Testing H-freeness on sparse graphs, the case of bounded expansion

TL;DR

This paper demonstrates that testing H-freeness in sparse graphs with bounded expansion can be achieved with a constant number of queries, simplifying previous proofs and broadening applicability to various graph classes.

Contribution

It provides a simpler proof that H-freeness testing with constant queries is possible in bounded expansion classes, extending prior results beyond minor-free graphs.

Findings

Constant-query H-freeness testing for bounded expansion classes.

Applicable to various sparse graph classes like bounded degree and cubic graphs.

Simplifies previous proofs using the sparsity toolkit.

Abstract

In property testing, a tester makes queries to (an oracle for) a graph and, on a graph having or being far from having a property P, it decides with high probability whether the graph satisfies P or not. Often, testers are restricted to a constant number of queries. While the graph properties for which there exists such a tester are somewhat well characterized in the dense graph model, it is not the case for sparse graphs. In this area, Czumaj and Sohler (FOCS'19) proved that H-freeness (i.e. the property of excluding the graph H as a subgraph) can be tested with constant queries on planar graphs as well as on graph classes excluding a minor. Using results from the sparsity toolkit, we propose a simpler alternative to the proof of Czumaj and Sohler, for a statement generalized to the broader notion of bounded expansion. That is, we prove that for any class C with bounded expansion and any graph H, testing H-freeness can be done with constant query complexity on any graph G in C, where the constant depends on H and C, but is independent of G. While classes excluding a minor are prime examples of classes with bounded expansion, so are, for example, cubic graphs, graph classes with bounded maximum degree, graphs of bounded book thickness, or random graphs of bounded average degree.