k-irreducible triangulations of orientable genus-g surfaces have O(k²g) triangles.
Untangling planar curves.Discrete Comput
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HCS is a discrete shortcut-and-tighten curve evolution that experimentally approaches the affine curve-shortening flow on fine obstacle sets, with proven monotonicity of curvature, crossings, and inflection counts.
Forests, trees, and matchings can always be reconfigured on the torus and higher-genus orientable surfaces by rerouting one edge at a time while maintaining crossing-free embeddings.
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On the size of k-irreducible triangulations
k-irreducible triangulations of orientable genus-g surfaces have O(k²g) triangles.