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oai:drops-oai.dagstuhl.de:22837 2025-10-02T13:35:05Z ddc:004 open_access

Results on H-Freeness Testing in Graphs of Bounded r-Admissibility Awofeso, Christine Greaves, Patrick Lachish, Oded Reidl, Felix Property Testing Sparse Graphs Degeneracy Admissibility We study the property of H-freeness in graphs with known bounded average degree, i.e. the property of a graph not containing some graph H as a subgraph. H-freeness is one of the fundamental graph properties that has been studied in the property testing framework. Levi [Reut Levi, 2021] showed that triangle-freeness is testable in graphs of bounded arboricity, which is a superset of e.g. planar graphs or graphs of bounded degree. Complementing this result is a recent preprint [Talya Eden et al., 2024] by Eden ηl which shows that, for every r ≥ 4, C_r-freeness is not testable in graphs of bounded arboricity. We proceed in this line of research by using the r-admissibility measure that originates from the field of structural sparse graph theory. Graphs of bounded 1-admissibility are identical to graphs of bounded arboricity, while graphs of bounded degree, planar graphs, graphs of bounded genus, and even graphs excluding a fixed graph as a (topological) minor have bounded r-admissibility for any value of r [Nešetřil and Ossona de Mendez, 2012]. In this work we show that H-freeness is testable in graphs with bounded 2-admissibility for all graphs H of diameter 2. Furthermore, we show the testability of C₄-freeness in bounded 2-admissible graphs directly (with better query complexity) and extend this result to C₅-freeness. Using our techniques it is also possible to show that C₆-freeness and C₇-freeness are testable in graphs with bounded 3-admissibility. The formal proofs will appear in the journal version of this paper. These positive results are supplemented with a lower bound showing that, for every r ≥ 4, C_r-freeness is not testable for graphs of bounded (⌊r/2⌋ - 1)-admissibility. This lower bound will appear in the journal version of this paper. This implies that, for every r > 0, there exists a graph H of diameter r+1, such that H-freeness is not testable on graphs with bounded r-admissibility. These results lead us to the conjecture that, for every r > 4, and t ≤ 2r+1, C_t-freeness is testable in graphs of bounded r-admissibility, and for every r > 2, H-freeness for graphs H of diameter r is testable in graphs with bounded r-admissibility. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Christine Awofeso and Patrick Greaves and Oded Lachish and Felix Reidl 2025 Is Part Of LIPIcs, Volume 327, 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.STACS.2025.12 urn:nbn:de:0030-drops-228378 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2025.12 eng https://creativecommons.org/licenses/by/4.0/legalcode