Combinatorica

, Volume 20, Issue 4, pp 451–476 | Cite as

Efficient Testing of Large Graphs

  • Noga Alon
  • Eldar Fischer
  • Michael Krivelevich
  • Mario Szegedy
Original Paper

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.

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

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Copyright information

© János Bolyai Mathematical Society, 2000

Authors and Affiliations

  • Noga Alon
    • 1
  • Eldar Fischer
    • 2
  • Michael Krivelevich
    • 3
  • Mario Szegedy
    • 4
  1. 1.Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University; Tel Aviv 69978, Israel; and AT&T Labs–Research; Florham Park, NJ 07932, USA; E-mail: noga@math.tau.ac.ilIL
  2. 2.NEC Research Institute; 4 Independence Way, Princeton NJ, 08540, USA; E-mail: fischer@research.nj.nec.comUS
  3. 3.Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University; Tel Aviv 69978, Israel; E-mail: krivelev@math.tau.ac.ilIL
  4. 4.School of Mathematics, Institute for Advanced Study; Olden Lane, Princeton, NJ 08540, USA; E-mail: szegedy@math.ias.eduUS

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