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Testability and Local Certification of Monotone Properties in Minor-Closed Classes

Authors Louis Esperet , Sergey Norin

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Louis Esperet
  • Univ. Grenoble Alpes, CNRS, Laboratoire G-SCOP, Grenoble, France
Sergey Norin
  • Department of Mathematics and Statistics, McGill University, Montreal, Canada


This work was initiated during the Graph Theory workshop in Oberwolfach, Germany, in January 2022. The authors would like to thank the organizers and participants for all the discussions and nice atmosphere (and in particular Gwenaël Joret, Chun-Hung Liu, and Ken-ichi Kawarabayashi for the discussions related to the topic of this paper). The authors would also like to thank Gábor Elek for his remarks on an earlier version of this manuscript, and his suggestion to replace minor-closed classes by classes of bounded asymptotic dimension in Theorem 4.

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Louis Esperet and Sergey Norin. Testability and Local Certification of Monotone Properties in Minor-Closed Classes. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 58:1-58:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


The main problem in the area of graph property testing is to understand which graph properties are 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 sparse model. We prove that for any proper minor-closed class 𝒢, any monotone property (i.e., any property that is closed under taking subgraphs) is testable for graphs from 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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Theory of computation → Streaming, sublinear and near linear time algorithms
  • Property testing
  • sparse model
  • local certification
  • minor-closed classes


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