On the computation of edit distance functions

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.Comment: 25 pages, 2 figure

The edit distance function of some graphs

“…We denote the limit in (2) by d * H . The edit distance functions of some kinds of graphs have been investigated in recent years, including complete graphs [13] and split graphs [12]. Actually, complete bipartite graphs are also studied.…”

On the edit distance of powers of cycles

“…The edit distance function of a hereditary property H is a function of p ∈ [0, 1] that measures, in the limit, the maximum normalized edit distance between a graph of density p and H. A principal hereditary property, denoted Forb(H), is a hereditary property that consists of the graphs with no induced copy of a single graph H. Most of the known edit distance functions are of the form Forb(H). These include the cases where H is a split graph [7] (including cliques and independent sets), complete bipartite graphs K 2,t [9] and K 3,3 [5] and cycles C h where h is small [6]. In this paper, we compute the edit distance function for powers of cycles.…”

A proof of the stability of extremal graphs, Simonovits' stability from Szemerédi's regularity

“…This notion was explicitly introduced in [3], Alon and Stav [2] proved connections with Turán theory. For more recent results see Martin [18].…”