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The Zarankiewicz Problem for Polygon Visibility Graphs

Eyal Ackerman, Balázs Keszegh

TL;DR

The paper investigates Zarankiewicz-type bounds for visibility graphs arising from polygons, proving a quasi-linear upper bound for $K_{t,t}$-free polygon visibility graphs and linear bounds for star-shaped and $x$-monotone polygons. In the curve setting, it establishes an $O(n\log n)$ upper bound for $K_{t,t}$-free curve pseudo-visibility graphs on $n$ vertices and constructs near-linear lower bounds of $\Omega(n\alpha(n))$ via Davenport–Schinzel sequences, showing a fundamental gap between upper and lower limits in this geometry-driven regime. The authors develop a rich toolkit including ordered/cyclic graphs, crossing sequences, 0-1 matrix representations, and pseudoline arrangements, and employ results from Marcus–Pardos (MT04), Klazar, Capoyleas–Pach, and Davenport–Schinzel theory to derive tight density bounds. These results advance understanding of degree-boundedness and extremal edge counts in geometric visibility graphs, with implications for incidence geometry and related algorithmic applications.

Abstract

We prove a quasi-linear upper bound on the size of $K_{t,t}$-free polygon visibility graphs. For visibility graphs of star-shaped and monotone polygons we show a linear bound. In the more general setting of $n$ points on a simple closed curve and visibility pseudo-segments, we provide an $O(n \log n)$ upper bound and an $Ω(nα(n))$ lower bound.

The Zarankiewicz Problem for Polygon Visibility Graphs

TL;DR

The paper investigates Zarankiewicz-type bounds for visibility graphs arising from polygons, proving a quasi-linear upper bound for -free polygon visibility graphs and linear bounds for star-shaped and -monotone polygons. In the curve setting, it establishes an upper bound for -free curve pseudo-visibility graphs on vertices and constructs near-linear lower bounds of via Davenport–Schinzel sequences, showing a fundamental gap between upper and lower limits in this geometry-driven regime. The authors develop a rich toolkit including ordered/cyclic graphs, crossing sequences, 0-1 matrix representations, and pseudoline arrangements, and employ results from Marcus–Pardos (MT04), Klazar, Capoyleas–Pach, and Davenport–Schinzel theory to derive tight density bounds. These results advance understanding of degree-boundedness and extremal edge counts in geometric visibility graphs, with implications for incidence geometry and related algorithmic applications.

Abstract

We prove a quasi-linear upper bound on the size of -free polygon visibility graphs. For visibility graphs of star-shaped and monotone polygons we show a linear bound. In the more general setting of points on a simple closed curve and visibility pseudo-segments, we provide an upper bound and an lower bound.

Paper Structure

This paper contains 16 sections, 23 theorems, 2 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Related work.
  3. Organization.
  4. Terminology and tools
  5. Bounds for $K_{t,t}$-free curve pseudo-visibility graphs
  6. An upper bound for $K_{t,t}$-free curve pseudo-visibility graphs
  7. A lower bound for $K_{t,t}$-free curve pseudo-visibility graphs
  8. Drawing $G'$:
  9. Drawing the non-edges of $G'$:
  10. Drawing the curve $K$:
  11. Bounds for $K_{t,t}$-free polygon visibility graphs
  12. $K_{t,t}$-free polygon visibility graphs
  13. A linear upper bound for star-shaped polygons
  14. A linear upper bound for $x$-monotone polygons
  15. Remarks.
  16. ...and 1 more sections

Key Result

Theorem 1.1

For every integer $t>0$ there is a constant $c_t$ such that the following holds. Let $P$ be a simple $n$-gon and let $G$ be the visibility graph of $P$. If $G$ is $K_{t,t}$-free, then it has $O(n2^{\alpha(n)^{c_t}})$ edges, where $\alpha(n)$ is the inverse Ackermann function.

Figures (11)

  • Figure 1: $e_1,e_2,e_3$ is a crossing sequence from $u$ to $v$ in this curve visibility graph. There is also a crossing sequence from $v$ to $u$ and therefore they must see each other. $x$ and $y$ form a double cherry.
  • Figure 2: The ordered graphs $H_0$ and $H_1$.
  • Figure 3: Drawing the non-edge visibility curves.
  • Figure 4: If $(a,b)$ is a non-edge, $a<b$ and $a$ is a 'right' vertex, then $\gamma_{a,b}$ runs along the edge of $P$ between $a$ and its following vertex along $P$. Thus, $\gamma_{a,b}$ is outside $\bigcup K_i$ locally near $a$.
  • Figure 5: If $\gamma_{a,b}$ and $\gamma_{c,d}$ intersect more than once and $a < c < b < d$, then there is $e \in E_{a,b} \setminus E_{c,d}$ and $f \in E_{c,d} \setminus E_{a,b}$ such that $f$ goes below $e$ which implies that $e$ and $f$ cross once more.
  • ...and 6 more figures

Theorems & Definitions (40)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.1
  • proof
  • Lemma 2.2: Coloring23
  • Theorem 2.3: CP92
  • Theorem 2.4
  • ...and 30 more