A characterization of one-sided error testable graph properties in bounded degeneracy graphs

TL;DR

This paper characterizes which graph properties can be tested with one-sided error in p-degenerate graphs using a random neighbor oracle, revealing that connectivity of forbidden structures determines testability.

Contribution

It provides a complete structural characterization of one-sided error testable properties in p-degenerate graphs, extending beyond minor-closed families.

Findings

Testability depends on the connectivity of forbidden structures.

Properties are testable if violations are not fragmented across high-degree neighborhoods.

Defines the exact structural boundary for testability in p-degenerate graphs.

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.