Summary: The maximum edit distance from hereditary graph properties
September 21, 2007
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 , 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 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