The ω(log n)–n^{o(1)} and ω(n^{1/(k+1)})–o(n^{1/k}) complexity gaps (with decidability) for LCL problems on trees extend to LPMSO problems on unbounded-degree rooted trees.
A Time Hierarchy Theorem for the LOCAL Model
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Generalizing LCL Complexity Gaps to Unbounded Degree via Monadic Second-Order Properties
The ω(log n)–n^{o(1)} and ω(n^{1/(k+1)})–o(n^{1/k}) complexity gaps (with decidability) for LCL problems on trees extend to LPMSO problems on unbounded-degree rooted trees.