Notes on Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs

TL;DR

This paper discusses properties of graphs with polynomial expansion and low-density segments, providing new insights into separators, divisions, and segment density relevant for approximation algorithms.

Contribution

It introduces a new condition for segment density and explores implications of hereditary sublinear separators on graph divisions.

Findings

Graphs with polynomial expansion have sublinear separators.

Hereditary sublinear separators imply small graph divisions.

A natural condition for low-density segments is proposed.

Abstract

This write-up contains some minor results and notes related to our work [HQ15] (some of them already known in the literature). In particular, it shows the following: - We show that a graph with polynomial expansion have sublinear separators. - We show that hereditary sublinear separators imply that a graph have small divisions. - We show a natural condition on a set of segments, such that they have low density. This might be of independent interest in trying to define a realistic input model for a set of segments. Unlike the previous two results, this is new. For context and more details, see the main paper.