On the query complexity of estimating the distance to hereditary graph properties

TL;DR

This paper investigates the query complexity involved in estimating how close a graph is to having a hereditary property, specifically being induced $$-free, using regularity and removal lemmas.

Contribution

It establishes that the normalized edit distance to hereditary properties is estimable with query complexity depending only on regularity and removal lemmas.

Findings

Query complexity depends only on regularity and removal lemmas.

Normalized edit distance to hereditary properties is estimable.

Results apply to graphs with respect to induced $$-freeness.

Abstract

Given a family of graphs $\mathcal{F}$, we prove that the normalized edit distance of any given graph $\Gamma$ to being induced $\mathcal{F}$-free is estimable with a query complexity that depends only on the bounds of the Frieze--Kannan Regularity Lemma and on a Removal Lemma for $\mathcal{F}$.