Problem
Research questions and friction points this paper is trying to address.
graph property testing
one-sided error
p-degenerate graphs
forbidden subgraphs
testability
A characterization of one-sided error testable graph properties in bounded degeneracy graphs
🤖 AI Summary
This work investigates the one-sided error testability of graph properties in bounded-degeneracy graphs, with particular attention to scenarios where highly connected hub nodes may obscure local violation structures. Operating within the random-neighbor oracle model, the study establishes the first complete structural characterization of such testable properties by analyzing the connectivity of violations across higher-order neighborhoods. Specifically, it proves that a graph property is one-sided error testable in bounded-degeneracy graphs if and only if its violations cannot be partitioned into disjoint higher-order neighborhoods. This result transcends prior limitations restricted to forbidden subgraph-type properties, revealing an intrinsic connection between violation connectivity and testability, and precisely delineating the boundary of testability in this model.
📝 Abstract
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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
one-sided error
graph property testing
p-degenerate graphs
forbidden subgraphs
connectivity
🔎 Similar Papers
No similar papers found.
