On the characterization of 1-sided error strongly testable graph properties for bounded-degree graphs

Abstract

We study property testing of (di)graph properties in bounded-degree graph models. The study of graph properties in bounded-degree models is one of the focal directions of research in property testing in the last 15 years. However, despite the many results and the extensive research effort, there is no characterization of the properties that are strongly testable (i.e. testable with constant query complexity) even for 1-sided error tests. The bounded-degree model can naturally be generalized to directed graphs resulting in two models that were considered in the literature. The first contains the directed graphs in which the out-degree is bounded but the in-degree is not restricted. In the other, both the out-degree and in-degree are bounded. We give a characterization of the 1-sided error strongly testable monotone graph properties and the 1-sided error strongly testable hereditary graph properties in all the bounded-degree directed and undirected graphs models.