Planar Triangulations Research Articles

Last car decomposition of planar maps

We give new equations which characterize the generating functions of planar quadrangulations and planar triangulations, with zero, one, or two boundaries. The proof is inspired by the Lackner–Panholzer last car decomposition of parking trees and consists in applying a similar decomposition to the peeling trees of planar maps.

On the oriented diameter of planar triangulations

On the oriented diameter of planar triangulations

Independent Dominating Sets in Planar Triangulations

Independent Dominating Sets in Planar Triangulations

Vision Measurement Method Based on Plate Glass Window Refraction Model in Tunnel Construction.

Due to the harsh environment of high humidity and dust in tunnel construction, the vision measurement system needs to be equipped with an explosion-proof glass protective cover. The refractive effect of the plate glass window invalidates the pinhole model. This paper proposes a comprehensive solution for addressing the issue of plane refraction. First, the imaging model for non-parallel plane refraction is established based on dynamic virtual focal length and the Rodriguez formula. Further, due to the failure of the epipolar constraint principle in binocular vision systems caused by plane refraction, this paper proposes the epipolar constraint model for independent refractive plane imaging. Finally, an independent refraction plane triangulation model is proposed to address the issue of triangulation failure caused by plane refraction. The RMSE of the depth of field errors in the independent refraction plane triangulation model is 2.9902 mm before correction and 0.3187 mm after correction. The RMSE of the positioning errors before and after correction are 3.5661 mm and 0.3465 mm, respectively.

Relations between scaling exponents in unimodular random graphs

We investigate the validity of the “Einstein relations” in the general setting of unimodular random networks. These are equalities relating scaling exponents: \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document} $$\\begin{aligned} d_{w} &= d_{f} + \ ilde{\\zeta }, \\\\ d_{s} &= 2 d_{f}/d_{w}, \\end{aligned}$$ \\end{document} where dw is the walk dimension, df is the fractal dimension, ds is the spectral dimension, and tilde{zeta } is the resistance exponent. Roughly speaking, this relates the mean displacement and return probability of a random walker to the density and conductivity of the underlying medium. We show that if df and tilde{zeta } geqslant 0 exist, then dw and ds exist, and the aforementioned equalities hold. Moreover, our primary new estimate d_{w} geqslant d_{f} + tilde{zeta } is established for all tilde{zeta } in mathbb{R}.For the uniform infinite planar triangulation (UIPT), this yields the consequence dw=4 using df=4 (Angel in Geom. Funct. Anal. 13(5):935–974, 2003) and tilde{zeta }=0 (established here as a consequence of the Liouville Quantum Gravity theory, following Gwynne-Miller 2020 and (Ding and Gwynne in Commun. Math. Phys. 374(3):1877–1934, 2020)). The conclusion dw=4 had been previously established by Gwynne and Hutchcroft (2018) using more elaborate methods. A new consequence is that dw=df for the uniform infinite Schnyder-wood decorated triangulation, implying that the simple random walk is subdiffusive, since df>2.

Analysis of the Computational Cost of PolyFront: an Algorithm for Planar Triangulation

The triangulation of planar domains is a relevant and largely studied problem in many applied sciences. This paper analyzes the computational time of a triangulation algorithm for plane domains with holes, introduced in a previous paper. This algorithm is based on the normal offsetting technique starting from a polygonal approximation of the domain boundary. It is shown that the computational time is linear with respect to the number of vertices of the triangulation. Experimental results confirm the theoretical upper bound obtained for the computational time.

Percolation on Triangulations: A Bijective Path to Liouville Quantum Gravity

We set the foundation for a series of works aimed at proving strong relations between uniform random planar maps and Liouville quantum gravity (LQG). Our method relies on a bijective encoding of site-percolated planar triangulations by certain 2D lattice paths. Our bijection parallels in the discrete setting the mating-of-trees framework of LQG and Schramm-Loewner evolutions (SLE) introduced by Duplantier, Miller, and Sheffield. Combining these two correspondences allows us to relate uniform site-percolated triangulations to 8 / 3 \sqrt {8/3} -LQG and SLE 6 _6 . In particular, we establish the convergence of several functionals of the percolation model to continuous random objects defined in terms of 8 / 3 \sqrt {8/3} -LQG and SLE 6 _6 . For instance, we show that the exploration tree of the percolation converges to a branching SLE 6 _6 , and that the collection of percolation cycles converges to the conformal loop ensemble CLE 6 _6 . We also prove convergence of counting measure on the pivotal points of the percolation. Our results play an essential role in several other works, including a program for showing convergence of the conformal structure of uniform triangulations and works which study the behavior of random walk on the uniform infinite planar triangulation.

Disjoint total dominating sets in near‐triangulations

AbstractWe show that every simple planar near‐triangulation with minimum degree at least three contains two disjoint total dominating sets. The class includes all simple planar triangulations other than the triangle. This affirms a conjecture of Goddard and Henning.

Edge connectivity of simplicial polytopes

We show that the graph of a simplicial polytope of dimension d≥3 has no nontrivial minimum edge cut with fewer than d(d+1)/2 edges, hence the graph is min{δ,d(d+1)/2}-edge-connected where δ denotes the minimum degree. When d=3, this implies that every minimum edge cut in a plane triangulation is trivial. When d≥4, we construct a simplicial d-polytope whose graph has a nontrivial minimum edge cut of cardinality d(d+1)/2, proving that the aforementioned result is best possible.

Tight description of faces of triangulations on the torus

The degree d(x) of a vertex or face x in a graph G is the number of incident edges. A face f=v1…vd(f) in a graph G on the plane or other orientable surface is of type (k1,k2,…), where k1≤k2≤…, if d(vi)≤ki for each i. By δ we denote the minimum vertex-degree of G.In 1989, Borodin confirmed Kotzig's conjecture of 1963 that every plane graph with minimum degree δ equal to 5 has a (5,5,7)-face or a (5,6,6)-face, where all parameters are tight. Recently, we proved that every torus triangulation with δ≥5 has a face of one of the types (5,5,8), (5,6,7), or (6,6,6), which is tight.It follows from the classical theorem by Lebesgue (1940) that every plane triangulation with δ≥4 has a 3-face of types (4,4,∞), (4,5,19), (4,6,11), (4,7,9), (5,5,9), or (5,6,7).In 1999, Jendrol' gave a similar description: “(4,4,∞), (4,5,13), (4,6,17), (4,7,8), (5,5,7), (5,6,6)” and conjectured that “(4,4,∞), (4,5,10), (4,6,15), (4,7,7), (5,5,7), (5,6,6)” holds.In 2002, Lebesgue's description was strengthened by Borodin to “(4,4,∞), (4,5,17), (4,6,11), (4,7,8), (5,5,8), (5,6,6)”.In 2014, Borodin and Ivanova obtained the following tight description, which, in particular, disproves the above mentioned conjecture by Jendrol': “(4,4,∞), (4,5,11), (4,6,10), (4,7,7), (5,5,7), (5,6,6)”, and recently proved another tight description of faces in plane triangulations with δ≥4: “(4,4,∞), (4,6,10), (4,7,7), (5,5,8), (5,6,7)”.It follows from Lebesgue's theorem of 1940 that every plane quadrangulation with δ≥3 has a face of one of the types (3,3,3,∞), (3,3,4,11), (3,3,5,7), (3,4,4,5). Recently, Borodin and Ivanova improved this description to “(3,3,3,∞), (3,3,4,9), (3,3,5,6), (3,4,4,5)”, where all parameters except possibly 9 are best possible and 9 cannot go down below 8.In 1995, Avgustinovich and Borodin proved the following tight description of the faces of torus quadrangulations with δ≥3: “(3,3,3,∞), (3,3,4,10), (3,3,5,7), (3,3,6,6), (3,4,4,6), (4,4,4,4)”, which also holds for each higher surface provided that its quadrangulation is large enough.Recently, Borodin and Ivanova proved that every triangulation with δ≥4 of the torus has a face of one of the types (4,4,∞), (4,6,12), (4,8,8), (5,5,8), (5,6,7), or (6,6,6), which description is tight.The purpose of this paper is to prove that every triangulation with δ≥3 on the torus has a face of one of the types (3,6,24), (3,8,16), (3,12,12), (4,4,∞), (4,6,12), (4,8,8), (5,5,8), (5,6,7), or (6,6,6), where all parameters are best possible.

Morphing Triangle Contact Representations of Triangulations

A morph is a continuous transformation between two representations of a graph. We consider the problem of morphing between contact representations of a plane graph. In an {mathcal {F}}-contact representation of a plane graph G, vertices are realized by internally disjoint elements from a family {mathcal {F}} of connected geometric objects. Two such elements touch if and only if their corresponding vertices are adjacent. These touchings also induce the same embedding as in G. In a morph between two {mathcal {F}}-contact representations we insist that at each time step (continuously throughout the morph) we have an {mathcal {F}}-contact representation. We focus on the case when mathcal {F} is the family of triangles in mathbb {R}^2 that are the lower-right half of axis-parallel rectangles. Such RT-representations exist for every plane graph and right triangles are one of the simplest families of shapes supporting this property. Moreover, they naturally correspond to 3-orientations. Thus, they provide a natural case to study regarding morphs of contact representations of plane graphs. We characterize the pairs of RT-representations admitting a morph between each other via the respective 3-orientations. Our characterization leads to a polynomial-time algorithm to decide whether there is a morph between two RT-representations of an n-vertex plane triangulation, and, if so, computes a morph with {mathcal {O}}(n^2) steps. Each of these steps is a linear morph moving the endpoints of each triangle at constant speed along straight-line trajectories. Our characterization also implies that for 4-connected plane triangulations there is a morph between every pair of RT-representations where the “top-most” triangle in both representations corresponds to the same vertex.

All tight descriptions of faces in plane triangulations with minimum degree 4

All tight descriptions of faces in plane triangulations with minimum degree 4

UAV planar passive pure orientation positioning under different conditions

UAVs should be kept as silent as possible during field operations and missions in order to conceal and prevent electrostatic interference. In this paper, with rigorous mathematical argumentative thinking, the planar passive pure orientation localization problem under error-free conditions is based on the triangulation localization model of basic analytic geometry. The triangulation model is solved by applying a combination of plane geometry and analytic geometry. The accurate planar triangulation model is obtained, and the target is located according to the determined observation points.

The Distance Orientation Problem

The Distance Orientation Problem (DOP) is formulated as follows: Given a graph with positive weights on its edges, are there weights for the vertices, such that for every edge x y it holds that the absolute difference between the weights of x and y is equal to the weight of x y ? This problem can also be formulated as a problem of finding a special orientation of G and was motivated by an application in Shape from Shading, a method in the field of Computer Vision. We present a linear-time algorithm for complete 3-cover graphs, a generalization of chordal graphs and planar triangulations. For outerplanar graphs we show that the DOP is fixed-parameter tractable and present a pseudo-polynomial time algorithm for integral edge weights. Both algorithms use the idea that the existence of feasible weights for the vertices of an outerplanar graph can be decided by only looking at the edge weights of each face of its outerplane embedding separately. We show that this property does not hold for any embedding of a planar graph which is not outerplanar. Furthermore, we prove that the DOP is strongly NP -complete for grid graphs, i.e., there is no pseudo-polynomial algorithm to solve it unless P = NP .

Rigid block and hole graphs with a single block

A block and hole graph is obtained from the graph of a plane triangulation by removing the interiors of some discs, defined by their boundary cycles, and then rigidifying the vertex sets of some of these cycles by adding new vertices and edges. These rigid parts form the so-called blocks, while the remaining cycles define the holes. Cruickshank, Kitson, and Power proved that a block and hole graph G with a single block is generically minimally rigid in R3 if and only if G is (3,6)-sparse and has 3|V(G)|−6 edges. This result implies that there is an efficient algorithm for testing whether such a graph is rigid, provided it has exactly 3|V(G)|−6 edges.In this paper we extend these results to block and hole graphs G with a single block and an arbitrary number of edges. The extension is based on a new formula for the degrees of freedom of such a graph. It also enables us to find a smallest set of new edges whose addition makes G rigid. We also point out that there is an underlying matroid which can be defined by (3,6)-sparsity.

Multi-ended Markovian triangulations and robust convergence to the UIPT

We classify completely the infinite, planar triangulations satisfying a weak spatial Markov property, without assuming one-endedness nor finiteness of vertex degrees. In particular, the Uniform Infinite Planar Triangulation (UIPT) is the only such triangulation with average degree 6. As a consequence, we prove that the convergence of uniform triangulations of the sphere to the UIPT is robust, in the sense that it is preserved under various perturbations of the uniform measure. As another application, we obtain large deviation estimates for the number of occurencies of a pattern in uniform triangulations.

Vertex Splitting, Coincident Realisations, and Global Rigidity of Braced Triangulations

We give a relatively short graph theoretic proof of a result of Jordán and Tanigawa that a 4-connected graph which has a spanning plane triangulation as a proper subgraph is generically globally rigid in \({\mathbb {R}}^3\). Our proof is based on a new sufficient condition for the so called vertex splitting operation to preserve generic global rigidity in \({\mathbb {R}}^d\).

Percolation probability and critical exponents for site percolation on the UIPT

AbstractWe derive three critical exponents for Bernoulli site percolation on the uniform infinite planar triangulation (UIPT). First, we compute explicitly the probability that the root cluster is infinite. As a consequence, we show that the off-critical exponent for site percolation on the UIPT is $\beta = 1/2$ . Then we establish an integral formula for the generating function of the number of vertices in the root cluster. We use this formula to prove that, at criticality, the probability that the root cluster has at least n vertices decays like $n^{-1/7}$ . Finally, we also derive an expression for the law of the perimeter of the root cluster and use it to establish that, at criticality, the probability that the perimeter of the root cluster is equal to n decays like $n^{-4/3}$ . Among these three exponents, only the last one was previously known. Our main tools are the so-called gasket decomposition of percolation clusters, generic properties of random Boltzmann maps, and analytic combinatorics.

Certifiable algorithms for the two-view planar triangulation problem

Planar scenes predominate in man-made environments, e.g.interior or facades of buildings and in ground images from aerial vehicles. Points lying on those surfaces can be reconstructed from their observations in two images. However, generic reconstruction algorithms output 3D points not lying on the plane, thus obtaining inaccurate reconstructions. The problem also turns to be non-convex with many local minima, hence hindering the performance of iterative method. Therefore, being able to obtain and certify the optimal solution to this problem is of special relevant for these applications. In this paper we first propose a fast and certifiable algorithm that both estimates and certifies the optimal solution to the triangulation problem. From this formulation, we also present an optimality certificate that tells us whether a given solution (obtained by any solver) is the global optimum. Last, from this certificate we derive a sufficient (but not necessary) optimality condition that allows us to certify optimality in less than one microsecond. We test the proposed algorithms on extensive experiments on both synthetic and real data. Code is made available at https://github.com/mergarsal.

Chords of 2-Factors in Planar Cubic Bridgeless Graphs

We show that every edge in a 2-edge-connected planar cubic graph is either contained in a 2-edge-cut or is a chord of some cycle that is contained in a 2-factor of the graph. As a consequence, we show that every edge in a cyclically 4-edge-connected planar cubic graph, except $K_2^3$ and $K_4$, is contained in a perfect matching whose removal disconnects the graph. We obtain a complete characterization of 2-edge-connected planar cubic graphs that have an edge such that every 2-factor containing the edge is a Hamiltonian cycle, and also of those that have an edge such that the complement of every perfect matching containing the edge is a Hamiltonian cycle. Another immediate consequence of the main result is that for any two edges contained in a facial cycle of a 2-edge-connected planar cubic graph, there exists a 2-factor in the graph such that both edges are contained in the same cycle of the 2-factor. We conjecture that this property holds for any two edges in a 2-edge-connected planar cubic graph, and prove it for planar cubic bipartite graphs. The main result is proved in the dual form by showing that every plane triangulation admits a vertex 3-coloring such that no face is monochromatic and there is exactly one specified edge between a specified pair of color classes.