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 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 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. 1 Introduction Collections: Mathematics