A sufficient condition for characterizing the one-sided testable properties of families of graphs in the Random Neighbour Oracle Model

2025-11-24

AI Summary

This paper addresses the characterization of one-sided error testability for graph properties under the random neighbor query model, focusing on $H$-freeness (i.e., subgraph-freeness). We introduce and rigorously prove a general sufficient condition for $H$-testability based on the $r$-admissibility measure from sparse graph theory—yielding the first unified criterion for $H$-testability. Our condition systematically generalizes prior results for planar graphs and $H$-minor-free families, and establishes $H$-testability for several new important sparse graph classes, including topologically $H$-closed graphs, low-density geometric intersection graphs, and graphs with bounded crossing number. Technically, our approach integrates $r$-admissibility analysis, combinatorial structural arguments, and constructive graph algorithm design. The result provides the first systematic classification framework for graph property testing that applies broadly across diverse sparse graph families, significantly extending the scope of testability theory beyond previously known regimes.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

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.

Problem

• Characterizing one-sided testable graph properties in random neighbor oracle model
• Establishing sufficient conditions for H-testable graph families
• Extending testability criteria to broader sparse graph families

Innovation

• Using r-admissibility measure for graph testability characterization
• Providing sufficient criterion for H-testable graph families
• Extending testability to topological minor closed families