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, or graphs of bounded book thickness. Additionally, random graphs with bounded average degree almost surely have bounded expansion.

The Kantorovich distance is a widely used metric between probability distributions. The Kantorovich-Rubinstein duality states that it can be defined in two equivalent ways: as a supremum, based on non-expansive functions into [0,1], and as an infimum, based on probabilistic couplings. Orthogonally, there are categorical generalisations of both presentations proposed in the literature, in the form of codensity liftings and what we refer to as coupling-based liftings. Both lift endofunctors on the category Set of sets and functions to that of pseudometric spaces, and both are parameterised by modalities from coalgebraic modal logic. A generalisation of the Kantorovich-Rubinstein duality has been more nebulous - it is known not to work in some cases. In this paper we propose a compositional approach for obtaining such generalised dualities for a class of functors, which is closed under coproducts and products. Our approach is based on an explicit construction of modalities and also applies to and extends known cases such as that of the powerset functor.

We provide a finite equational presentation of graphs of treewidth at most three, solving an instance of an open problem by Courcelle and Engelfriet. We use a syntax generalising series-parallel expressions, denoting graphs with a small interface. We introduce appropriate notions of connectivity for such graphs (components, cutvertices, separation pairs). We use those concepts to analyse the structure of graphs of treewidth at most three, showing how they can be decomposed recursively, first canonically into connected parallel components, and then non-deterministically. The main difficulty consists in showing that all non-deterministic choices can be related using only finitely many equational axioms.