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A Sharp Threshold Phenomenon in String Graphs.

István Tomon¹

¹ETH Zürich, Zürich, Switzerland.

Discrete & Computational Geometry

|January 13, 2022

Summary

This summary is machine-generated.

Sparse string graphs, intersection graphs of curves, guarantee large vertex subsets with no edges. This study identifies a sharp density threshold (1/4) for this structural property in string graphs.

Keywords:

Separator String graph

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Area of Science:

Graph Theory
Computational Geometry

Background:

String graphs are intersection graphs of curves in the plane.
Understanding structural properties of sparse graphs is crucial in graph theory.

Purpose of the Study:

To determine the sharp threshold for edge density below which a string graph must contain two large, disjoint vertex subsets with no edges between them.
To generalize a known result for x-monotone string graphs to general string graphs.

Main Methods:

Leveraging a recent result on separators by Lee for string graphs.
Analyzing the edge density of string graphs to establish the threshold.
Building upon previous work on x-monotone string graphs.

Main Results:

Proved that for any string graph with edge density below 1/4, there exist two linear-sized vertex subsets with no edges between them.
Established 1/4 as the sharp threshold, demonstrating string graphs with density less than 1/4 can have edges between any two logarithmic-sized subsets.
Extended the result for x-monotone string graphs to the general case.

Conclusions:

The structural property of having large, edge-free vertex subsets holds for string graphs below a specific, sharp edge density threshold.
This research resolves a conjecture regarding the general case of string graphs.