Independence number of intersection graphs of axis-parallel segments

TL;DR

This paper establishes a tight lower bound on the independence number of triangle-free intersection graphs of axis-parallel segments, showing it is at least roughly a quarter of the total segments with an additive square root term.

Contribution

It provides a new bound on the independence number for this class of graphs and constructs examples showing the bound's optimality.

Findings

Lower bound: independence number ≥ n/4 + Ω(√n)

Upper bound example: independence number ≤ n/4 + c√n

Bound is tight and optimal for the class

Abstract

We prove that for any triangle-free intersection graph of $n$ axis-parallel segments in the plane, the independence number $\alpha$ of this graph is at least $\alpha \ge n/4 + \Omega(\sqrt{n})$. We complement this with a construction of a graph in this class satisfying $\alpha \le n/4 + c \sqrt{n}$ for an absolute constant $c$, which demonstrates the optimality of our result.