 
Summary: Efficient testing of large graphs
Noga Alon
Eldar Fischer
Michael Krivelevich §
Mario Szegedy ¶
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 n2
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 kcolorability 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.
Our results are proven using a combinatorial lemma, a special case of which, that may be of
independent interest, is the following. A graph H is called unavoidable in G if all graphs that differ
