The edit distance function and symmetrization

TL;DR

This paper introduces a localization method to compute the edit distance function for hereditary graph properties, providing explicit results for properties forbidding cycles of up to nine vertices.

Contribution

It develops a new localization technique for calculating the edit distance function and applies it to specific hereditary properties, including those forbidding small cycles.

Findings

Developed a method called localization for computing edit distance functions.

Provided explicit edit distance functions for properties forbidding cycles up to 9 vertices.

Estimated the edit distance function for general hereditary properties.

Abstract

The edit distance between two graphs on the same labeled vertex set is the size of the symmetric difference of the edge sets. The distance between a graph, $G$, and a hereditary property, ${\cal H}$, is the minimum of the distance between $G$ and each $G'\in{\cal H}$. The edit distance function of ${\cal H}$ is a function of $p\in[0,1]$ and is the limit of the maximum normalized distance between a graph of density $p$ and ${\cal H}$. This paper develops a method, called localization, for computing the edit distance function of various hereditary properties. For any graph $H$, ${\rm Forb}(H)$ denotes the property of not having an induced copy of $H$. This paper gives some results regarding estimation of the function for an arbitrary hereditary property. This paper also gives the edit distance function for ${\rm Forb}(H)$, where $H$ is a cycle on 9 or fewer vertices.