Related papers: Simultaneous Diagonal Flips in Plane Triangulation…
A triangulation of a polygon is a subdivision of it into triangles, using diagonals between its vertices. Two different triangulations of a polygon can be related by a sequence of flips: a flip replaces a diagonal by the unique other…
Given two triangulations of a convex polygon, computing the minimum number of flips required to transform one to the other is a long-standing open problem. It is not known whether the problem is in P or NP-complete. We prove that two…
A flipturn is an operation that transforms a nonconvex simple polygon into another simple polygon, by rotating a concavity 180 degrees around the midpoint of its bounding convex hull edge. Joss and Shannon proved in 1973 that a sequence of…
We investigate the minimum number of cycles of specified lengths in planar $n$-vertex triangulations $G$. It is proven that this number is $\Omega(n)$ for any cycle length at most $3 + \max \{ {\rm rad}(G^*), \lceil…
We prove that for a given flat surface with conical singularities, any pair of geometric triangulations can be connected by a chain of flips.
In this thesis, we study two different graph problems. The first problem revolves around geometric spanners. Here, we have a set of points in the plane and we want to connect them with straight line segments, such that there is a path…
We study the configuration space of rectangulations and convex subdivisions of $n$ points in the plane. It is shown that a sequence of $O(n\log n)$ elementary flip and rotate operations can transform any rectangulation to any other…
Whitney proved in 1931 that 4-connected planar triangulations are Hamiltonian. Hakimi, Schmeichel, and Thomassen conjectured in 1979 that if $G$ is a 4-connected planar triangulation with $n$ vertices then $G$ contains at least…
We study flips in hypertriangulations of planar points sets. Here a level-$k$ hypertriangulation of $n$ points in the planes is a subdivision induced by the projection of a $k$-hypersimplex, which is the convex hull of the barycenters of…
A triangle $T'$ is $\varepsilon$-similar to another triangle $T$ if their angles pairwise differ by at most $\varepsilon$. Given a triangle $T$, $\varepsilon>0$ and $n\in\mathbb{N}$, B\'ar\'any and F\"uredi asked to determine the maximum…
Hakimi, Schmeichel, and Thomassen in 1979 conjectured that every $4$-connected planar triangulation $G$ on $n$ vertices has at least $2(n-2)(n-4)$ Hamiltonian cycles, with equality if and only if $G$ is a double wheel. In this paper, we…
We give an overview of the 2026 Computational Geometry Challenge targeting the problem of finding a Central Triangulation under Parallel Flip Operations in triangulations of point sets. A flip is the parallel exchange of a set of edges in a…
Let T be a triangulation of a simple polygon. A flip in T is the operation of removing one diagonal of T and adding a different one such that the resulting graph is again a triangulation. The flip distance between two triangulations is the…
Let $\mathcal{P}$ be a set of $n=2m+1$ points in the plane in general position. We define the graph $GM_\mathcal{P}$ whose vertex set is the set of all plane matchings on $\mathcal{P}$ with exactly $m$ edges. Two vertices in…
We show that 4-connected plane triangulations can be redrawn such that edges are represented by straight segments and the vertices are covered by a set of at most $\sqrt{2n}$ lines each of them horizontal or vertical. The same holds for all…
Flip-graph connectedness is established here for the vertex set of the 4-dimensional cube. It is found as a consequence that this vertex set has 92 487 256 triangulations, partitioned into 247 451 symmetry classes.
In this paper we investigate the structure of flip graphs on non-crossing perfect matchings in the plane. Specifically, consider all non-crossing straight-line perfect matchings on a set of $2n$ points that are placed equidistantly on the…
Let $P$ be a convex polygon in the plane, and let $T$ be a triangulation of $P$. An edge $e$ in $T$ is called a diagonal if it is shared by two triangles in $T$. A flip of a diagonal $e$ is the operation of removing $e$ and adding the…
We report on the implementation of an algorithm for computing the set of all regular triangulations of finitely many points in Euclidean space. This algorithm, which we call down-flip reverse search, can be restricted, e.g., to computing…
Delaunay flip is an elegant, simple tool to convert a triangulation of a point set to its Delaunay triangulation. The technique has been researched extensively for full dimensional triangulations of point sets. However, an important case of…