Efficient Testing of Bipartite Graphs for Forbidden Induced Subgraphs

Alon et. al. [3], showed that every property that is characterized by a finite collection of forbidden induced subgraphs is -testable. However, the complexity of the test is double-tower with respect to 1/ , as the only tool known to construct such tests uses a variant of Szemerédi’s Regularity Lemma. Here we show that any property of bipartite graphs that is characterized by a finite collection of forbidden induced subgraphs is -testable, with a number of queries that is polynomial in 1/ . Our main tool is a new ‘conditional’ version of the regularity lemma for binary matrices, which may be interesting on its own. ∗A preliminary (and weaker) version of these results formed part of [10]. †Schools of Mathematics and Computer Science, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel, and IAS, Princeton, NJ 08540, USA. Email: [email protected] Research supported in part by a grant from the Israel Science Foundation, by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University, and by the Von Neumann Fund. ‡Faculty of Computer Science, Technion – Israel Institute of Technology, Haifa 32000, Israel. Email: [email protected] Research supported in part by grant number 55/03 from the Israel Science Foundation. §Department of Computer Science, University of Haifa, Haifa 31905, Israel. Email: [email protected] Research supported in part by grant number 55/03 from the Israel Science Foundation.