Summary: What is the furthest graph from a hereditary property?
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 edge-modification 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.