Related papers: A Bound on the Edge-Flipping Distance between Tria…
Given a finite point set P in general position in the plane, a full triangulation is a maximal straight-line embedded plane graph on P. A partial triangulation is a full triangulation of some subset P' of P containing all extreme points in…
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 show upper and lower bounds for angles in iterations of trisections of certain triangulations.
Plane perfect matchings of $2n$ points in convex position are in bijection with triangulations of convex polygons of size $n+2$. Edge flips are a classic operation to perform local changes both structures have in common. In this work, we…
Given a triangulation of a point set in the plane, a \emph{flip} deletes an edge $e$ whose removal leaves a convex quadrilateral, and replaces $e$ by the opposite diagonal of the quadrilateral. It is well known that any triangulation of a…
A (multi)set of segments in the plane may form a TSP tour, a matching, a tree, or any multigraph. If two segments cross, then we can reduce the total length with the following flip operation. We remove a pair of crossing segments, and…
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…
Given two combinatorial triangulations, how many edge flips are necessary and sufficient to convert one into the other? This question has occupied researchers for over 75 years. We provide a comprehensive survey, including full proofs, of…
The rhombus tilings of a simply connected domain of the Euclidean plane are known to form a flip-connected space (a flip is the elementary operation on rhombus tilings which rotates 180{\deg} a hexagon made of three rhombi). Motivated by…
We prove that for a given flat surface with conical singularities, any pair of geometric triangulations can be connected by a chain of flips.
A flip is a minimal move between two triangulations of a polytope. An open question is whether any two triangulations of the product of two simplices can be connected through a series of flips. This was proven in the case where one of the…
Given a set $\cal P$ of points in the Euclidean plane and two triangulations of $\cal P$, the flip distance between these two triangulations is the minimum number of flips required to transform one triangulation into the other.…
The flip-graph of a convex polygon $\pi$ is the graph whose vertices are the triangulations of $\pi$ and whose edges correspond to flips between them. The eccentricity of a triangulation $T$ of $\pi$ is the largest possible distance in this…
We study the problem of finding a triangulation T of a planar point set S such as to minimize the expected distance between two points x and y chosen uniformly at random from S. By distance we mean the length of the shortest path between x…
The flip graph is the graph whose nodes correspond to non-isomorphic combinatorial triangulations and whose edges connect pairs of triangulations that can be obtained one from the other by flipping a single edge. In this note we show that…
Using existing technology, we prove a Masur-Minsky style distance formula for flip- graph distance between two triangulations, expressed as a sum of the distances of the projections of these triangulations into arc graphs of the suitable…
We show that the triangulations of a finite point set form a flip graph that can be embedded isometrically into a hypercube, if and only if the point set has no empty convex pentagon. Point sets of this type include convex subsets of…
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…
A triangulation of a point configuration is regular if it can be given by a height function, that is every point gets lifted to a certain height and projecting the lower convex hull gives the triangulation. Checking regularity of a…
We show that if a disc triangulation has all internal vertex degrees at least 6, then the full triangulation may be determined from the pairwise graph distance between boundary vertices. A similar result holds for quadrangulations with all…