### Abstract

Let 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 ∈n^{2} 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.

Original language | English (US) |
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Pages (from-to) | 451-476 |

Number of pages | 26 |

Journal | Combinatorica |

Volume | 20 |

Issue number | 4 |

DOIs | |

State | Published - Jan 1 2000 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Discrete Mathematics and Combinatorics
- Computational Mathematics

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## Cite this

*Combinatorica*,

*20*(4), 451-476. https://doi.org/10.1007/s004930070001