On the edit distance from $K_{2,t}$-free graphs (Extended Version)

TL;DR

This paper determines the edit distance function for graphs avoiding certain bipartite subgraphs, revealing new maximum values and connections to extremal graph theory, especially for odd t, using advanced combinatorial techniques.

Contribution

It provides the first complete characterization of the edit distance function for orb(K_{2,t}) for t=3,4 and extends known results for all t, highlighting maximum values for odd t.

Findings

Exact edit distance functions for t=3,4

Maximum values of the function for all odd t

Connections to extremal graph theory problems

Abstract

The edit distance between two graphs on the same vertex set is defined to be the size of the symmetric difference of their edge sets. The edit distance function of a hereditary property, $\mathcal{H}$, is a function of $p$, and measures, asymptotically, the furthest graph of edge density $p$ from $\mathcal{H}$ under this metric. In this paper, we address the hereditary property $\forb(K_{2,t})$, the property of having no induced copy of the complete bipartite graph with 2 vertices in one class and $t$ in the other. Employing an assortment of techniques and colored regularity graph constructions, we are able to determine the edit distance function over the entire domain $p\in [0,1]$ when $t=3,4$ and extend the interval over which the edit distance function for $\forb(K_{2,t})$ is known for all values of $t$, determining its maximum value for all odd $t$. We also prove that the function for odd $t$ has a nontrivial interval on which it achieves its maximum. These are the only known principal hereditary properties for which this occurs. In the process of studying this class of functions, we encounter some surprising connections to extremal graph theory problems, such as strongly regular graphs and the problem of Zarankiewicz. This is an extended version of a paper with the same name now published in the Journal of Graph Theory \cite{jgt_version}. In particular, this version contains Appendix A, which has tables and graphs pertaining to the hereditary property $\forb(K_{2,t})$ for small $t$, and Appendix B, which has the proofs of Lemma 31, Proposition 32, Proposition 33, and Lemma 34.