Efficient Testing of Large Graphs


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.

This is a preview of subscription content, access via your institution.

Author information



Additional information

Received December 6, 1999

Rights and permissions

Reprints and Permissions

About this article

Cite this article

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

Download citation

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