Flip-graph non-convexity for once-punctured polygons
Abstract
The set of the triangulations with vertex set X of a simple polygon P can be structured into a flip-graph ℱ(P,X) whose edges connect two triangulations that differ by a single arc. The geometry of flip-graphs has been thoroughly studied and it is known that the subgraph ℱε(P,X) induced by the triangulations that contain a given arc ε is strongly convex in ℱ(P,X) when P is convex and X contains no puncture (points in the interior of P) and at most one flat vertex (points in the interior of an edge). When X contains at least two punctures or flat vertices, it is also known that this strong convexity property fails. Here, we close the last open case by showing that, for any convex polygon with sufficiently many vertices, one can always place a single puncture in X in such a way that ℱε(P,X) is not strongly convex in ℱ(P,X). We prove a similar result for simple polygons with a single reflex vertex. The main ingredients in our proofs are a decomposition lemma for a class of $3$-dimensional triangulations and a hyperbolic volume argument regarding their embedding into ℍ3.
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BibTeX
@misc{pournin2026flipgraph,
title = {{Flip-graph non-convexity for once-punctured polygons}},
author = {Lionel Pournin and Zili Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01412},
eprint = {2609.01412},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01412},
}
RIS
TY - UNPB
TI - Flip-graph non-convexity for once-punctured polygons
AU - Lionel Pournin
AU - Zili Wang
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.01412
UR - https://arxiv.org/abs/2609.01412
AB - The set of the triangulations with vertex set X of a simple polygon P can be structured into a flip-graph ℱ(P,X) whose edges connect two triangulations that differ by a single arc. The geometry of flip-graphs has been thoroughly studied and it is known that the subgraph ℱ_ε(P,X) induced by the triangulations that contain a given arc ε is strongly convex in ℱ(P,X) when P is convex and X contains no puncture (points in the interior of P) and at most one flat vertex (points in the interior of an edge). When X contains at least two punctures or flat vertices, it is also known that this strong convexity property fails. Here, we close the last open case by showing that, for any convex polygon with sufficiently many vertices, one can always place a single puncture in X in such a way that ℱ_ε(P,X) is not strongly convex in ℱ(P,X). We prove a similar result for simple polygons with a single reflex vertex. The main ingredients in our proofs are a decomposition lemma for a class of $3$-dimensional triangulations and a hyperbolic volume argument regarding their embedding into ℍ^3.
ER -