Testability and local certification of monotone properties in minor-closed classes

TL;DR

This paper proves that any monotone property is testable in sparse minor-closed graph classes, extending previous results and removing degree restrictions, with implications for property testing and certification.

Contribution

It extends testability results for monotone properties to all minor-closed classes in the sparse model, removing bounded degree assumptions.

Findings

Monotone properties are testable in minor-closed classes within the sparse model.

Extension of prior results from finitely forbidden subgraphs to all monotone properties.

Removal of bounded degree restrictions in property verification schemes.

Abstract

The main problem in the area of graph property testing is to understand which graph properties are \emph{testable}, which means that with constantly many queries to any input graph $G$, a tester can decide with good probability whether $G$ satisfies the property, or is far from satisfying the property. Testable properties are well understood in the dense model and in the bounded degree model, but little is known in sparse graph classes when graphs are allowed to have unbounded degree. This is the setting of the \emph{sparse model}. We prove that for any proper minor-closed class $\mathcal{G}$, any monotone property (i.e., any property that is closed under taking subgraphs) is testable for graphs from $\mathcal{G}$ in the sparse model. This extends a result of Czumaj and Sohler (FOCS'19), who proved it for monotone properties with finitely many forbidden subgraphs. Our result implies for instance that for any integers $k$ and $t$, $k$-colorability of $K_t$-minor free graphs is testable in the sparse model. Elek recently proved that monotone properties of bounded degree graphs from minor-closed classes that are closed under disjoint union can be verified by an approximate proof labeling scheme in constant time. We show again that the assumption of bounded degree can be omitted in his result.