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arXiv · 0903.2184

Flip Graphs of Degree-Bounded (Pseudo-)Triangulations

Oswin Aichholzer·Thomas Hackl·David Orden·Pedro Ramos·Günter Rote·André Schulz·Bettina Speckmann
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Abstract

We study flip graphs of triangulations whose maximum vertex degree is bounded by a constant $k$. In particular, we consider triangulations of sets of $n$ points in convex position in the plane and prove that their flip graph is connected if and only if $k > 6$; the diameter of the flip graph is $O(n^2)$. We also show that, for general point sets, flip graphs of pointed pseudo-triangulations can be disconnected for $k \leq 9$, and flip graphs of triangulations can be disconnected for any $k$. Additionally, we consider a relaxed version of the original problem. We allow the violation of the degree bound $k$ by a small constant. Any two triangulations with maximum degree at most $k$ of a convex point set are connected in the flip graph by a path of length $O(n \log n)$, where every intermediate triangulation has maximum degree at most $k+4$.

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Oswin Aichholzer, Thomas Hackl, David Orden, Pedro Ramos, Günter Rote, André Schulz, Bettina Speckmann. 2012-09-11. Flip Graphs of Degree-Bounded (Pseudo-)Triangulations. https://doi.org/10.1007/s00373-012-1229-0

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