On the edit distance of powers of cycles

Abstract

The edit distance between two graphs on the same labeled vertex set is defined to be the size of the symmetric difference of their edge sets. The edit distance function of a hereditary property H is a function of p∈[0,1] that measures, in the limit, the maximum normalized edit distance between a graph of density p and H. The expression H=Forb(H) denotes the property of having no induced subgraph isomorphic to H.In this paper, we address the edit distance function for the hereditary property ForbCht, where Cht denotes the tth power of the cycle of length h. For h≥2t(t+1)+1 and h not divisible by t+1, we determine the function for all values of p. For h≥2t(t+1)+1 and h divisible by t+1, the function is obtained for all but small values of p. We also obtain edit distance functions for some smaller values of h.