 
Summary: The maximum edit distance from hereditary graph properties
Noga Alon
Uri Stav
September 21, 2007
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 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 [7], 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) = EP (G(n, p(P))) + o(n2
) 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 hered
itary 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 Hfree 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
