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

We study property testing of (di)graph properties inbounded-degree graph models. The study of graph properties inbounded-degree models is one of the focal directions of research inproperty testing in the last 15 years. However, despite the many resultsand the extensive research effort, there is no characterization ofthe properties that are strongly testable (i.e. testable with constantquery complexity) even for 1-sided error tests. The bounded-degree model can naturally be generalized to directedgraphs resulting in two models that were considered in the literature.The first contains the directed graphs in which the out-degreeis bounded but the in-degree is not restricted. In the other, both theout-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 undirectedgraphs models.