scholarly journals The number of corner polyhedra graphs

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Clement Dervieux ◽  
Dominique Poulalhon ◽  
Gilles Schaeffer
Keyword(s):  
Binary Trees ◽  
Recursive Decomposition ◽  
Cycle Cover ◽  
Rational Expression ◽  
Compositional Approach ◽  
Polyhedral Graph ◽  
International Audience ◽  
Simply Connected ◽  
Corner Polyhedra

International audience Corner polyhedra were introduced by Eppstein and Mumford (2014) as the set of simply connected 3D polyhedra such that all vertices have non negative integer coordinates, edges are parallel to the coordinate axes and all vertices but one can be seen from infinity in the direction (1, 1, 1). These authors gave a remarkable characterization of the set of corner polyhedra graphs, that is graphs that can be skeleton of a corner polyhedron: as planar maps, they are the duals of some particular bipartite triangulations, which we call hereafter corner triangulations.In this paper we count corner polyhedral graphs by determining the generating function of the corner triangulations with respect to the number of vertices: we obtain an explicit rational expression for it in terms of the Catalan gen- erating function. We first show that this result can be derived using Tutte's classical compositional approach. Then, in order to explain the occurrence of the Catalan series we give a direct algebraic decomposition of corner triangu- lations: in particular we exhibit a family of almond triangulations that admit a recursive decomposition structurally equivalent to the decomposition of binary trees. Finally we sketch a direct bijection between binary trees and almond triangulations. Our combinatorial analysis yields a simpler alternative to the algorithm of Eppstein and Mumford for endowing a corner polyhedral graph with the cycle cover structure needed to realize it as a polyhedral graph.

scholarly journals The b-chromatic number of power graphs

2003 ◽  
Vol Vol. 6 no. 1 ◽  
Author(s):  
Brice Effantin ◽  
Hamamache Kheddouci
Keyword(s):  
Chromatic Number ◽  
Binary Trees ◽  
Proper Coloring ◽  
Power Cycles ◽  
International Audience

International audience The b-chromatic number of a graph G is defined as the maximum number k of colors that can be used to color the vertices of G, such that we obtain a proper coloring and each color i, with 1 ≤ i≤ k, has at least one representant x_i adjacent to a vertex of every color j, 1 ≤ j ≠ i ≤ k. In this paper, we discuss the b-chromatic number of some power graphs. We give the exact value of the b-chromatic number of power paths and power complete binary trees, and we bound the b-chromatic number of power cycles.

scholarly journals p-box: a new graph model

2015 ◽  
Vol Vol. 17 no. 1 (Graph Theory) ◽  
Author(s):  
Mauricio Soto ◽  
Christopher Thraves-Caro
Keyword(s):  
Graph Theory ◽  
Graph Model ◽  
Directed Path ◽  
Outerplanar Graphs ◽  
New Model ◽  
Model Based ◽  
Representative Element ◽  
International Audience ◽  
Graph Families

Graph Theory International audience In this document, we study the scope of the following graph model: each vertex is assigned to a box in ℝd and to a representative element that belongs to that box. Two vertices are connected by an edge if and only if its respective boxes contain the opposite representative element. We focus our study on the case where boxes (and therefore representative elements) associated to vertices are spread in ℝ. We give both, a combinatorial and an intersection characterization of the model. Based on these characterizations, we determine graph families that contain the model (e. g., boxicity 2 graphs) and others that the new model contains (e. g., rooted directed path). We also study the particular case where each representative element is the center of its respective box. In this particular case, we provide constructive representations for interval, block and outerplanar graphs. Finally, we show that the general and the particular model are not equivalent by constructing a graph family that separates the two cases.

scholarly journals Classes of graphs with restricted interval models

1999 ◽  
Vol Vol. 3 no. 4 ◽  
Author(s):  
Andrzej Proskurowski ◽  
Jan Arne Telle
Keyword(s):  
Maximum Clique ◽  
Interval Graphs ◽  
Induced Subgraph ◽  
Graph Theoretic ◽  
Graph Classes ◽  
International Audience ◽  
Forbidden Induced Subgraph ◽  
Nested Hierarchy ◽  
Graph Families

International audience We introduce q-proper interval graphs as interval graphs with interval models in which no interval is properly contained in more than q other intervals, and also provide a forbidden induced subgraph characterization of this class of graphs. We initiate a graph-theoretic study of subgraphs of q-proper interval graphs with maximum clique size k+1 and give an equivalent characterization of these graphs by restricted path-decomposition. By allowing the parameter q to vary from 0 to k, we obtain a nested hierarchy of graph families, from graphs of bandwidth at most k to graphs of pathwidth at most k. Allowing both parameters to vary, we have an infinite lattice of graph classes ordered by containment.

scholarly journals Special Types of Locally Conformal Closed G2-Structures

Axioms ◽  
2018 ◽  
Vol 7 (4) ◽  
pp. 90 ◽  
Author(s):  
Giovanni Bazzoni ◽  
Alberto Raffero
Keyword(s):  
Lie Groups ◽  
Symplectic Geometry ◽  
Complete Characterization ◽  
Compact Manifolds ◽  
Different Types ◽  
Simply Connected ◽  
Locally Conformal

Motivated by known results in locally conformal symplectic geometry, we study different classes of G 2 -structures defined by a locally conformal closed 3-form. In particular, we provide a complete characterization of invariant exact locally conformal closed G 2 -structures on simply connected Lie groups, and we present examples of compact manifolds with different types of locally conformal closed G 2 -structures.

scholarly journals Exponential bounds and tails for additive random recursive sequences

2007 ◽  
Vol Vol. 9 no. 1 (Analysis of Algorithms) ◽  
Author(s):  
Ludger Rüschendorf ◽  
Eva-Maria Schopp
Keyword(s):  
Analysis Of Algorithms ◽  
Sufficient Conditions ◽  
Random Trees ◽  
Divide And Conquer ◽  
Recursive Sequences ◽  
Recursive Structures ◽  
International Audience ◽  
Exponential Bounds ◽  
The Moment

Analysis of Algorithms International audience Exponential bounds and tail estimates are derived for additive random recursive sequences, which typically arise as functionals of recursive structures, of random trees or in recursive algorithms. In particular they arise as parameters of divide and conquer type algorithms. We derive tail bounds from estimates of the Laplace transforms and of the moment sequences. For the proof we use some classical exponential bounds and some variants of the induction method. The paper generalizes results of Rösler (% \citeyearNPRoesler:91, % \citeyearNPRoesler:92) and % \citeNNeininger:05 on subgaussian tails to more general classes of additive random recursive sequences. It also gives sufficient conditions for tail bounds of the form \exp(-a t^p) which are based on a characterization of \citeNKasahara:78.

scholarly journals An optimal cardinality estimation algorithm based on order statistics and its full analysis

2010 ◽  
Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽  
Author(s):  
Jérémie Lumbroso
Keyword(s):  
Order Statistics ◽  
Estimation Algorithm ◽  
Fine Tuning ◽  
Simple Analysis ◽  
Relative Precision ◽  
Asymptotic Regime ◽  
Cardinality Estimation ◽  
International Audience ◽  
Full Analysis

International audience Building on the ideas of Flajolet and Martin (1985), Alon et al. (1987), Bar-Yossef et al. (2002), Giroire (2005), we develop a new algorithm for cardinality estimation, based on order statistics which, according to Chassaing and Gerin (2006), is optimal among similar algorithms. This algorithm has a remarkably simple analysis that allows us to take its $\textit{fine-tuning}$ and the $\textit{characterization of its properties}$ further than has been done until now. We prove that, asymptotically, it is $\textit{strictly unbiased}$ (contrarily to Probabilistic Counting, Loglog, Hyperloglog), we verify that its relative precision is about $1/\sqrt{m-2}$ when $m$ words of storage are used, and we fully characterize the limit law of the estimates it provides, in terms of gamma distribution―-this is the first such algorithm for which the limit law has been established. We also develop a Poisson analysis for the pre-asymptotic regime. In this way, we are able to devise a complete algorithm, covering all cardinalities ranges from $0$ to very large.

scholarly journals A Characterization of GRW Spacetimes

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2209
Author(s):  
Ibrahim Al-Dayel ◽  
Sharief Deshmukh ◽  
Mohd. Danish Siddiqi
Keyword(s):  
Vector Field ◽  
Potential Function ◽  
Laplace Operator ◽  
Einstein Manifold ◽  
Lorentzian Manifold ◽  
De Rham ◽  
Associated Function ◽  
Simply Connected ◽  
The Way

We show presence a special torse-forming vector field (a particular form of torse-forming of a vector field) on generalized Robertson–Walker (GRW) spacetime, which is an eigenvector of the de Rham–Laplace operator. This paves the way to showing that the presence of a time-like special torse-forming vector field ξ with potential function ρ on a Lorentzian manifold (M,g), dimM>5, which is an eigenvector of the de Rham Laplace operator, gives a characterization of a GRW-spacetime. We show that if, in addition, the function ξ(ρ) is nowhere zero, then the fibers of the GRW-spacetime are compact. Finally, we show that on a simply connected Lorentzian manifold (M,g) that admits a time-like special torse-forming vector field ξ, there is a function f called the associated function of ξ. It is shown that if a connected Lorentzian manifold (M,g), dimM>4, admits a time-like special torse-forming vector field ξ with associated function f nowhere zero and satisfies the Fischer–Marsden equation, then (M,g) is a quasi-Einstein manifold.

scholarly journals Self-complementing permutations of k-uniform hypergraphs

2009 ◽  
Vol Vol. 11 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Artur Szymański ◽  
Adam Pawel Wojda
Keyword(s):  
Positive Integer ◽  
Uniform Hypergraph ◽  
Uniform Hypergraphs ◽  
Unique Integer ◽  
International Audience ◽  
Sigma E

Graphs and Algorithms International audience A k-uniform hypergraph H = ( V; E) is said to be self-complementary whenever it is isomorphic with its complement (H) over bar = ( V; ((V)(k)) - E). Every permutation sigma of the set V such that sigma(e) is an edge of (H) over bar if and only if e is an element of E is called self-complementing. 2-self-comlementary hypergraphs are exactly self complementary graphs introduced independently by Ringel ( 1963) and Sachs ( 1962). <br> For any positive integer n we denote by lambda(n) the unique integer such that n = 2(lambda(n)) c, where c is odd. <br> In the paper we prove that a permutation sigma of [1, n] with orbits O-1,..., O-m O m is a self-complementing permutation of a k-uniform hypergraph of order n if and only if there is an integer l >= 0 such that k = a2(l) + s, a is odd, 0 <= s <= 2(l) and the following two conditions hold: <br> (i)n = b2(l+1) + r,r is an element of {0,..., 2(l) - 1 + s}, and <br> (ii) Sigma(i:lambda(vertical bar Oi vertical bar)<= l) vertical bar O-i vertical bar <= r. <br> For k = 2 this result is the very well known characterization of self-complementing permutation of graphs given by Ringel and Sachs.

scholarly journals Brick polytopes, lattices and Hopf algebras

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Vincent Pilaud
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Transitive Closure ◽  
Weak Order ◽  
Binary Trees ◽  
International Audience ◽  
Congruence Classes

International audience Generalizing the connection between the classes of the sylvester congruence and the binary trees, we show that the classes of the congruence of the weak order on Sn defined as the transitive closure of the rewriting rule UacV1b1 ···VkbkW ≡k UcaV1b1 ···VkbkW, for letters a < b1,...,bk < c and words U,V1,...,Vk,W on [n], are in bijection with acyclic k-triangulations of the (n + 2k)-gon, or equivalently with acyclic pipe dreams for the permutation (1,...,k,n + k,...,k + 1,n + k + 1,...,n + 2k). It enables us to transport the known lattice and Hopf algebra structures from the congruence classes of ≡k to these acyclic pipe dreams, and to describe the product and coproduct of this algebra in terms of pipe dreams. Moreover, it shows that the fan obtained by coarsening the Coxeter fan according to the classes of ≡k is the normal fan of the corresponding brick polytope

scholarly journals Solving equations over small unary algebras

2005 ◽  
Vol DMTCS Proceedings vol. AF,... (Proceedings) ◽  
Author(s):  
Przemyslaw Broniek
Keyword(s):  
Polynomial Time ◽  
Partial Characterization ◽  
Polynomial Equations ◽  
Solving Equations ◽  
International Audience ◽  
Np Complete ◽  
System Of Polynomial Equations

International audience We consider the problem of solving a system of polynomial equations over fixed algebra $A$ which we call MPolSat($A$). We restrict ourselves to unary algebras and give a partial characterization of complexity of MPolSat($A$). We isolate a preorder $P(A)$ to show that when $A$ has at most 3 elements then MPolSat($A$) is in $P$ when width of $P(A)$ is at most 2 and is NP-complete otherwise. We show also that if $P ≠ NP$ then the class of unary algebras solvable in polynomial time is not closed under homomorphic images.

Sign in / Sign up

Export Citation Format

Share Document