Every minor-closed property of sparse graphs is testable

Suppose G is a graph of bounded degree d, and one needs to remove epsilon n of its edges ill Order to make it planar. We show that ill this case the statistics of local neighborhoods around vertices of G is far from the statistics of local neighborhoods around vertices of any planar graph G'. In fact, a similar result is proved for any minor-closed property of bounded degree graphs. The main motivation of the above result comes from theoretical computer-science. Using our main result is a Constant time algorithm for detecting if a graph we infer that for any minor-closed property P. there a is "far" from satisfying P. This. ill particular, answers an open problem of Goldreich and Ron [STOC 1997] [20], who asked if such an algorithm exists when P is the graph property of being planar. The proof combines results from the theory of graph minors with results on convergent sequences of sparse graphs, which rely on martingale arguments. (c) 2009 Elsevier Inc. All rights reserved.