The edit distance of word-representable and comparability graphs

TL;DR

This paper determines the maximum edit distances for graphs from properties of word-representability and comparability, providing exact asymptotic bounds and analyzing edit distance functions across various densities.

Contribution

It establishes precise asymptotic bounds for the maximum edit distance from hereditary properties of word-representable and comparability graphs, including detailed edit distance functions.

Findings

Maximum edit distance from word-representable graphs is approximately n^2/8.

Maximum edit distance from comparability graphs is approximately 5n^2/32.

The edit distance functions are determined over all edge densities p in [0,1].

Abstract

In this paper, we establish that the maximum edit distance of an $n$-vertex graph from the hereditary property of word-representable graphs is $n^2/8-o(n^2)$. In addition, we establish that the maximum edit distance of an $n$-vertex graph from the hereditary property of poset comparability graphs is $5n^2/32-o(n^2)$. In fact, we determine the edit distance function over all edge densities $p\in [0,1]$ for the property of word-representable graphs, for the property of $k$-word-representable graphs for each $k\geq 2$, and for the property comparability graphs. The latter has a peculiar structure that requires an infinite sequence of colored regularity graphs.