertices lie on a circle, and every edge is crossed by at most $k$ other edges. Wood and Telle [New York J. Math., 2007] showed that every outer $k$-planar graph has treewidth at most $3k + 11$ using so-called planar decompositions, and later, Auer et al. [Algorithmica, 2016] proved that the treewidth of outer $1$-planar graphs is at most $3$, which is tight. In this paper, we improve the general upper bound to $1.5k + 2$ and give a tight bound of $4$ for $k = 2$. We also establish a lower bound: we show that, for every even $k$, there is an outer $k$-planar graph with treewidth $k+2$. Our new bound immediately implies a better bound on the cop number, which answers an open question of Durocher et al. [GD 2023] in the affirmative. Our treewidth bound relies on a new and simple triangulation method for outer $k$-planar graphs that yields few crossings with graph edges per edge of the triangulation. Our method also enables us to obtain a tight upper bound of $k + 2$ for the separation number of outer $k$-planar graphs, improving an upper bound of $2k + 3$ by Chaplick et al. [GD 2017]. We also consider outer min-$k$-planar graphs, a generalization of outer $k$-planar graphs, where we achieve smaller improvements.

Bounding the Treewidth of Outer $k$-Planar Graphs via Triangulations

TL;DR

We address bounding the treewidth of outer -planar graphs via a new triangulation method of the outer cycle that limits edge piercings. The core approach yields a tighter treewidth bound of (tight for with ) and a matching lower bound of for even , while also improving the separation number. The constructions are constructive, enabling efficient tree decompositions and balanced separators, and extend to outer min--planar graphs with analogous gains. The results have direct implications for algorithmic applications and the cop-number, and raise open questions about the exact constant and broader applicability of the triangulation technique.

Abstract

The treewidth is a structural parameter that measures the tree-likeness of a graph. Many algorithmic and combinatorial results are expressed in terms of the treewidth. In this paper, we study the treewidth of outer -planar graphs, that is, graphs that admit a straight-line drawing where all the vertices lie on a circle, and every edge is crossed by at most other edges. Wood and Telle [New York J. Math., 2007] showed that every outer -planar graph has treewidth at most using so-called planar decompositions, and later, Auer et al. [Algorithmica, 2016] proved that the treewidth of outer -planar graphs is at most , which is tight. In this paper, we improve the general upper bound to and give a tight bound of for . We also establish a lower bound: we show that, for every even , there is an outer -planar graph with treewidth . Our new bound immediately implies a better bound on the cop number, which answers an open question of Durocher et al. [GD 2023] in the affirmative. Our treewidth bound relies on a new and simple triangulation method for outer -planar graphs that yields few crossings with graph edges per edge of the triangulation. Our method also enables us to obtain a tight upper bound of for the separation number of outer -planar graphs, improving an upper bound of by Chaplick et al. [GD 2017]. We also consider outer min--planar graphs, a generalization of outer -planar graphs, where we achieve smaller improvements.
Paper Structure (8 sections, 14 theorems, 3 figures)

This paper contains 8 sections, 14 theorems, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Triangulations
  4. Applications
  5. Treewidth
  6. Separation Number
  7. Lower Bounds
  8. Conclusion and Open Problems

Key Result

Lemma 4

Given an outer $k$-planar drawing $\Gamma$ of a graph $G$ and an active link $\{v_i,v_k\}$, there exists an outer $k$-planar drawing $\Gamma'$ with the same cyclic order as $\Gamma$, and all crossings among the edges piercing the active link are drawn to the left of the active link.

Figures (3)

  • Figure 1: Steps of the splitting procedure. The active edge is purple, the (new/old) links are (dark/light) blue, the left/right side of the active edge is red/green. The split vertex is big.
  • Figure 5: $\lambda$ and $\hat{e}$ share the endpoint $v$; the new triangle $\triangle$ is bounded by $\lambda$, $c'$, and $c"$.
  • Figure 9: Two drawings of the stacked prism $Y_{m,n}$.

Theorems & Definitions (15)

  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • Lemma 9
  • Remark 10
  • Lemma 11
  • Theorem 12
  • Theorem 13
  • ...and 5 more