68 pages, 36 figures. To appear in "Proceedings of the Joint Summer Research Conference on Discrete and Computational Geometry", Edited by J. E. Goodman, J. Pach, and R. Pollack, Contemporary Mathematics, AMS
发表信息：
In: Surveys on Discrete and Computational Geometry-Twenty Years Later. Herausgeber: Jacob E. Goodman, János Pach und Richard Pollack, Contemporary Mathematics, Band 453, American Mathematical Society, 2008, pp. 343-410
A pseudo-triangle is a simple polygon with three convex vertices, and a pseudo-triangulation is a face-to-face tiling of a planar region into pseudo-triangles. Pseudo-triangulations appear as data structures in computational geometry, as planar bar-and-joint frameworks in rigidity theory and as projections of locally convex surfaces.
This survey of current literature includes combinatorial properties and counting of special classes, rigidity theoretical results, representations as polytopes, straight-line drawings from abstract versions called combinatorial pseudo-triangulations, algorithms and applications of pseudo-triangulations.
We wish to draw attention to an interesting and promising interaction of two theories. On the one hand, it is the theory of \textbf{pseudo-triangulations} which was useful for implicit solution of thecarpenter's rule problem and proved later to give a nice tool for graph embeddings. On the other hand, it is the theory of hyperbolic virtual polytopes which arose from an old uniqueness conjecture for convex bodies (A. D. Alexandrov's problem): suppose that a constant $C$ separates (non-strictly) everywhere the principal curvature radii of a smooth 3-dimensional convex body $K$. Then $K$ is neces…
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 degr…
We prove that a planar graph is generically rigid in the plane if and only if it can be embedded as a pseudo-triangulation. This generalizes the main result of math.CO/0307347 which treats the minimally generically rigid case. The proof uses the concept of combinatorial pseudo-triangulation, CPT, in the plane and has two main steps: showing that a certain ``generalized Laman property'' is a necessary and sufficient condition for a CPT to be ``stretchable'', and showing that all generically rigid plane graphs admit a CPT assignment with that property. Additionally, we propose the study of combi…
Pointed pseudo-triangulations are planar minimally rigid graphs embedded in the plane with pointed vertices (adjacent to an angle larger than 180 degrees. In this paper we prove that the opposite statement is also true, namely that planar minimally rigid graphs always admit pointed embeddings, even under certain natural topological and combinatorial constraints. We provide two proofs, which both yield efficient embedding algorithms. One based on Henneberg inductive constructions from combinatorial rigidity theory, the other on a generalization of Tutte's barycentric embeddings to directed grap…
We study non-crossing frameworks in the plane for which the classical reciprocal on the dual graph is also non-crossing. We give a complete description of the self-stresses on non-crossing frameworks whose reciprocals are non-crossing, in terms of: the types of faces (only pseudo-triangles and pseudo-quadrangles are allowed); the sign patterns in the self-stress; and a geometric condition on the stress vectors at some of the vertices. As in other recent papers where the interplay of non-crossingness and rigidity of straight-line plane graphs is studied, pseudo-triangulations show up as objects…
We study the maximum numbers of pseudo-triangulations and pointed pseudo-triangulations that can be embedded over a specific set of points in the plane or contained in a specific triangulation. We derive the bounds $O(5.45^N)$ and $Ω(2.41^N)$ for the maximum number of pointed pseudo-triangulations that can be contained in a specific triangulation over a set of $N$ points. For the number of all pseudo-triangulations contained in a triangulation we derive the bounds $O^*(6.54^N)$ and $Ω(3.30^N)$. We also prove that $O^*(89.1^N)$ pointed pseudo-triangulations can be embedded over any specific set…
We introduce the polytope of pointed pseudo-triangulations of a point set in the plane, defined as the polytope of infinitesimal expansive motions of the points subject to certain constraints on the increase of their distances. Its 1-skeleton is the graph whose vertices are the pointed pseudo-triangulations of the point set and whose edges are flips of interior pseudo-triangulation edges. For points in convex position we obtain a new realization of the associahedron, i.e., a geometric representation of the set of triangulations of an n-gon, or of the set of binary trees on n vertices, or of ma…
This paper introduces even triangulations of n-dimensional pseudo-manifolds and links their combinatorics to the topology of the pseudo-manifolds. This is done via normal hypersurface theory and the study of certain symmetric representation. In dimension 3, necessary and sufficient conditions for the existence of even triangulations having one or two vertices are given. For Haken n-manifolds, an interesting connection between very short hierarchies and even triangulations is observed.
We pose a monotonicity conjecture on the number of pseudo-triangulations of any planar point set, and check it on two prominent families of point sets, namely the so-called double circle and double chain. The latter has asymptotically $12^n n^{Θ(1)}$ pointed pseudo-triangulations, which lies significantly above the maximum number of triangulations in a planar point set known so far.
Let $P\subset\mathbb{R}^{2}$ be a set of $n$ points. In this paper we show two new algorithms, one to compute the number of triangulations of $P$, and one to compute the number of pseudo-triangulations of $P$. We show that our algorithms run in time $O^{*}(t(P))$ and $O^{*}(pt(P))$ respectively, where $t(P)$ and $pt(P)$ are the largest number of triangulation paths (T-paths) and pseudo-triangulations paths (PT-paths), respectively, that the algorithms encounter during their execution. Moreover, we show that $t(P) = O^{*}(9^{n})$, which is the first non-trivial bound on $t(P)$ to be known. Whil…
In this paper we consider the flip operation for combinatorial pointed pseudo-triangulations where faces have size 3 or 4, so-called combinatorial 4-PPTs. We show that every combinatorial 4-PPT is stretchable to a geometric pseudo-triangulation, which in general is not the case if faces may have size larger than 4. Moreover, we prove that the flip graph of combinatorial 4-PPTs is connected and has diameter $O(n^2)$, even in the case of labeled vertices with fixed outer face. For this case we provide an $Ω(n\log n)$ lower bound.
For any finite set $\A$ of $n$ points in $\R^2$, we define a $(3n-3)$-dimensional simple polyhedron whose face poset is isomorphic to the poset of ``non-crossing marked graphs'' with vertex set $\A$, where a marked graph is defined as a geometric graph together with a subset of its vertices. The poset of non-crossing graphs on $\A$ appears as the complement of the star of a face in that polyhedron. The polyhedron has a unique maximal bounded face, of dimension $2n_i +n -3$ where $n_i$ is the number of points of $\A$ in the interior of $\conv(\A)$. The vertices of this polytope are all the pseu…
We prove that properly rescaled large planar Eulerian triangulations converge to the Brownian map. This result requires more than a standard application of the methods that have been used to obtain the convergence of other families of planar maps to the Brownian map, as the natural distance for Eulerian triangulations is a canonical oriented pseudo-distance. To circumvent this difficulty, we adapt the layer decomposition method established by Curien and Le Gall, which yields asymptotic proportionality between three natural distances on planar Eulerian triangulations: the usual graph distance, …
In this survey article, we are interested on minimal triangulations of closed pl manifolds. We present a brief survey on the works done in last 25 years on the following: (i) Finding the minimal number of vertices required to triangulate a given pl manifold. (ii) Given positive integers $n$ and $d$, construction of $n$-vertex triangulations of different $d$-dimensional pl manifolds. (iii) Classifications of all the triangulations of a given pl manifold with same number of vertices. In Section 1, we have given all the definitions which are required for the remaining part of this article. In Sec…
The enumerative theory of simplicial subdivisions (triangulations) of simplicial complexes was developed by Stanley in order to understand the effect of such subdivisions on the $h$-vector of a simplicial complex. A key role there is played by the concept of a local $h$-vector. This paper surveys some of the highlights of this theory and some recent developments, concerning subdivisions of flag homology spheres and their $γ$-vectors. Several interesting examples and open problems are discussed.
In this survey on combinatorial properties of triangulated manifolds we discuss various lower bounds on the number of vertices of simplicial and combinatorial manifolds. Moreover, we give a list of all known examples of vertex-minimal triangulations.
Let $M$ be a closed triangulable manifold, and let $Δ$ be a triangulation of $M$. What is the smallest number of vertices that $Δ$ can have? How big or small can the number of edges of $Δ$ be as a function of the number of vertices? More generally, what are the possible face numbers ($f$-numbers, for short) that $Δ$ can have? In other words, what restrictions does the topology of $M$ place on the possible $f$-numbers of triangulations of $M$? To make things even more interesting, we can add some combinatorial conditions on the triangulations we are considering (e.g., flagness, balancedness, et…
We survey current term-wise techniques for quadratizing high-degree pseudo-Boolean functions and introduce a new one, which allows multiple splits of terms. We also introduce the first aggregative approach, which splits a collection of terms based on their common parts.
