Planar Triangulations Research Articles

Toric origami structures on quasitoric manifolds

We construct quasitoric manifolds of dimension 6 and higher which are not equivariantly homeomorphic to any toric origami manifold. All necessary topological definitions and combinatorial constructions are given, and the statement is reformulated in discrete geometrical terms. The problem reduces to the existence of planar triangulations with certain coloring and metric properties.

Random Walk on Random Infinite Looptrees

Looptrees have recently arisen in the study of critical percolation on the uniform infinite planar triangulation. Here we consider random infinite looptrees defined as the local limit of the looptree associated with a critical Galton--Watson tree conditioned to be large. We study simple random walk on these infinite looptrees by means of providing estimates on volume and resistance growth. We prove that if the offspring distribution of the Galton--Watson process is in the domain of attraction of a stable distribution with index $\alpha\in(1,2]$ then the spectral dimension of the looptree is $2\alpha/(\alpha+1)$.

Plane Triangulations Without a Spanning Halin Subgraph: Counterexamples to the Lovász--Plummer Conjecture on Halin Graphs

A \sl Halin graph is a simple plane graph consisting of a tree without degree 2 vertices and a cycle induced by the leaves of the tree. In 1975, Lovász and Plummer conjectured that every 4-connected plane triangulation has a spanning Halin subgraph. In this paper, we construct an infinite family of counterexamples to the conjecture.

Percolation on random triangulations and stable looptrees

We study site percolation on Angel and Schramm’s uniform infinite planar triangulation. We compute several critical and near-critical exponents, and describe the scaling limit of the boundary of large percolation clusters in all regimes (subcritical, critical and supercritical). We prove in particular that the scaling limit of the boundary of large critical percolation clusters is the random stable looptree of index $$3=2$$ , which was introduced in Curien and Kortchemski (Random stable looptrees. arXiv:1304.1044 , 2014). We also give a conjecture linking looptrees of any index $$\alpha \in (1,2)$$ with scaling limits of cluster boundaries in random triangulations decorated with $${O(N)}$$ models.

On longest paths and diameter in random apollonian networks

ABSTRACTWe consider the following iterative construction of a random planar triangulation. Start with a triangle embedded in the plane. In each step, choose a bounded face uniformly at random, add a vertex inside that face and join it to the vertices of the face. After n – 3 steps, we obtain a random triangulated plane graph with n vertices, which is called a Random Apollonian Network (RAN). We show that asymptotically almost surely (a.a.s.) a longest path in a RAN has length o(n), refuting a conjecture of Frieze and Tsourakakis. We also show that a RAN always has a cycle (and thus a path) of length , and that the expected length of its longest cycles (and paths) is . Finally, we prove that a.a.s. the diameter of a RAN is asymptotic to , where is the solution of an explicit equation. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 45, 703–725, 2014

Construction of planar 4-connected triangulations

In this article we describe a recursive structure for the class of 4-connected triangulations or – equivalently – cyclically 4-connected plane cubic graphs.

A Glimpse of the Conformal Structure of Random Planar Maps

We present a way to study the conformal structure of random planar maps. The main idea is to explore the map along an SLE (Schramm–Loewner evolution) process of parameter \({ \kappa = 6}\) and to combine the locality property of the SLE6 together with the spatial Markov property of the underlying lattice in order to get a non-trivial geometric information. We follow this path in the case of the conformal structure of random triangulations with a boundary.Under a reasonable assumption called (*) that we have unfortunately not been able to verify, we prove that the limit of uniformized random planar triangulations has a fractal boundary measure of Hausdorff dimension \({\frac{1}{3}}\) almost surely. This agrees with the physics KPZ predictions and represents a first step towards a rigorous understanding of the links between random planar maps and the Gaussian free field (GFF).

Incremental and batch planar simplification of dense point cloud maps

Dense RGB-D SLAM techniques and high-fidelity LIDAR scanners are examples from an abundant set of systems capable of providing multi-million point datasets. These datasets quickly become difficult to process due to the sheer volume of data, typically containing significant redundant information, such as the representation of planar surfaces with millions of points. In order to exploit the richness of information provided by dense methods in real-time robotics, techniques are required to reduce the inherent redundancy of the data. In this paper we present a method for incremental planar segmentation of a gradually expanding point cloud map and a method for efficient triangulation and texturing of planar surface segments. Experimental results show that our incremental segmentation method is capable of running in real-time while producing a segmentation faithful to what would be achieved using a batch segmentation method. Our results also show that the proposed planar simplification and triangulation algorithm removes more than 90% of the input planar points, leading to a triangulation with only 10% of the original quantity of triangles per planar segment. Additionally, our texture generation algorithm preserves all colour information contained within planar segments, resulting in a visually appealing and geometrically accurate simplified representation.

Edge Proximity Conditions for Extendability in Planar Triangulations

AbstractA graph G with at least vertices is said to be distance d m‐extendable if for any matching M of G with m edges in which the edges lie at distance at least d pairwise, there exists a perfect matching of G containing M. In this article we prove that every 5‐connected triangulation on the plane of even order is distance 3 9‐extendable and distance 4 m‐extendable for any m.

Scalable and Detail-Preserving Ground Surface Reconstruction from Large 3D Point Clouds Acquired by Mobile Mapping Systems

Abstract. The currently existing mobile mapping systems equipped with active 3D sensors allow to acquire the environment with high sampling rates at high vehicle velocities. While providing an effective solution for environment sensing over large scale distances, such acquisition provides only a discrete representation of the geometry. Thus, a continuous map of the underlying surface must be built. Mobile acquisition introduces several constraints for the state-of-the-art surface reconstruction algorithms. Smoothing becomes a difficult task for recovering sharp depth features while avoiding mesh shrinkage. In addition, interpolation-based techniques are not suitable for noisy datasets acquired by Mobile Laser Scanning (MLS) systems. Furthermore, scalability is a major concern for enabling real-time rendering over large scale distances while preserving geometric details. This paper presents a fully automatic ground surface reconstruction framework capable to deal with the aforementioned constraints. The proposed method exploits the quasi-flat geometry of the ground throughout a morphological segmentation algorithm. Then, a planar Delaunay triangulation is applied in order to reconstruct the ground surface. A smoothing procedure eliminates high frequency peaks, while preserving geometric details in order to provide a regular ground surface. Finally, a decimation step is applied in order to cope with scalability constraints over large scale distances. Experimental results on real data acquired in large urban environments are presented and a performance evaluation with respect to ground truth measurements demonstrate the effectiveness of our method.

Light [formula omitted] and [formula omitted] in 3-polytopes with minimum degree 5

Let wP(Cl) (wT(Cl)) be the minimum integer k with the property that every 3-polytope (respectively, every plane triangulation) with minimum degree 5 has an l-cycle with weight, defined as the degree-sum of all vertices, at most k.In 1998, O.V. Borodin and D.R. Woodall proved wT(C4)=25 and wT(C5)=30. We prove that wP(C4)=26 and wP(C5)=30.

Rainbow numbers for matchings in plane triangulations

Given two graphs G and H, let f(G,H) denote the maximum number c for which there is a way to color the edges of G with c colors such that every subgraph H of G has at least two edges of the same color. Equivalently, any edge-coloring of G with at least rb(G,H)=f(G,H)+1 colors contains a rainbow copy of H, where a rainbow subgraph of an edge-colored graph is such that no two edges of it have the same color. The number rb(G,H) is called the rainbow number of Hwith respect to G. If G is a complete graph Kn, the numbers f(Kn,H) and rb(Kn,H) are called anti-Ramsey numbers and rainbow numbers, respectively.In this paper we will study the existence of rainbow matchings in plane triangulations. Denote by kK2 a matching of size k and Tn the class of all plane triangulations of order n. The rainbow numberrb(Tn,kK2) is the minimum number of colors c such that, if kK2⊆Tn∈Tn, then any edge-coloring of Tn with at least c colors contains a rainbow copy of kK2. We give lower and upper bounds on rb(Tn,kK2) for all k≥3 and n≥2k. Furthermore, we obtain the exact values of rb(Tn,kK2) for 2≤k≤4 and n≥2k.

Rainbow Numbers for Cycles in Plane Triangulations

AbstractIn the article, the existence of rainbow cycles in edge colored plane triangulations is studied. It is shown that the minimum number of colors that force the existence of a rainbow C3 in any n‐vertex plane triangulation is equal to . For a lower bound and for an upper bound of the number is determined.

Some Results on Open-Edge and Open Mobile Guarding of Polygons and Triangulations

This paper focuses on a variation of the Art Gallery problem that considers open-edge guards and open mobile-guards. A mobile guard can be placed on edges and diagonals of a polygon, and the ‘open’ prefix means that the endpoints of such an edge or diagonal are not taken into account for visibility purposes. This paper studies the number of guards that are sufficient and sometimes necessary to guard some classes of simple polygons for both open-edge and open mobile-guards. A wide range of polygons is studied, which include orthogonal polygons with or without holes, spirals, orthogonal spirals and monotone polygons. Moreover, this problem is also considered for planar triangulation graphs using open-edge guards.

Combinatorial structure of faces in triangulated 3-polytopes with minimum degree 4

In 1940, Lebesgue proved that every 3-polytope with minimum degree at least 4 contains a 3-face for which the set of degrees of its vertices is majorized by one of the entries: (4, 4,∞), (4, 5, 19), (4, 6, 11), (4, 7, 9), (5, 5, 9), and (5, 6, 7). This description was strengthened by Borodin (2002) to (4, 4,∞), (4, 5, 17), (4, 6, 11), (4, 7, 8), (5, 5, 8), and (5, 6, 6). For triangulations with minimum degree at least 4, Jendrol’ (1999) gave a description of faces: (4, 4,∞), (4, 5, 13), (4, 6, 17), (4, 7, 8), (5, 5, 7), and (5, 6, 6). We obtain the following description of faces in plane triangulations (in particular, for triangulated 3-polytopes) with minimum degree at least 4 in which all parameters are best possible and are attained independently of the others: (4, 4,∞), (4, 5, 11), (4, 6, 10), (4, 7, 7), (5, 5, 7), and (5, 6, 6). In particular, we disprove a conjecture by Jendrol’ (1999) on the combinatorial structure of faces in triangulated 3-polytopes.

Linear-Time Compression of Bounded-Genus Graphs into Information-Theoretically Optimal Number of Bits

A $\textit{compression scheme}$ $A$ for a class $\mathbb{G}$ of graphs consists of an encoding algorithm $\textit{Encode}_A$ that computes a binary string $\textit{Code}_A(G)$ for any given graph $G$ in $\mathbb{G}$ and a decoding algorithm $\textit{Decode}_A$ that recovers $G$ from $\textit{Code}_A(G)$. A compression scheme $A$ for $\mathbb{G}$ is $\textit{optimal}$ if both $\textit{Encode}_A$ and $\textit{Decode}_A$ run in linear time and the number of bits of $\textit{Code}_A(G)$ for any $n$-node graph $G$ in $\mathbb{G}$ is information-theoretically optimal to within lower-order terms. Trees and plane triangulations were the only known nontrivial graph classes that admit optimal compression schemes. Based upon Goodrich's separator decomposition for planar graphs and Djidjev and Venkatesan's planarizers for bounded-genus graphs, we give an optimal compression scheme for any hereditary (i.e., closed under taking subgraphs) class $\mathbb{G}$ under the premise that any $n$-node graph of $\mathbb{G}$ to be encoded comes with a genus-$o(\frac{n}{\log^2 n})$ embedding. By Mohar's linear-time algorithm that embeds a bounded-genus graph on a genus-$O(1)$ surface, our result implies that any hereditary class of genus-$O(1)$ graphs admits an optimal compression scheme. For instance, our result yields the first-known optimal compression schemes for planar graphs, plane graphs, graphs embedded on genus-$1$ surfaces, graphs with genus $2$ or less, $3$-colorable directed plane graphs, $4$-outerplanar graphs, and forests with degree at most $5$. For non-hereditary graph classes, we also give a methodology for obtaining optimal compression schemes. From this methodology, we give the first known optimal compression schemes for triangulations of genus-$O(1)$ surfaces and floorplans.

Geometry and percolation on half planar triangulations

We analyze the geometry of domain Markov half planar triangulations. In [5] it is shown thatthere exists a one-parameter family ofmeasures supported on half planar triangulations satisfying translation invariance and domain Markov property. We study the geometry of these maps and show that they exhibit a sharp phase-transition inview of their geometry atα = 2/3. For α < 2/3, the maps form atree-like stricture with infinitely many small cut-sets.For α > 2/3,we obtain maps of hyperbolic nature with exponential growth andanchoredexpansion. Some results about the geometry of percolation clusters on such maps and random walk on them are also obtained.

On the Upward Planarity of Mixed Plane Graphs

A mixed plane graph is a plane graph whose edge set is partitioned into a set of directed edges and a set of undirected edges. An orientation of a mixed plane graph G is an assignment of directions to the undirected edges of G resulting in a directed plane graph $\vec G$ . In this paper, we study the computational complexity of testing whether a given mixed plane graph G is upward planar, i.e., whether it admits an orientation resulting in a directed plane graph G such that G admits a planar drawing in which each edge is represented by a curve monotonically increasing in the y-direction according to its orientation. Our contribution is threefold. First, we show that the upward planarity testing problem is solvable in cubic time for mixed outerplane graphs. Second, we show that the problem of testing the upward planarity of mixed plane graphs reduces in quadratic time to the problem of testing the upward planarity of mixed plane triangulations. Third, we exhibit linear-time testing algorithms for two classes of mixed plane triangulations, namely mixed plane 3-trees and mixed plane triangulations in which the undirected edges induce a forest.

Describing faces in plane triangulations

Lebesgue (1940) proved that every plane triangulation contains a face with the vertex-degrees majorized by one of the following triples: (3,6,∞),(3,7,41),(3,8,23),(3,9,17),(3,10,14),(3,11,13),(4,4,∞),(4,5,19),(4,6,11),(4,7,9),(5,5,9),(5,6,7). Jendrol’ (1999) improved this description, except for (4,4,∞) and (4,6,11), to (3,4,35),(3,5,21),(3,6,20),(3,7,16),(3,8,14),(3,9,14),(3,10,13),(4,4,∞),(4,5,13),(4,6,17),(4,7,8),(5,5,7),(5,6,6) and conjectured that the tight description is (3,4,30),(3,5,18),(3,6,20),(3,7,14),(3,8,14),(3,9,12),(3,10,12),(4,4,∞),(4,5,10),(4,6,15),(4,7,7),(5,5,7),(5,6,6). We prove that in fact every plane triangulation contains a face with the vertex-degrees majorized by one of the following triples, where every parameter is tight: (3,4,31),(3,5,21),(3,6,20),(3,7,13),(3,8,14),(3,9,12),(3,10,12),(4,4,∞),(4,5,11),(4,6,10),(4,7,7),(5,5,7),(5,6,6).

Proof of a conjecture of Bowlin and Brin on four-colouring triangulations

We prove a conjecture of Bowlin and Brin that for all n≥5, the n-vertex biwheel is the planar triangulation with n vertices admitting the largest number of four-colourings.