A note on the flip distance between non-crossing spanning trees

03/14/2023

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by   Nicolas Bousquet, et al.

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We consider spanning trees of n points in convex position whose edges are pairwise non-crossing. Applying a flip to such a tree consists in adding an edge and removing another so that the result is still a non-crossing spanning tree. Given two trees, we investigate the minimum number of flips required to transform one into the other. The naive 2n-Ω(1) upper bound stood for 25 years until a recent breakthrough from Aichholzer et al. yielding a 2n-Ω(log n) bound. We improve their result with a 2n-Ω(√(n)) upper bound, and we strengthen and shorten the proofs of several of their results.