eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-06-28
58:1
58:15
10.4230/LIPIcs.ICALP.2022.58
article
Testability and Local Certification of Monotone Properties in Minor-Closed Classes
Esperet, Louis
1
https://orcid.org/0000-0001-6200-0514
Norin, Sergey
2
https://orcid.org/0000-0003-4833-7983
Univ. Grenoble Alpes, CNRS, Laboratoire G-SCOP, Grenoble, France
Department of Mathematics and Statistics, McGill University, Montreal, Canada
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol229-icalp2022/LIPIcs.ICALP.2022.58/LIPIcs.ICALP.2022.58.pdf
Property testing
sparse model
local certification
minor-closed classes