Testing $C_k$-freeness in Bounded Admissibility Graphs

Awofeso, Christine and Greaves, Patrick and Lachish, Oded and Levi, Amit and Reidl, Felix (2025) Testing $C_k$-freeness in Bounded Admissibility Graphs. In: International Colloquium on Automata, Languages, and Programming, 9-11 Jul 2025, Aarhus, Denmark.

Abstract

We study $C_k$-freeness in sparse graphs from a property testing perspective, specifically for graph classes with bounded $r$-admissibility. Our work is motivated by the large gap between upper and lower bounds in this area: $C_k$-freeness is known to be testable in planar graphs (2019), but not in graphs with bounded arboricity for $k > 3$~\cite{edenCycles2024}. There are a large number of interesting graph classes that include planar graphs and have bounded arboricity (\eg classes excluding a minor), calling for a more fine-grained approach to the question of testing $C_k$-freeness in sparse graph classes. One such approach, inspired by the work of Nesetril and Ossona de Mendez (2012), is to consider the graph measure of $r$-admissibility, which naturally forms a hierarchy of graph families $\mathcal A_1 \supset \mathcal A_2 \supset \ldots \supset \mathcal A_\infty$ where $\mathcal A_r$ contains all graph classes whose $r$-admissibility is bounded by some constant. The family $\mathcal A_1$ contains classes with bounded arboricity, the class $\mathcal A_\infty$ contains classes like planar graphs, graphs of bounded degree, and minor-free graphs. Awofeso \etal~(2025) recently made progress in this direction. They showed that $C_4$- and $C_5$-freeness is testable in $\mathcal A_2$. They further showed that $C_k$-freeness is \emph{not} testable in $\mathcal A_{\lfloor k/2\rfloor -1}$ and conjectured that $C_k$-freeness is testable in $\mathcal A_{\lfloor k/2 \rfloor}$. In this work, we prove this conjecture: $C_k$-freeness is indeed testable in graphs of bounded $\lfloor k/2\rfloor$-admissibility.

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