On anonymous paths local certification has a gap between O(1) and Θ(log log n), and a natural property with optimal Θ(log log n) certificates exists; on cycles the gap is between O(1) and Θ(log n).
Locally checkable problems in rooted trees
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Complexity landscape for local certification
On anonymous paths local certification has a gap between O(1) and Θ(log log n), and a natural property with optimal Θ(log log n) certificates exists; on cycles the gap is between O(1) and Θ(log n).