Property testing is concerned with constant-time algorithms for deciding whether a given object satisfies a predetermined property or is far from satisfying it. In this paper, we consider testing properties related to the connectivity of two vertices in sparse graphs.We present one-sided error testers for (s,t)-disconnectivity with query complexity 2O(1/ϵ) for digraphs and O(1/ϵ2) for graphs, where ϵ is an error parameter. Furthermore, we show that these algorithms are the best possible in view of query complexity, i.e., we give matching lower bounds for two-sided error testers for both cases.We also give a constant-time algorithm for testing the (s,t)-disconnectivity of a directed bounded-degree hypergraph, which can be used to test the satisfiability of Horn SAT.