Accumulation points of the edit distance function

TL;DR

This paper investigates the accumulation points of the edit distance function for hereditary graph properties, revealing known and novel accumulation points, and analyzing the structure of associated colored regularity graphs.

Contribution

It characterizes accumulation points of the edit distance function, including a new example at p=1/4, and analyzes the structure of CRGs at these points.

Findings

p=0 and p=1 are accumulation points with known slopes

Constructed a hereditary property with an accumulation point at p=1/4

Derived structural properties of CRGs at accumulation points

Abstract

Given a hereditary property $\mathcal H$ of graphs and some $p\in[0,1]$, the edit distance function $\operatorname{ed}_{\mathcal H}(p)$ is (asymptotically) the maximum proportion of "edits" (edge-additions plus edge-deletions) necessary to transform any graph of density $p$ into a member of $\mathcal H$. For any fixed $p\in[0,1]$, $\operatorname{ed}_{\mathcal H}(p)$ can be computed from an object known as a colored regularity graph (CRG). This paper is concerned with those points $p\in[0,1]$ for which infinitely many CRGs are required to compute $\operatorname{ed}_{\mathcal H}$ on any open interval containing $p$; such a $p$ is called an accumulation point. We show that, as expected, $p=0$ and $p=1$ are indeed accumulation points for some hereditary properties; we additionally determine the slope of $\operatorname{ed}_{\mathcal H}$ at these two extreme points. Unexpectedly, we construct a hereditary property with an accumulation point at $p=1/4$. Finally, we derive a significant structural property about those CRGs which occur at accumulation points.