Results on H-Freeness testing in graphs of Bounded r-Admissibility

Awofeso, C. and Greaves, P. and Lachish, Oded and Reidl, Felix (2025) Results on H-Freeness testing in graphs of Bounded r-Admissibility. In: 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025), 04-07 Mar 2025, Jena, Germany.

Abstract

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.

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