Every minor-closed property of sparse graphs is testable

Abstract

Testing a property P of graphs in the bounded degree model deals with the following problem: given a graph G of bounded degree d we should distinguish (with probability 0.9, say) between the case that G satisfies P and the case that one should add/remove at least ε d n edges of G to make it satisfy P. In sharp contrast to property testing of dense graphs, which is relatively well understood, very few properties are known to be testable in bounded degree graphs with a constant number of queries. In this paper we identify for the first time a large (and natural) family of properties that can be efficiently tested in bounded degree graphs, by showing that every minor-closed graph property can be tested with a constant number of queries. As a special case, we infer that many well studied graph properties, like being planar, outer-planar, series-parallel, bounded genus, bounded tree-width and several others, are testable with a constant number of queries. None of these properties was previously known to be testable even with o(n) queries. The proof combines results from the theory of graph minors with results on convergent sequences of sparse graphs, which rely on martingale arguments.