The maximum edit distance from hereditary graph properties

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 largest possible edit distance of a graph on n vertices from P? Denote this distance by ed(n,P). A graph property is hereditary if it is closed under removal of vertices. In a previous work, the authors show that for any hereditary property, a random graph G(n,p(P)) essentially achieves the maximal distance from P, proving: ed(n,P)=E"P(G(n,p(P)))+o(n^2) with high probability. The proof implicitly asserts the existence of such p(P), but it does not supply a general tool for determining its value or the edit distance. In this paper, we determine the values of p(P) and ed(n,P) for some subfamilies of hereditary properties including sparse hereditary properties, complement invariant properties, (r,s)-colorability and more. We provide methods for analyzing the maximum edit distance from the graph properties of being induced H-free for some graphs H, and use it to show that in some natural cases G(n,1/2) is not the furthest graph. Throughout the paper, the various tools let us deduce the asymptotic maximum edit distance from some well studied hereditary graph properties, such as being Perfect, Chordal, Interval, Permutation, Claw-Free, Cograph and more. We also determine the edit distance of G(n,1/2) from any hereditary property, and investigate the behavior of E"P(G(n,p)) as a function of p. The proofs combine several tools in Extremal Graph Theory, including strengthened versions of the Szemeredi Regularity Lemma, Ramsey Theory and properties of random graphs.

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