Testing graphs against an unknown distribution

The classical model of graph property testing, introduced by Goldreich, Goldwasser and Ron, assumes that the algorithm can obtain uniformly distributed vertices from the input graph. Goldreich introduced a more general model, called the Vertex-Distribution-Free model (or VDF for short) in which the testing algorithm obtains vertices drawn from an arbitrary and unknown distribution. The main motivation for this investigation is that it can allow one to give different weight/importance to different parts of the input graph, as well as handle situations where one cannot obtain uniformly selected vertices from the input. Goldreich proved that any property which is testable in this model must (essentially) be hereditary¹, and that several hereditary properties can indeed be tested in this model. He further asked which properties are testable in this model. In this paper we completely solve Goldreich’s problem by giving a precise characterization of the graph properties that are testable in the VDF model. Somewhat surprisingly this characterization takes the following clean form: say that a graph property P is extendable if given any graph G satisfying P, one can add one more vertex to G, and connect it to some of the vertices of G in a way that the resulting graph satisfies P. Then a property P is testable in the VDF model if and only if P is hereditary and extendable.