In a recent work (ECCC, TR18-171, 2018), we introduced models of testing graph properties in which, in addition to answers to the usual graph-queries, the tester obtains random vertices drawn according to an arbitrary distribution $D$. Such a tester is required to distinguish between graphs that have the property and graphs that are far from having the property, where the distance between graphs is defined based on the unknown vertex distribution $D$. These (``vertex-distribution free'' (VDF)) models generalize the standard models in which $D$ is postulated to be uniform on the vertex-set, and they were studies both in the dense graph model and in the bounded-degree graph model.
The focus of the aforementioned work was on testers, called strong, whose query complexity depends only on the proximity parameter $\epsilon$. Unfortunately, in the standard bounded-degree graph model, some natural properties such as Bipartiteness do not have strong testers, and others (like cycle-freeness) do not have strong testers of one-sided error (whereas one-sided error was shown inherent to the VDF model). Hence, it was suggested to study general (i.e., non-strong) testers of ``sub-linear'' complexity.
In this work, we pursue the foregoing suggestion, but do so in a model that augments the model presented in the aforementioned work. Specifically, we provide the tester with an evaluation oracle to the unknown distribution $D$, in addition to samples of $D$ and oracle access to the tested graph. Our main results are testers for Bipartitness and cycle-freeness, in this augmented model, having complexity that is almost-linear in the square root of the ``effective support size'' of $D$.