Contractions of Polygons in Abstract Polytopes
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On Finding Two Posets That Cover Given Linear Orders
algorithms Article On Finding Two Posets that Cover Given Linear Orders Ivy Ordanel 1,*, Proceso Fernandez, Jr. 2 and Henry Adorna 1 1 Department of Computer Science, University of the Philippines Diliman, Quezon City 1101, Philippines; [email protected] 2 Department of Information Sytems and Computer Science, Ateneo De Manila University, Quezon City 1108, Philippines; [email protected] * Correspondence: [email protected] Received: 13 August 2019; Accepted: 17 October 2019; Published: 19 October 2019 Abstract: The Poset Cover Problem is an optimization problem where the goal is to determine a minimum set of posets that covers a given set of linear orders. This problem is relevant in the field of data mining, specifically in determining directed networks or models that explain the ordering of objects in a large sequential dataset. It is already known that the decision version of the problem is NP-Hard while its variation where the goal is to determine only a single poset that covers the input is in P. In this study, we investigate the variation, which we call the 2-Poset Cover Problem, where the goal is to determine two posets, if they exist, that cover the given linear orders. We derive properties on posets, which leads to an exact solution for the 2-Poset Cover Problem. Although the algorithm runs in exponential-time, it is still significantly faster than a brute-force solution. Moreover, we show that when the posets being considered are tree-posets, the running-time of the algorithm becomes polynomial, which proves that the more restricted variation, which we called the 2-Tree-Poset Cover Problem, is also in P. -
Confluent Hasse Diagrams
Confluent Hasse diagrams David Eppstein Joseph A. Simons Department of Computer Science, University of California, Irvine, USA. October 29, 2018 Abstract We show that a transitively reduced digraph has a confluent upward drawing if and only if its reachability relation has order dimension at most two. In this case, we construct a confluent upward drawing with O(n2) features, in an O(n) × O(n) grid in O(n2) time. For the digraphs representing series-parallel partial orders we show how to construct a drawing with O(n) fea- tures in an O(n) × O(n) grid in O(n) time from a series-parallel decomposition of the partial order. Our drawings are optimal in the number of confluent junctions they use. 1 Introduction One of the most important aspects of a graph drawing is that it should be readable: it should convey the structure of the graph in a clear and concise way. Ease of understanding is difficult to quantify, so various proxies for readability have been proposed; one of the most prominent is the number of edge crossings. That is, we should minimize the number of edge crossings in our drawing (a planar drawing, if possible, is ideal), since crossings make drawings harder to read. Another measure of readability is the total amount of ink required by the drawing [1]. This measure can be formulated in terms of Tufte’s “data-ink ratio” [22,35], according to which a large proportion of the ink on any infographic should be devoted to information. Thus given two different ways to present information, we should choose the more succinct and crossing-free presentation. -
Petrie Schemes
Canad. J. Math. Vol. 57 (4), 2005 pp. 844–870 Petrie Schemes Gordon Williams Abstract. Petrie polygons, especially as they arise in the study of regular polytopes and Coxeter groups, have been studied by geometers and group theorists since the early part of the twentieth century. An open question is the determination of which polyhedra possess Petrie polygons that are simple closed curves. The current work explores combinatorial structures in abstract polytopes, called Petrie schemes, that generalize the notion of a Petrie polygon. It is established that all of the regular convex polytopes and honeycombs in Euclidean spaces, as well as all of the Grunbaum–Dress¨ polyhedra, pos- sess Petrie schemes that are not self-intersecting and thus have Petrie polygons that are simple closed curves. Partial results are obtained for several other classes of less symmetric polytopes. 1 Introduction Historically, polyhedra have been conceived of either as closed surfaces (usually topo- logical spheres) made up of planar polygons joined edge to edge or as solids enclosed by such a surface. In recent times, mathematicians have considered polyhedra to be convex polytopes, simplicial spheres, or combinatorial structures such as abstract polytopes or incidence complexes. A Petrie polygon of a polyhedron is a sequence of edges of the polyhedron where any two consecutive elements of the sequence have a vertex and face in common, but no three consecutive edges share a commonface. For the regular polyhedra, the Petrie polygons form the equatorial skew polygons. Petrie polygons may be defined analogously for polytopes as well. Petrie polygons have been very useful in the study of polyhedra and polytopes, especially regular polytopes. -
Regular Polyhedra Through Time
Fields Institute I. Hubard Polytopes, Maps and their Symmetries September 2011 Regular polyhedra through time The greeks were the first to study the symmetries of polyhedra. Euclid, in his Elements showed that there are only five regular solids (that can be seen in Figure 1). In this context, a polyhe- dron is regular if all its polygons are regular and equal, and you can find the same number of them at each vertex. Figure 1: Platonic Solids. It is until 1619 that Kepler finds other two regular polyhedra: the great dodecahedron and the great icosahedron (on Figure 2. To do so, he allows \false" vertices and intersection of the (convex) faces of the polyhedra at points that are not vertices of the polyhedron, just as the I. Hubard Polytopes, Maps and their Symmetries Page 1 Figure 2: Kepler polyhedra. 1619. pentagram allows intersection of edges at points that are not vertices of the polygon. In this way, the vertex-figure of these two polyhedra are pentagrams (see Figure 3). Figure 3: A regular convex pentagon and a pentagram, also regular! In 1809 Poinsot re-discover Kepler's polyhedra, and discovers its duals: the small stellated dodecahedron and the great stellated dodecahedron (that are shown in Figure 4). The faces of such duals are pentagrams, and are organized on a \convex" way around each vertex. Figure 4: The other two Kepler-Poinsot polyhedra. 1809. A couple of years later Cauchy showed that these are the only four regular \star" polyhedra. We note that the convex hull of the great dodecahedron, great icosahedron and small stellated dodecahedron is the icosahedron, while the convex hull of the great stellated dodecahedron is the dodecahedron. -
Operations on Partially Ordered Sets and Rational Identities of Type a Adrien Boussicault
Operations on partially ordered sets and rational identities of type A Adrien Boussicault To cite this version: Adrien Boussicault. Operations on partially ordered sets and rational identities of type A. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2013, Vol. 15 no. 2 (2), pp.13–32. hal- 00980747 HAL Id: hal-00980747 https://hal.inria.fr/hal-00980747 Submitted on 18 Apr 2014 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Discrete Mathematics and Theoretical Computer Science DMTCS vol. 15:2, 2013, 13–32 Operations on partially ordered sets and rational identities of type A Adrien Boussicault Institut Gaspard Monge, Universite´ Paris-Est, Marne-la-Valle,´ France received 13th February 2009, revised 1st April 2013, accepted 2nd April 2013. − −1 We consider the family of rational functions ψw = Q(xwi xwi+1 ) indexed by words with no repetition. We study the combinatorics of the sums ΨP of the functions ψw when w describes the linear extensions of a given poset P . In particular, we point out the connexions between some transformations on posets and elementary operations on the fraction ΨP . We prove that the denominator of ΨP has a closed expression in terms of the Hasse diagram of P , and we compute its numerator in some special cases. -
LNCS 7034, Pp
Confluent Hasse Diagrams DavidEppsteinandJosephA.Simons Department of Computer Science, University of California, Irvine, USA Abstract. We show that a transitively reduced digraph has a confluent upward drawing if and only if its reachability relation has order dimen- sion at most two. In this case, we construct a confluent upward drawing with O(n2)features,inanO(n) × O(n)gridinO(n2)time.Forthe digraphs representing series-parallel partial orders we show how to con- struct a drawing with O(n)featuresinanO(n)×O(n)gridinO(n)time from a series-parallel decomposition of the partial order. Our drawings are optimal in the number of confluent junctions they use. 1 Introduction One of the most important aspects of a graph drawing is that it should be readable: it should convey the structure of the graph in a clear and concise way. Ease of understanding is difficult to quantify, so various proxies for it have been proposed, including the number of crossings and the total amount of ink required by the drawing [1,18]. Thus given two different ways to present information, we should choose the more succinct and crossing-free presentation. Confluent drawing [7,8,9,15,16] is a style of graph drawing in which multiple edges are combined into shared tracks, and two vertices are considered to be adjacent if a smooth path connects them in these tracks (Figure 1). This style was introduced to re- duce crossings, and in many cases it will also Fig. 1. Conventional and confluent improve the ink requirement by represent- drawings of K5,5 ing dense subgraphs concisely. -
Convex Polytopes and Tilings with Few Flag Orbits
Convex Polytopes and Tilings with Few Flag Orbits by Nicholas Matteo B.A. in Mathematics, Miami University M.A. in Mathematics, Miami University A dissertation submitted to The Faculty of the College of Science of Northeastern University in partial fulfillment of the requirements for the degree of Doctor of Philosophy April 14, 2015 Dissertation directed by Egon Schulte Professor of Mathematics Abstract of Dissertation The amount of symmetry possessed by a convex polytope, or a tiling by convex polytopes, is reflected by the number of orbits of its flags under the action of the Euclidean isometries preserving the polytope. The convex polytopes with only one flag orbit have been classified since the work of Schläfli in the 19th century. In this dissertation, convex polytopes with up to three flag orbits are classified. Two-orbit convex polytopes exist only in two or three dimensions, and the only ones whose combinatorial automorphism group is also two-orbit are the cuboctahedron, the icosidodecahedron, the rhombic dodecahedron, and the rhombic triacontahedron. Two-orbit face-to-face tilings by convex polytopes exist on E1, E2, and E3; the only ones which are also combinatorially two-orbit are the trihexagonal plane tiling, the rhombille plane tiling, the tetrahedral-octahedral honeycomb, and the rhombic dodecahedral honeycomb. Moreover, any combinatorially two-orbit convex polytope or tiling is isomorphic to one on the above list. Three-orbit convex polytopes exist in two through eight dimensions. There are infinitely many in three dimensions, including prisms over regular polygons, truncated Platonic solids, and their dual bipyramids and Kleetopes. There are infinitely many in four dimensions, comprising the rectified regular 4-polytopes, the p; p-duoprisms, the bitruncated 4-simplex, the bitruncated 24-cell, and their duals. -
Lecture Notes on Algebraic Combinatorics Jeremy L. Martin
Lecture Notes on Algebraic Combinatorics Jeremy L. Martin [email protected] December 3, 2012 Copyright c 2012 by Jeremy L. Martin. These notes are licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 2 Foreword The starting point for these lecture notes was my notes from Vic Reiner's Algebraic Combinatorics course at the University of Minnesota in Fall 2003. I currently use them for graduate courses at the University of Kansas. They will always be a work in progress. Please use them and share them freely for any research purpose. I have added and subtracted some material from Vic's course to suit my tastes, but any mistakes are my own; if you find one, please contact me at [email protected] so I can fix it. Thanks to those who have suggested additions and pointed out errors, including but not limited to: Logan Godkin, Alex Lazar, Nick Packauskas, Billy Sanders, Tony Se. 1. Posets and Lattices 1.1. Posets. Definition 1.1. A partially ordered set or poset is a set P equipped with a relation ≤ that is reflexive, antisymmetric, and transitive. That is, for all x; y; z 2 P : (1) x ≤ x (reflexivity). (2) If x ≤ y and y ≤ x, then x = y (antisymmetry). (3) If x ≤ y and y ≤ z, then x ≤ z (transitivity). We'll usually assume that P is finite. Example 1.2 (Boolean algebras). Let [n] = f1; 2; : : : ; ng (a standard piece of notation in combinatorics) and let Bn be the power set of [n]. We can partially order Bn by writing S ≤ T if S ⊆ T . -
Representing the Sporadic Archimedean Polyhedra As Abstract Polytopes$
CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector Discrete Mathematics 310 (2010) 1835–1844 Contents lists available at ScienceDirect Discrete Mathematics journal homepage: www.elsevier.com/locate/disc Representing the sporadic Archimedean polyhedra as abstract polytopesI Michael I. Hartley a, Gordon I. Williams b,∗ a DownUnder GeoSolutions, 80 Churchill Ave, Subiaco, 6008, Western Australia, Australia b Department of Mathematics and Statistics, University of Alaska Fairbanks, PO Box 756660, Fairbanks, AK 99775-6660, United States article info a b s t r a c t Article history: We present the results of an investigation into the representations of Archimedean Received 29 August 2007 polyhedra (those polyhedra containing only one type of vertex figure) as quotients of Accepted 26 January 2010 regular abstract polytopes. Two methods of generating these presentations are discussed, Available online 13 February 2010 one of which may be applied in a general setting, and another which makes use of a regular polytope with the same automorphism group as the desired quotient. Representations Keywords: of the 14 sporadic Archimedean polyhedra (including the pseudorhombicuboctahedron) Abstract polytope as quotients of regular abstract polyhedra are obtained, and summarised in a table. The Archimedean polyhedron Uniform polyhedron information is used to characterize which of these polyhedra have acoptic Petrie schemes Quotient polytope (that is, have well-defined Petrie duals). Regular cover ' 2010 Elsevier B.V. All rights reserved. Flag action Exchange map 1. Introduction Much of the focus in the study of abstract polytopes has been on the regular abstract polytopes. A publication of the first author [6] introduced a method for representing any abstract polytope as a quotient of regular polytopes. -
Lecture Notes: Combinatorics, Aalto, Fall 2014 Instructor
Lecture notes: Combinatorics, Aalto, Fall 2014 Instructor: Alexander Engstr¨om TA & scribe: Oscar Kivinen Contents Preface ix Chapter 1. Posets 1 Chapter 2. Extremal combinatorics 11 Chapter 3. Chromatic polynomials 17 Chapter 4. Acyclic matchings on posets 23 Chapter 5. Complete (perhaps not acyclic) matchings 29 Bibliography 33 vii Preface These notes are from a course in Combinatorics at Aalto University taught during the first quarter of the school year 14-15. The intended structure is five separate chapters on topics that are fairly independent. The choice of topics could have been done in many other ways, and we don't claim the included ones to be in any way more important than others. There is another course on combinatorics at Aalto, towards computer science. Hence, we have selected topics that go more towards pure mathematics, to reduce the overlap. A particular feature about all of the topics is that there are active and interesting research going on in them, and some of the theorems we present are not usually mentioned at the undergraduate level. We should end with a warning: These are lecture notes. There are surely many errors and lack of references, but we have tried to eliminate these. Please ask if there is any incoherence, and feel free to point out outright errors. References to better and more comprehensive texts are given in the course of the text. ix CHAPTER 1 Posets Definition 1.1. A poset (or partially ordered set) is a set P with a binary relation ≤⊆ P × P that is (i) reflexive: p ≤ p for all p 2 P ; (ii) antisymmetric: if p ≤ q and q ≤ p, then p = q; (iii) transitive: if p ≤ q and q ≤ r, then p ≤ r Definition 1.2. -
Regular Incidence Complexes, Polytopes, and C-Groups
Regular Incidence Complexes, Polytopes, and C-Groups Egon Schulte∗ Department of Mathematics Northeastern University, Boston, MA 02115, USA March 13, 2018 Abstract Regular incidence complexes are combinatorial incidence structures generalizing regu- lar convex polytopes, regular complex polytopes, various types of incidence geometries, and many other highly symmetric objects. The special case of abstract regular poly- topes has been well-studied. The paper describes the combinatorial structure of a regular incidence complex in terms of a system of distinguished generating subgroups of its automorphism group or a flag-transitive subgroup. Then the groups admitting a flag-transitive action on an incidence complex are characterized as generalized string C-groups. Further, extensions of regular incidence complexes are studied, and certain incidence complexes particularly close to abstract polytopes, called abstract polytope complexes, are investigated. Key words. abstract polytope, regular polytope, C-group, incidence geometries MSC 2010. Primary: 51M20. Secondary: 52B15; 51E24. arXiv:1711.02297v1 [math.CO] 7 Nov 2017 1 Introduction Regular incidence complexes are combinatorial incidence structures with very high combina- torial symmetry. The concept was introduced by Danzer [12, 13] building on Gr¨unbaum’s [17] notion of a polystroma. Regular incidence complexes generalize regular convex polytopes [7], regular complex polytopes [8, 42], various types of incidence geometries [4, 5, 21, 44], and many other highly symmetric objects. The terminology and notation is patterned after con- vex polytopes [16] and was ultimately inspired by Coxeter’s work on regular figures [7, 8]. ∗Email: [email protected] 1 The first systematic study of incidence complexes from the discrete geometry perspective occurred in [33] and the related publications [13, 34, 35, 36]. -
Locally Spherical Hypertopes from Generalized Cubes
LOCALLY SPHERICAL HYPERTOPES FROM GENERLISED CUBES ANTONIO MONTERO AND ASIA IVIĆ WEISS Abstract. We show that every non-degenerate regular polytope can be used to construct a thin, residually-connected, chamber-transitive incidence geome- try, i.e. a regular hypertope, with a tail-triangle Coxeter diagram. We discuss several interesting examples derived when this construction is applied to gen- eralised cubes. In particular, we produce an example of a rank 5 finite locally spherical proper hypertope of hyperbolic type. 1. Introduction Hypertopes are particular kind of incidence geometries that generalise the no- tions of abstract polytopes and of hypermaps. The concept was introduced in [7] with particular emphasis on regular hypertopes (that is, the ones with highest de- gree of symmetry). Although in [6, 9, 8] a number of interesting examples had been constructed, within the theory of abstract regular polytopes much more work has been done. Notably, [20] and [22] deal with the universal constructions of polytopes, while in [3, 17, 18] the constructions with prescribed combinatorial conditions are explored. In another direction, in [2, 5, 11, 16] the questions of existence of poly- topes with prescribed (interesting) groups are investigated. Much of the impetus to the development of the theory of abstract polytopes, as well as the inspiration with the choice of problems, was based on work of Branko Grünbaum [10] from 1970s. In this paper we generalise the halving operation on polyhedra (see 7B in [13]) on a certain class of regular abstract polytopes to construct regular hypertopes. More precisely, given a regular non-degenerate n-polytope P, we construct a reg- ular hypertope H(P) related to semi-regular polytopes with tail-triangle Coxeter diagram.