On the computation of edit distance functions

TL;DR

This paper computes the edit distance functions for hereditary graph properties, especially for classes forbidding certain subgraphs, using Sidorenko's symmetrization method, advancing understanding of graph similarity measures.

Contribution

It introduces a method to compute edit distance functions for hereditary properties, including complex cases like split graphs and specific challenging graphs.

Findings

Computed edit distance functions for ${ m Forb}(H)$ with split graphs

Analyzed the graph $H_9$ for edit distance complexities

Extended symmetrization techniques to new 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 edit distance function of the hereditary property, $\mathcal{H}$, is a function of $p\in[0,1]$ and is the limit of the maximum normalized distance between a graph of density $p$ and $\mathcal{H}$. This paper uses the symmetrization method of Sidorenko in order to compute 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 $H$. We compute the edit distance function for ${\rm Forb}(H)$, where $H$ is any split graph, and the graph $H_9$, a graph first used to describe the difficulties in computing the edit distance function.