A property is one-sided error testable in p-degenerate graphs if and only if its forbidden-subgraph violations cannot be fragmented across disjoint high-degree neighborhoods.
On the characterization of 1-sided error strongly testable graph properties for bounded-degree graphs
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A characterization of one-sided error testable graph properties in bounded degeneracy graphs
A property is one-sided error testable in p-degenerate graphs if and only if its forbidden-subgraph violations cannot be fragmented across disjoint high-degree neighborhoods.