Constrained Flips in Plane Spanning Trees
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913999890677760 |
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| author | Aichholzer, Oswin Dorfer, Joseph Vogtenhuber, Birgit |
| author_facet | Aichholzer, Oswin Dorfer, Joseph Vogtenhuber, Birgit |
| contents | A flip in a plane spanning tree $T$ is the operation of removing one edge from $T$ and adding another edge such that the resulting structure is again a plane spanning tree. For trees on a set of points in convex position we study two classic types of constrained flips: (1)~Compatible flips are flips in which the removed and inserted edge do not cross each other. We relevantly improve the previous upper bound of $2n-O(\sqrt{n})$ on the diameter of the compatible flip graph to~$\frac{5n}{3}-O(1)$, by this matching the upper bound for unrestricted flips by Bjerkevik, Kleist, Ueckerdt, and Vogtenhuber [SODA~2025] up to an additive constant of $1$. We further show that no shortest compatible flip sequence removes an edge that is already in its target position. Using this so-called happy edge property, we derive a fixed-parameter tractable algorithm to compute the shortest compatible flip sequence between two given trees. (2)~Rotations are flips in which the removed and inserted edge share a common vertex. Besides showing that the happy edge property does not hold for rotations, we improve the previous upper bound of $2n-O(1)$ for the diameter of the rotation graph to~$\frac{7n}{4}-O(1)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_15520 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Constrained Flips in Plane Spanning Trees Aichholzer, Oswin Dorfer, Joseph Vogtenhuber, Birgit Computational Geometry Discrete Mathematics Combinatorics A flip in a plane spanning tree $T$ is the operation of removing one edge from $T$ and adding another edge such that the resulting structure is again a plane spanning tree. For trees on a set of points in convex position we study two classic types of constrained flips: (1)~Compatible flips are flips in which the removed and inserted edge do not cross each other. We relevantly improve the previous upper bound of $2n-O(\sqrt{n})$ on the diameter of the compatible flip graph to~$\frac{5n}{3}-O(1)$, by this matching the upper bound for unrestricted flips by Bjerkevik, Kleist, Ueckerdt, and Vogtenhuber [SODA~2025] up to an additive constant of $1$. We further show that no shortest compatible flip sequence removes an edge that is already in its target position. Using this so-called happy edge property, we derive a fixed-parameter tractable algorithm to compute the shortest compatible flip sequence between two given trees. (2)~Rotations are flips in which the removed and inserted edge share a common vertex. Besides showing that the happy edge property does not hold for rotations, we improve the previous upper bound of $2n-O(1)$ for the diameter of the rotation graph to~$\frac{7n}{4}-O(1)$. |
| title | Constrained Flips in Plane Spanning Trees |
| topic | Computational Geometry Discrete Mathematics Combinatorics |
| url | https://arxiv.org/abs/2508.15520 |