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### Pairs of trees in tree–tree triangulations

Günter Schaar, Zdzisław Skupień
2007 Discrete Mathematics
In particular, for a pair made up of any tree and any long enough path, there is a spherical triangulation whose graph is partitionable into that pair.  ...  Hence each tree-tree triangulation is a triangulation of the 2-sphere. Recognizing tree-tree triangulations among all simple spherical ones can be seen to be an NP-complete problem.  ...  In other words, the graph of a 0 -1 triangulation admits of a bipartition into trees isomorphic to 0 and 1 . It is easily seen that each tree-tree triangulation T is spherical.  ...

### On the Representation of Triangulated Graphs in Trees

R. Halin
1984 European journal of combinatorics (Print)
The intersection graph of a family of subtrees of a tree is always a triangulated graph, and vice versa every finite triangulated graph can be represented in this way.  ...  The analogous statement is in general not true for infinite triangulated graphs.  ...  In the present note the same problem is treated if the hypothesis that G be finite is omitted. We shall see that in general an infinite triangulated graph is not tree-representable.  ...

### Common Edges in Rooted Trees and Polygonal Triangulations

Sean Cleary, Andrew Rechnitzer, Thomas Wong
2013 Electronic Journal of Combinatorics
Rotation distance between rooted binary trees measures the degree of similarity of two trees with ordered leaves and is equivalent to edge-flip distance between triangular subdivisions of regular polygons  ...  Here we describe the distribution of common edges between randomly-selected triangulations and measure the sizes of the remaining pieces into which the common edges separate the polygons.  ...  Thus, a triangulation of size n means a triangulation on an n + 1-gon or with n leaves in the underlying binary tree.  ...

### Triangulated edge intersection graphs of paths in a tree

Maciej M. Sysło
1985 Discrete Mathematics
in T, and G is called an edge intersection graph of paths in a tree (EPT graph) if G = O(E(T), 9) for some path collection 9 in a tree T.  ...  A graph G is a vertex intersection graph of paths in a tree (shortly, VPT graph) if G = f2( V(T), 9) for a certain tree T and a path collection 9?  ...  in T, and G is called an edge intersection graph of paths in a tree (EPT graph) if G = O(E(T), 9) for some path collection 9 in a tree T.  ...

### Trees and spatial topology change in causal dynamical triangulations

J Ambjørn, T G Budd
2013 Journal of Physics A: Mathematical and Theoretical
Generalized causal dynamical triangulations (generalized CDT) is a model of two-dimensional quantum gravity in which a limited number of spatial topology changes is allowed to occur.  ...  We solve the model at the discretized level using bijections between quadrangulations and trees.  ...  Causal triangulations A similar bijection between causal triangulations and trees has been used in [36, 32, 27] and earlier in a slightly different form in [26] .  ...

### Computation of Chromatic Polynomials Using Triangulations and Clique Trees [chapter]

Pascal Berthomé, Sylvain Lebresne, Kim Nguyễn
2005 Lecture Notes in Computer Science
To achieve our goal, we use the properties of triangulations and clique-trees with respect to the previous operations, and guide our algorithm to efficiently divide the original problem.  ...  In this paper, we present a new algorithm for computing the chromatic polynomial of a general graph G.  ...  Triangulation and clique trees A chordal graph G is a graph in which there are no induced cycles of length > 3.  ...

### A criterion for sharpness in tree enumeration and the asymptotic number of triangulations in Kuperberg's G2 spider [article]

Robert Scherer
2020 arXiv   pre-print
Kuperberg proved in that this identity holds in the case that b_n=Inv_G_2 (V(λ_1)^⊗ n), where V(λ_1) is the 7-dimensional fundamental representation of G_2, and a_n is the number of triangulations of a  ...  Given a non-negative sequence (a_n)_n≥ 1, the identity B(x)=A(xB(x)) for generating functions A(x)=1+∑_n≥ 1 a_n x^n and B(x)=1+∑_n≥ 1 b_n x^n determines the number b_n of rooted planar trees with n vertices  ...  The author is also grateful to Greg Kuperberg for his encouragement and for several helpful discussions about various aspects of the subject matter in this paper.  ...

### Common edges in rooted trees and polygonal triangulations

Sean Cleary, Andrew Rechnitzer, Thomas Wong
2013 unpublished
Rotation distance between rooted binary trees measures the degree of similarity of two trees with ordered leaves and is equivalent to edge-flip distance between triangular subdivisions of regular polygons  ...  Here we describe the distribution of common edges between randomly-selected triangulations and measure the sizes of the remaining pieces into which the common edges separate the polygons.  ...  Thus, a triangulation of size n means a triangulation on an n + 1-gon or with n leaves in the underlying binary tree.  ...

### Convex polygon triangulation based on planted trivalent binary tree and ballot problem

Muzafer SARAČEVIĆ, Aybeyan SELIMI
2019 Turkish Journal of Electrical Engineering and Computer Sciences
The properties of the Catalan numbers were examined and their decomposition and application in developing the hierarchy and triangulation trees were analyzed.  ...  This paper presents a new technique of generation of convex polygon triangulation based on planted trivalent binary tree and ballot notation.  ...  Tree of triangulations and expression of the Catalan numbers The first step in our method is generating a complete triangulation tree from the initial basic triangle to the given n -gon.  ...

### A Simple Linear Time Algorithm for Triangulating Three-Colored Graphs

H. Bodlaender, T. Kloks
1993 Journal of Algorithms
In this paper we consider the problem of determining whether a given colored graph can be triangulated, such that no edges between vertices of the same color are added.  ...  In this paper we give a simple linear time algorithm that solves the problem when there are three colors.  ...  We must show that G is a tree of cycles. Consider a triangulation of G into a 2-tree H. Consider the edges in H that are not in G.  ...

### A simple linear time algorithm for triangulating three-colored graphs [chapter]

Hans Bodlaender, Ton Kloks
1992 Lecture Notes in Computer Science
In this paper we consider the problem of determining whether a given colored graph can be triangulated, such that no edges between vertices of the same color are added.  ...  In this paper we give a simple linear time algorithm that solves the problem when there are three colors.  ...  We must show that G is a tree of cycles. Consider a triangulation of G into a 2-tree H. Consider the edges in H that are not in G.  ...

### JAVA Implementation for Triangulation of Convex Polygon Based on Lukasiewicz's Algorithm and Binary Trees

Muzafer Saračević, Sead H.Mašović, Danijela G. Milošević
2013 Southeast Europe Journal of Soft Computing
The notation that is obtained is expressed in the form of binary records. A presented method of triangulations is based on Lukasiewicz's algorithm and binary trees.  ...  Triangulation of the polygon is one of the fundamental algorithm computational geometry. This paper describes one method triangulations of a convex polygon.  ...  In paper is described graph of triangulations of a convex polygon and tree of triangulations, while the authors of the paper (Devroye, 1999) of random triangulations and trees.  ...

### Triangulations without pointed spanning trees

Oswin Aichholzer, Clemens Huemer, Hannes Krasser
2008 Computational geometry
Problem 50 in the Open Problems Project of the computational geometry community asks whether any triangulation on a point set in the plane contains a pointed spanning tree as a subgraph.  ...  As a consequence we show that there exist triangulations which require a linear number of edge flips to become Hamiltonian.  ...  Acknowledgements The innocent looking conjecture that any triangulation contains a pointed spanning tree as a subgraph has fascinated several people.  ...

### Hamiltonian-connectedness of triangulations with few separating triangles [article]

Nico Van Cleemput
2016 arXiv   pre-print
In order to show bounds on the strongest form of this theorem, we proved that for any s≥4 there are 3-connected triangulation with s separating triangles that are not hamiltonian-connected.  ...  We prove that 3-connected triangulations with at most one separating triangle are hamiltonian-connected.  ...  The author would also like to thank Jasper Souffriau for providing him with the programs to determine the decomposition tree of a triangulation which were developed for [1] .  ...

### Diagonal-Flip Distance Algorithms of Three Type Triangulations

Deqiang Wang, Xian Wang, Shaoxi Li, Shaofang Zhang
2008 2008 International Conference on Computer Science and Software Engineering
In this paper we study the diagonal flipping problem in three special type triangulations of Ò (Ò ) vertex convex polygons (and rotations in three type binary trees).  ...  trees).  ...  A rotation in a binary tree is a local restructuring of the tree that changes the position of an internal node and one of its children while the symmetric order in the tree is preserved (see Fig.1 ).  ...
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