Accumulation points of the edit distance function

Abstract

Given a hereditary property H of graphs and some p∈[0,1], the edit distance function edH(p) is (asymptotically) the maximum proportion of “edits” (edge-additions plus edge-deletions) necessary to transform any graph of density p into a member of H. For any fixed p∈[0,1], edH(p) can be computed from an object known as a colored regularity graph (CRG). This paper is concerned with those points p∈[0,1] for which infinitely many CRGs are required to compute edH on any open interval containing p; such a p is called an accumulation point. We show that, as expected, p=0 and p=1 are indeed accumulation points for some hereditary properties; we additionally determine the slope of edH at these two extreme points. Unexpectedly, we construct a hereditary property with an accumulation point at p=1/4. Finally, we derive a significant structural property about those CRGs which occur at accumulation points.