Testing H-freeness on sparse graphs, the case of bounded expansion
Samuel Humeau, Mamadou Moustapha Kanté, Daniel Mock, Timothé Picavet, Alexandre Vigny
TL;DR
The paper tackles the problem of testing whether a graph excludes a fixed subgraph H (H-freeness) in sparse graph classes, aiming for constant-query testers in the random-neighbor model. The authors extend the landmark Czumaj–Sohler result from minor-free classes to bounded-expansion classes via a sequence of reductions: from general graphs to subgraphs induced by many edge-disjoint copies of H, then to graphs with bounded treedepth, and finally to a self-contained treatment for bounded-treedepth graphs. A key contribution is a self-contained, simpler proof for H-freeness testing on graphs of bounded treedepth that avoids heavy machinery, while preserving the core probabilistic guarantees. The results yield constant-query, one-sided testers for any fixed H on classes of bounded expansion, with broad implications for graph families such as cubic graphs, bounded-degree classes, and graphs with bounded book thickness, thereby advancing sparse-property testing theory and its practical reach in algorithmic graph analysis.
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.