Testing Subdivision-Freeness: - Property Testing Meets Structural Graph Theory -

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 2/3, say) between the case that G satisfies P and the case that one should add/remove at least εdn edges of G to make it satisfy P. In sharp contrast to property testing of dense graphs, which is relatively well understood, only few properties are known to be testable with a constant number of queries in the bounded-degree model. In particular, no global monotone (i.e., closed under edge deletions) property that expander graphs can satisfy has been shown to be testable in constant time so far. In this paper, we identify for the first time a natural family of global monotone property that expander graphs can satisfy and can be efficiently tested in the bounded degree model. Specifically, we show that, for any integer t?1, Kt-subdivision-freeness is testable with a constant number of queries in the bounded-degree model. This property was not previously known to be testable even with o(n) queries. Note that an expander graph with all degree less than t-1 does not have a Kt-subdivision. The proof is based on a novel combination of some results that develop the framework of partitioning oracles, together with structural graph theory results that develop the seminal graph minor theory by Robertson and Seymour. As far as we aware, this is the first result that bridges property testing and structural graph theory. Although we know a rough structure for graphs without H-minors from the famous graph minor theory by Robertson and Seymour, there is no corresponding structure theorem for graphs without H-subdivisions so far, even K5-subdivision-free graphs. Therefore, subdivisions and minors are very different in a graph structural sense.