 
Summary: What is the furthest graph from a hereditary property?
Noga Alon
Uri Stav
Abstract
For a graph property P, the edit distance of a graph G from P, denoted EP (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 edgemodification problems, in which one wishes to
find or approximate the value of EP (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) = EP (G(n, p(P))) + o(n2
) with high probability. The proofs combine several
tools, including strengthened versions of the Szemer´edi Regularity Lemma, properties of random
graphs and probabilistic arguments.
1 Introduction
