A sufficient condition for characterizing the one-sided testable properties of families of graphs in the Random Neighbour Oracle Model
Christine Awofeso, Patrick Greaves, Oded Lachish, Amit Levi, Felix Reidl
TL;DR
This work investigates property testing in the random neighbour oracle model and provides a sufficient criterion for a graph family to be $H$-testable: universally bounded $r$-admissibility. By constructing a trimming procedure that yields a sparse subgraph and designing a growth-based tester that explores limited neighborhoods via admissible paths and prefix–nadir–strata structures, the authors show that every graph family with universally bounded $r$-admissibility has testable $H$-freeness for all fixed $H$. The main technical contribution is the PBFS-based tester whose query complexity does not depend on the input size, together with a detailed trimming analysis ensuring accuracy under the one-sided error regime. The results unify and extend known testability results (e.g., planar and minor-free families) and provide a broadly applicable framework for identifying $H$-testable graph families in sparse settings, signaling a path toward a full characterization in this model.
Abstract
We study property testing in the \emph{random neighbor oracle} model for graphs, originally introduced by Czumaj and Sohler [STOC 2019]. Specifically, we initiate the study of characterizing the graph families that are $H$-\emph{testable} in this model. A graph family $\mathcal{F}$ is $H$-testable if, for every graph $H$, $H$-\emph{freeness} (that is, not having a subgraph isomorphic to $H$) is testable with one-sided error on all inputs from $\mathcal{F}$. Czumaj and Sohler showed that for any $H$-testable family of graphs $\mathcal{F}$, the family of testable properties of $\mathcal{F}$ has a known characterization, a major goal in the study of property testing. Consequently, characterizing the collection of $H$-testable graph families will not only result in new characterizations, but will also exhaust this method of characterizing testable properties. We believe that our result is a substantial step towards this goal. Czumaj and Sohler further showed that the family of planar graphs is $H$-testable, as is any family of minor-free graphs. In this paper, we provide a sufficient and much broader criterion under which a family of graphs is $H$-testable. As a corollary, we obtain new characterizations for many families of graphs including: families that are closed under taking topological minors or immersions, geometric intersection graphs of low-density objects, euclidean nearest-neighbour graphs with bounded clique number, graphs with bounded crossing number (per edge), graphs with bounded queue- and stack number, and more. The criterion we provide is based on the \emph{$r$-admissibility} graph measure from the theory of sparse graph families initiated by Nesetril and Ossona de Mendez. Proving that specific families of graphs satisfy this criterion is an active area of research, consequently, the implications of this paper may be strengthened in the future.