A Characterization of Easily Testable Induced Subgraphs

Let H be a fixed graph on h vertices. We say that a graph G is induced H-free if it does not contain any induced copy of H. Let G be a graph on n vertices and suppose that at least en² edges have to be added to or removed from it in order to make it induced H-free. It was shown in [5] that in this case G contains at least f(ε,h)n^h induced copies of H, where 1/f(ε, h) is an extremely fast growing function in 1/ε, that is independent of n. As a consequence, it follows that for every H, testing induced H-freeness with one-sided error has query complexity independent of n. A natural question, raised by the first author in [1], is to decide for which graphs H the function 1/f(ε,H) can be bounded from above by a polynomial in 1/ε. An equivalent question is for which graphs H, can one design a one-sided error property tester for testing induced H-freeness, whose query complexity is polynomial in 1/ε. We settle this question almost completely by showing that, quite surprisingly, for any graph other than the paths of lengths 1, 2 and 3, the cycle of length 4, and their complements, no such property tester exists. We further show that a similar result also applies to the case of directed graphs, thus answering a question raised by the authors in [9]. We finally show that the same results hold even in the case of two-sided error property testers. The proofs combine combinatorial, graph theoretic and probabilistic arguments with results from additive number theory.