What is the furthest graph from a hereditary property?

Abstract

For a graph property P, the edit distance of a graph G from P, denoted E P (G), is the minimum number of edge modifications (additions or deletions) one needs to apply to G in order to turn it into a graph satisfying P. What is the furthest graph on n vertices from P and what is the largest possible edit distance from P? Denote this maximal distance by ed(n, P). This question is motivated by algorithmic edge-modification problems, in which one wishes to find or approximate the value of E P (G) given an input graph G.A monotone graph property is closed under removal of edges and vertices. Trivially, for any monotone property, the largest edit distance is attained by a complete graph. We show that this is a simple instance of a much broader phenomenon. A hereditary graph property is closed under removal of vertices. We prove that for any hereditary graph property P, a random graph with an edge density that depends on P essentially achieves the maximal distance from P, that is: ed(n, P) = E P (G(n, p(P))) + o(n 2 ) with high probability. The proofs combine several tools, including strengthened versions of the Szemerédi Regularity Lemma, properties of random graphs and probabilistic arguments.

Stability‐type results for hereditary properties

“…[4], [5], [9], [13], [18], [19]). It was shown that various classical extremal results for monotone graph properties have a natural analogue for some hereditary properties.…”

“…The usage of these definitions is to model regular partitions of graphs. This is done via Lemma 3.4 which is proved in [4]. This powerful lemma allows us to omit some of the tedious details that usually accompany the usage of the regularity lemma.…”

“…That is, G does not contain an induced copy of any member of H. The following lemma shows that every induced H-free graph is close to conforming to some H-homomorphism-free colored regularity graph. This lemma is implicitly proved in [4], following the method of [2].…”

“…An appropriate choice of the parameters, depending on P, together with Ramsey Theorem, then determines the choice of colors for the vertices and edges of K for which the lemma holds. The details of the proof appear as steps 1 to 3 in the proof of Theorem 1.1 in [4].…”

Edit distance measure for graphs

“…The study of the edit distance in graphs was originated independently by Axenovich, Kézdy and Martin [4] and Alon and Stav [3]. Since then, there has been a great deal of study on the edit distance itself and on the so-called edit distance function (see for example [1], [5]).…”