Every Minor-Closed Property of Sparse Graphs is Testable

TL;DR

This paper proves that minor-closed properties of sparse graphs are testable with a constant number of queries, by showing local neighborhood statistics distinguish graphs far from having these properties.

Contribution

It establishes the testability of various minor-closed properties in bounded-degree graphs, a result not previously known even with sublinear queries.

Findings

Minor-closed properties are testable with constant queries.

Graphs far from having these properties have distinguishable local neighborhood statistics.

Many well-studied properties like planarity and bounded tree-width are testable.

Abstract

Suppose $G$ is a graph with degrees bounded by $d$, and one needs to remove more than $\epsilon n$ of its edges in order to make it planar. We show that in 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 $G'$ with the same degree bound. In fact, a similar result is proved for any minor-closed property of bounded degree graphs. As an immediate corollary of the above result 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, where the constant may depend on $\epsilon$ and $d$, but not on the graph size. None of these properties was previously known to be testable even with $o(n)$ queries.