一类退化图中单边误差可测试图性质的特征化
A characterization of one-sided error testable graph properties in bounded degeneracy graphs
AI总结：
本文研究了在退化图中单边误差可测试图性质的结构特征，证明了可测试性由禁止子图的连通性决定，即违反性质不能被分割到高度数邻居中。
摘要：
我们考虑在p-退化图中基于随机邻居 oracle 模型的图性质测试。在该框架下，测试器通过采样顶点的均匀邻居来探索图，且若查询复杂度与图大小无关，则性质具有单边误差可测试性。已知对于minor-closed家族，单边误差可测试性质恰好是那些可由有限大小禁止子图定义的性质。然而，更广泛的p-退化图类允许存在高度数的'hub'，这些结构可能隐藏禁止子图以避免局部探索。在本文中，我们为p-退化图中所有单边误差可测试性质提供了完整的结构特征化。我们证明可测试性本质上由禁止子图的连通性决定：一个性质可测试当且仅当其违反不能被分割到不同的高度数邻居中。我们的结果定义了在这些约束下可测试性的精确结构边界，考虑了单个禁止子图的连通性以及它们所定义的性质的集体行为。
英文摘要：
We consider graph property testing in $p$-degenerate graphs under the random neighbor oracle model (Czumaj and Sohler, FOCS 2019). In this framework, a tester explores a graph by sampling uniform neighbors of vertices, and a property is testable with one-sided error if its query complexity is independent of the graph size. It is known that one-sided error testable properties for minor-closed families are exactly those that can be defined by forbidden subgraphs of bounded size. However, the much broader class of $p$-degenerate graphs allows for high-degree ``hubs" that can structurally hide forbidden subgraphs from local exploration. In this work, we provide a complete structural characterization of all properties testable with one-sided error in $p$-degenerate graphs. We show that testability is determined by how copies of the forbidden graphs can share their high-degree vertices: $\mathcal{F}$-freeness is testable if and only if, for every member $H$, every set of roles of $H$ that may be played by high-degree vertices, and every pattern of dependencies between these roles, some member of $\mathcal F$ can be assembled from pieces of copies of $H$ glued according to this pattern. Our results define the exact structural boundary for testability under these constraints, accounting for both the connectivity of individual forbidden subgraphs and the collective behavior of the properties they define.
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