Property-Testing in Sparse Directed Graphs: 3-Star-Freeness and Connectivity

Frank Hellweg, Christian Sohler

We study property testing in directed graphs in the bounded degree model, where we assume that an algorithm may only query the outgoing edges of a vertex, a model proposed by Bender and Ron in 2002. As our first main result, we we present a property testing algorithm for strong connectivity in this model, having a query complexity of $\mathcal{O}(n^{1-\epsilon/(3+\alpha)})$ for arbitrary $\alpha>0$; it is based on a reduction to estimating the vertex indegree distribution. For subgraph-freeness we give a property testing algorithm with a query complexity of $\mathcal{O}(n^{1-1/k})$, where $k$ is the number of connected componentes in the queried subgraph which have no incoming edge. We furthermore take a look at the problem of testing whether a weakly connected graph contains vertices with a degree of least $3$, which can be viewed as testing for freeness of all orientations of $3$-stars; as our second main result, we show that this property can be tested with a query complexity of $\mathcal{O}(\sqrt{n})$ instead of, what would be expected, $\Omega(n^{2/3})$.

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