Abstract:
In this paper we consider two refined questions regarding the query complexity of testing graph properties in the adjacency matrix model. The first question refers to the relation between adaptive and non-adaptive testers, whereas the second question refers to testability within complexity that is inversely proportional to the proximity parameter, denoted $\e$. The study of these questions reveals the importance of algorithmic design (also) in this model. The highlights of our study are:

• A gap between the complexity of adaptive and non-adaptive testers. Specifically, there exists a (natural) graph property that can be tested using $\tildeO(\e^{-1})$ adaptive queries, but cannot be tested using $o(\e^{-3/2})$ non-adaptive queries.
• In contrast, there exist natural graph properties that can be tested using $\tildeO(\e^{-1})$ non-adaptive queries, whereas $\Omega(\e^{-1})$ queries are required even in the adaptive case.

We mention that the properties used in the foregoing conflicting results have a similar flavor, although they are of course different.