Efficient Testing of Large Graphs

P

be a property of graphs. An -test for P is a randomized algorithm which, given the ability to make queries whether a desired pair of vertices of an input graph G with n vertices are adjacent or not, distinguishes, with high probability, between the case of G satisfying P and the case that it has to be modified by adding and removing more than edges to make it satisfy P. The property P is called testable, if for every there exists an -test for P whose total number of queries is independent of the size of the input graph. Goldreich, Goldwasser and Ron [8] showed that certain individual graph properties, like k-colorability, admit an -test. In this paper we make a first step towards a complete logical characterization of all testable graph properties, and show that properties describable by a very general type of coloring problem are testable. We use this theorem to prove that first order graph properties not containing a quantifier alternation of type ``'' are always testable, while we show that some properties containing this alternation are not.

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Received December 6, 1999

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Alon, N., Fischer, E., Krivelevich, M. et al. Efficient Testing of Large Graphs. Combinatorica 20, 451–476 (2000). https://doi.org/10.1007/s004930070001

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  • AMS Subject Classification (1991) Classes:  68R10, 05C85, 05C35.