Edit distance and its computation

TL;DR

This paper introduces a new method to compute the asymptotic maximum edit distance for hereditary graph properties without relying on Szemerédi's Regularity Lemma, enabling calculations for previously unknown cases.

Contribution

The paper presents a novel approach to determine the asymptotic edit distance for hereditary properties, including new computations for specific graphs and a generalization of Turán's theorem.

Findings

Computed edit distance for graphs with forbidden subgraphs like $K_a+E_b$ and $K_{3,3}$.

Developed weighted Turán's theorem generalizations.

Identified a graph where edit distance cannot be derived from simple vertex partitioning.

Abstract

In this paper, we provide a method for determining the asymptotic value of the maximum edit distance from a given hereditary property. This method permits the edit distance to be computed without using Szemer\'edi's Regularity Lemma directly. Using this new method, we are able to compute the edit distance from hereditary properties for which it was previously unknown. For some graphs $H$, the edit distance from ${\rm Forb}(H)$ is computed, where ${\rm forb}(H)$ is the class of graphs which contain no induced copy of graph $H$. Those graphs for which we determine the edit distance asymptotically are $H=K_a+E_b$, an $a$-clique with $b$ isolated vertices, and $H=K_{3,3}$, a complete bipartite graph. We also provide a graph, the first such construction, for which the edit distance cannot be determined just by considering partitions of the vertex set into cliques and cocliques. In the process, we develop weighted generalizations of Tur\'an's theorem, which may be of independent interest.