Awofeso, Christine and Greaves, Patrick 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.
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Abstract
We study the property of \emph{$H$-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 showed that triangle-freeness is testable in graphs of bounded \emph{arboricity}, which is a superset of e.g. planar graphs or graphs of bounded degree. Complementing this results is a recent preprint by Eden et al which shows that, for every $r\geq 4$, $C_r$-freeness is not testable in graphs of bounded arboricity. We proceed in this line of research by drawing on results from the field of structural sparse graph theory and identify the \emph{$r$-admissibility} as a useful measure: 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~$r$~\cite{sparsity}. In this work we show that~$H$-freeness is testable in graphs with bounded $2$-admissibility for all graphs~$H$ of diameter~$2$. Further, we show the testability of~$C_4$-freeness in bounded $2$-admissible graphs directly (with better query complexity) and extend this result to~$C_5$s. Extending our technique further, we are able to show that~$C_6$- and $C_7$-freeness are testable in graphs with bounded $3$-admissibility. We supplement this positive results with a lower bound showing that, for every $r\geq 4$, $C_r$-freeness is not testable for graphs of bounded $(\lfloor r/2 \rfloor - 1)$-admissibility. 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 $r> 2$, $H$-freeness for graphs $H$ of diameter $r$ is testable in graphs with bounded $r$-admissibility.
Metadata
| Item Type: | Conference or Workshop Item (Paper) |
|---|---|
| Additional Information: | Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 12:1-12:16 |
| Keyword(s) / Subject(s): | property testing, sparse graphs, degeneracy, admissibility |
| School: | Birkbeck Faculties and Schools > Faculty of Science > School of Computing and Mathematical Sciences |
| Depositing User: | Oded Lachish |
| Date Deposited: | 25 Mar 2025 15:18 |
| Last Modified: | 02 Apr 2025 13:58 |
| URI: | https://eprints.bbk.ac.uk/id/eprint/54729 |
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