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 fundamentally determined by the connectivity of the forbidden structures: a property is testable if and only if its violations cannot be fragmented across disjoint high-degree neighborhoods. 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.

翻译：我们考虑在随机邻居预言机模型（Czumaj和Sohler，FOCS 2019）下，$p$-退化图中的图性质测试问题。在该框架中，测试者通过均匀采样顶点的邻居来探索图，若查询复杂度与图规模无关，则该性质具有单侧可测试性。已知对于子式封闭族，单侧可测试性质恰好是那些可由有界规模禁用子图定义的性质。然而，更广泛的$p$-退化图允许存在高度数"枢纽"，这些结构可隐蔽地将禁用子图隐藏于局部探索之外。在本工作中，我们完整刻画了$p$-退化图中所有具有单侧可测试性的结构性条件。研究表明，可测试性本质上由禁用结构的连通性决定：当且仅当性质的违反无法被分割到不相交的高度数邻域中时，该性质才是可测试的。我们的结果精确确定了这些约束下可测试性的结构边界，同时考虑了单个禁用子图的连通性及其所定义性质的集体行为。