The Zarankiewicz Problem for Polygon Visibility Graphs

2025-03-12

AI Summary

This work investigates a geometric variant of the Zarankiewicz problem for polygon visibility graphs—specifically, the maximum number of edges in a visibility graph on $n$ vertices that contains no $K_{t,t}$ subgraph. Employing tools from combinatorial geometry, Davenport–Schinzel sequence theory, and divide-and-conquer analysis, we establish the first quasi-linear upper bound $ ilde{O}(n)$ for general simple polygons. For star-shaped and monotone polygons, we obtain tight linear bounds $Theta(n)$. Furthermore, in the more general setting of pseudo-segment visibility graphs induced by $n$ points on a simple closed curve, we prove an $O(n log n)$ upper bound and an $Omega(n alpha(n))$ lower bound, where $alpha(n)$ denotes the inverse Ackermann function. These results determine the asymptotic growth rate of the Zarankiewicz function for visibility graphs: $ ilde{O}(n)$ in the general case, $Theta(n)$ for special polygon classes, and—crucially—reveal its intrinsic connection to $alpha(n)$ for pseudo-segment arrangements.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Computer Vision: Other Foundations of Computer Vision
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Web data visualization

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 $Omega(nalpha(n))$ lower bound.

Problem

• Quasi-linear upper bound for $K_{t,t}$-free polygon visibility graphs
• Linear bound for star-shaped and monotone polygon visibility graphs
• $O(n log n)$ upper and $Omega(nalpha(n))$ lower bounds for visibility pseudo-segments

Innovation

• Quasi-linear bound for K_{t,t}-free graphs
• Linear bound for star-shaped polygons
• O(n log n) bound for visibility pseudo-segments