Antimatroids and Convex Geometries MPRI 2017–2018
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A Forbidden Subgraph Characterization of Some Graph Classes Using Betweenness Axioms
A forbidden subgraph characterization of some graph classes using betweenness axioms Manoj Changat Department of Futures Studies University of Kerala Trivandrum - 695034, India e-mail: [email protected] Anandavally K Lakshmikuttyamma ∗ Department of Futures Studies University of Kerala, Trivandrum-695034, India e-mail: [email protected] Joseph Mathews Iztok Peterin C-GRAF, Department of Futures Studies University of Maribor University of Kerala, Trivandrum-695034, India FEECS, Smetanova 17, e-mail:[email protected] 2000 Maribor, Slovenia [email protected] Prasanth G. Narasimha-Shenoi Geetha Seethakuttyamma Department of Mathematics Department of Mathematics Government College, Chittur College of Engineering Palakkad - 678104, India Karungapally e-mail: [email protected] e-mail:[email protected] Simon Špacapan University of Maribor, FME, Smetanova 17, 2000 Maribor, Slovenia. e-mail: [email protected]. July 27, 2012 ∗ The work of this author (On leave from Mar Ivanios College, Trivandrum) is supported by the University Grants Commission (UGC), Govt. of India under the FIP scheme. 1 Abstract Let IG(x,y) and JG(x,y) be the geodesic and induced path interval between x and y in a connected graph G, respectively. The following three betweenness axioms are considered for a set V and R : V × V → 2V (i) x ∈ R(u,y),y ∈ R(x,v),x 6= y, |R(u,v)| > 2 ⇒ x ∈ R(u,v) (ii) x ∈ R(u,v) ⇒ R(u,x) ∩ R(x,v)= {x} (iii)x ∈ R(u,y),y ∈ R(x,v),x 6= y, ⇒ x ∈ R(u,v). We characterize the class of graphs for which IG satisfies (i), and the class for which JG satisfies (ii) and the class for which IG or JG satisfies (iii). -
(Eds.) Innovations in Bayesian Networks
Dawn E. Holmes and Lakhmi C. Jain (Eds.) InnovationsinBayesianNetworks Studies in Computational Intelligence,Volume 156 Editor-in-Chief Prof. Janusz Kacprzyk Systems Research Institute Polish Academy of Sciences ul. Newelska 6 01-447 Warsaw Poland E-mail: [email protected] Further volumes of this series can be found on our homepage: Vol. 145. Mikhail Ju. Moshkov, Marcin Piliszczuk springer.com and Beata Zielosko Partial Covers, Reducts and Decision Rules in Rough Sets, 2008 Vol. 134. Ngoc Thanh Nguyen and Radoslaw Katarzyniak (Eds.) ISBN 978-3-540-69027-6 New Challenges in Applied Intelligence Technologies, 2008 Vol. 146. Fatos Xhafa and Ajith Abraham (Eds.) ISBN 978-3-540-79354-0 Metaheuristics for Scheduling in Distributed Computing Vol. 135. Hsinchun Chen and Christopher C.Yang (Eds.) Environments, 2008 Intelligence and Security Informatics, 2008 ISBN 978-3-540-69260-7 ISBN 978-3-540-69207-2 Vol. 147. Oliver Kramer Self-Adaptive Heuristics for Evolutionary Computation, 2008 Vol. 136. Carlos Cotta, Marc Sevaux ISBN 978-3-540-69280-5 and Kenneth S¨orensen (Eds.) Adaptive and Multilevel Metaheuristics, 2008 Vol. 148. Philipp Limbourg ISBN 978-3-540-79437-0 Dependability Modelling under Uncertainty, 2008 ISBN 978-3-540-69286-7 Vol. 137. Lakhmi C. Jain, Mika Sato-Ilic, Maria Virvou, George A. Tsihrintzis,Valentina Emilia Balas Vol. 149. Roger Lee (Ed.) and Canicious Abeynayake (Eds.) Software Engineering, Artificial Intelligence, Networking and Computational Intelligence Paradigms, 2008 Parallel/Distributed Computing, 2008 ISBN 978-3-540-79473-8 ISBN 978-3-540-70559-8 Vol. 138. Bruno Apolloni,Witold Pedrycz, Simone Bassis Vol. 150. Roger Lee (Ed.) and Dario Malchiodi Software Engineering Research, Management and The Puzzle of Granular Computing, 2008 Applications, 2008 ISBN 978-3-540-79863-7 ISBN 978-3-540-70774-5 Vol. -
Discrete Mathematics Rooted Directed Path Graphs Are Leaf Powers
Discrete Mathematics 310 (2010) 897–910 Contents lists available at ScienceDirect Discrete Mathematics journal homepage: www.elsevier.com/locate/disc Rooted directed path graphs are leaf powers Andreas Brandstädt a, Christian Hundt a, Federico Mancini b, Peter Wagner a a Institut für Informatik, Universität Rostock, D-18051, Germany b Department of Informatics, University of Bergen, N-5020 Bergen, Norway article info a b s t r a c t Article history: Leaf powers are a graph class which has been introduced to model the problem of Received 11 March 2009 reconstructing phylogenetic trees. A graph G D .V ; E/ is called k-leaf power if it admits a Received in revised form 2 September 2009 k-leaf root, i.e., a tree T with leaves V such that uv is an edge in G if and only if the distance Accepted 13 October 2009 between u and v in T is at most k. Moroever, a graph is simply called leaf power if it is a Available online 30 October 2009 k-leaf power for some k 2 N. This paper characterizes leaf powers in terms of their relation to several other known graph classes. It also addresses the problem of deciding whether a Keywords: given graph is a k-leaf power. Leaf powers Leaf roots We show that the class of leaf powers coincides with fixed tolerance NeST graphs, a Graph class inclusions well-known graph class with absolutely different motivations. After this, we provide the Strongly chordal graphs largest currently known proper subclass of leaf powers, i.e, the class of rooted directed path Fixed tolerance NeST graphs graphs. -
Arxiv:1508.05446V2 [Math.CO] 27 Sep 2018 02,5B5 16E10
CELL COMPLEXES, POSET TOPOLOGY AND THE REPRESENTATION THEORY OF ALGEBRAS ARISING IN ALGEBRAIC COMBINATORICS AND DISCRETE GEOMETRY STUART MARGOLIS, FRANCO SALIOLA, AND BENJAMIN STEINBERG Abstract. In recent years it has been noted that a number of combi- natorial structures such as real and complex hyperplane arrangements, interval greedoids, matroids and oriented matroids have the structure of a finite monoid called a left regular band. Random walks on the monoid model a number of interesting Markov chains such as the Tsetlin library and riffle shuffle. The representation theory of left regular bands then comes into play and has had a major influence on both the combinatorics and the probability theory associated to such structures. In a recent pa- per, the authors established a close connection between algebraic and combinatorial invariants of a left regular band by showing that certain homological invariants of the algebra of a left regular band coincide with the cohomology of order complexes of posets naturally associated to the left regular band. The purpose of the present monograph is to further develop and deepen the connection between left regular bands and poset topology. This allows us to compute finite projective resolutions of all simple mod- ules of unital left regular band algebras over fields and much more. In the process, we are led to define the class of CW left regular bands as the class of left regular bands whose associated posets are the face posets of regular CW complexes. Most of the examples that have arisen in the literature belong to this class. A new and important class of ex- amples is a left regular band structure on the face poset of a CAT(0) cube complex. -
Graph Convexity and Vertex Orderings by Rachel Jean Selma Anderson B.Sc., Mcgill University, 2002 B.Ed., University of Victoria
Graph Convexity and Vertex Orderings by Rachel Jean Selma Anderson B.Sc., McGill University, 2002 B.Ed., University of Victoria, 2007 AThesisSubmittedinPartialFulfillmentofthe Requirements for the Degree of MASTER OF SCIENCE in the Department of Mathematics and Statistics c Rachel Jean Selma Anderson, 2014 University of Victoria All rights reserved. This thesis may not be reproduced in whole or in part, by photocopying or other means, without the permission of the author. ii Graph Convexity and Vertex Orderings by Rachel Jean Selma Anderson B.Sc., McGill University, 2002 B.Ed., University of Victoria, 2007 Supervisory Committee Dr. Ortrud Oellermann, Co-supervisor (Department of Mathematics and Statistics, University of Winnipeg) Dr. Gary MacGillivray, Co-supervisor (Department of Mathematics and Statistics, University of Victoria) iii Supervisory Committee Dr. Ortrud Oellermann, Co-supervisor (Department of Mathematics and Statistics, University of Winnipeg) Dr. Gary MacGillivray, Co-supervisor (Department of Mathematics and Statistics, University of Victoria) ABSTRACT In discrete mathematics, a convex space is an ordered pair (V, )where is a M M family of subsets of a finite set V ,suchthat: , V ,and is closed under ∅∈M ∈M M intersection. The elements of are called convex sets.ForasetS V ,theconvex M ⊆ hull of S is the smallest convex set that contains S.Apointx of a convex set X is an extreme point of X if X x is also convex. A convex space (V, )withtheproperty \{ } M that every convex set is the convex hull of its extreme points is called a convex geome- try.AgraphG has a P-elimination ordering if an ordering v1,v2,...,vn of the vertices exists such that vi has property P in the graph induced by vertices vi,vi+1,...,vn for all i =1, 2,...,n. -
What Convex Geometries Tell About Shattering-Extremal Systems Bogdan Chornomaz
What convex geometries tell about shattering-extremal systems Bogdan Chornomaz To cite this version: Bogdan Chornomaz. What convex geometries tell about shattering-extremal systems. 2020. hal- 02869292 HAL Id: hal-02869292 https://hal.archives-ouvertes.fr/hal-02869292 Preprint submitted on 15 Jun 2020 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. What convex geometries tell about shattering-extremal systems Bogdan Chornomaz [email protected] Vanderbilt University 1 Introduction Convex geometries admit many seemingly distinct yet equivalent characteriza- tions. Among other things, they are known to be exactly shattering-extremal closure systems [2]. In this paper we exploit this connection and generalize some known characterizations of convex geometries to shattering-extremal set families, which are a subject of intensive study in their own right. Our first main result is Theorem 2 in Section 4, which characterizes shattering- extremal set families in terms of forbidden projections, similar to the character- ization of convex geometries by Dietrich [3], discussed in Section 3. Another known characterization of convex geometries, given in Theorem 3, is that they are exactly closure systems in which any non-maximal set can be extended by one element. -
Clustering on Antimatroids and Convex Geometries
Clustering on antimatroids and convex geometries YULIA KEMPNER1 , ILYA MUCHNIK2 1Department of Computer Science Holon Academic Institute of Technology 52 Golomb Str., P.O. Box 305, Holon 58102 ISRAEL 2Department of Computer Science Rutgers University, NJ, DIMACS 96 Frelinghuysen Road , Piscataway, NJ 08854-8018 US Abstract: - The clustering problem as a problem of set function optimization with constraints is considered. The behavior of quasi-concave functions on antimatroids and on convex geometries is investigated. The duality of these two set function optimizations is proved. The greedy type Chain algorithm, which allows to find an optimal cluster, both as the “most distant” group on antimatroids and as a dense cluster on convex geometries, is described. Key-Words: - Quasi-concave function, antimatroid, convex geometry, cluster, greedy algorithm 1 Introduction clustering. We found that such situations can be In this paper we consider the problem of clustering characterized as constructively defined families of on antimatroids and on convex geometries. There subsets, such that only elements from these families are two approaches to clustering. In the first one we can be considered as feasible clusters. For example, try to find extreme dense clusters (either as a let us consider a boolean matrix of data in which partition of the considered set of objects into variables (columns of matrix) are divided into homogeneous groups, or as a single tight set of several groups. Variables, belonging to one group, objects). The second approach finds a cluster which are ordered. In this case every group of variables can is the “most distant” from its complementary subset be interpreted as a complex variable which represent in the whole set (as in the single linkage clustering an ordinal scale for some property: value 1 of the method) without a control how close are objects first (according to the order) boolean variable means within a cluster. -
Characterization and Recognition of Some Opposition and Coalition
Characterization and recognition of some opposition and coalition graph classes Van Bang Le Universit¨at Rostock, Institut fur¨ Informatik, Rostock, Germany Thomas Podelleck Universit¨at Rostock, Institut fur¨ Informatik, Rostock, Germany Abstract A graph is an opposition graph, respectively, a coalition graph, if it admits an acyclic orientation which puts the two end-edges of every chordless 4-vertex path in opposition, respectively, in the same direction. Opposition and coali- tion graphs have been introduced and investigated in connection to perfect graphs. Recognizing and characterizing opposition and coalition graphs are long-standing open problems. This paper gives characterizations for opposi- tion graphs and coalition graphs on some restricted graph classes. Implicit in our arguments are polynomial time recognition algorithms for these graphs. We also give a good characterization for the so-called generalized opposition graphs. Keywords: Perfectly orderable graph, one-in-one-out graph, opposition graph, coalition graph, perfect graph 2010 MSC: 05C75, 05C17, 05C62 arXiv:1507.00557v1 [cs.DM] 2 Jul 2015 1. Introduction and preliminaries Chv´atal [4] proposed to call a linear order < on the vertex set of an undirected graph G perfect if the greedy coloring algorithm applied to each induced subgraph H of G gives an optimal coloring of H: Consider the vertices of H sequentially by following the order < and assign to each vertex Email addresses: [email protected] (Van Bang Le), [email protected] (Thomas Podelleck) To appear in Discrete Appl. Math. November 8, 2017 v the smallest color not used on any neighbor u of v, u < v. A graph is perfectly orderable if it admits a perfect order. -
On Antimatroids of Infinite Character
Mathematica Pannonica 23/2 (2012), 257–266 ON ANTIMATROIDS OF INFINITE CHARACTER Hua Mao Department of Mathematics, Hebei University, Baoding 071002, China Sanyang Liu Department of Mathematics, Xidian University, Xi’an 710071, China Received: March 2012 MSC 2010 : 06 A 07,06 A 06,52 A 01,05E 99 Keywords: Antimatroid of infinite character, infinite antimatroid, poset, lifting, reduction. Abstract: This article provides a new construction of infinite antimatroids– antimatroid of infinite character. Comparing the new construction with a fa- mous construction of infinite antimatroids–antimatroids of finite character, it obtains that there is no one-to-one corresponding between antimatroids of in- finite character and antimatroids of finite character. It also deals with some properties for antimatroids of infinite character. At the final part, with the as- sistance of poset theory, it introduces a one-element extension of antimatroids of infinite character and discover methods to build up antimatroids of infinite character. 1. Introduction and preliminaries Many researchers are interested in infinite antimatroids though there is no unique construction that is called an infinite antimatroid. Generally, people refer to a finite antimatroid-like model defined on in- E-mail address: [email protected] 258 H. Mao and S. Liu finite set as an infinite antimatroid. Therefore, there are many infinite antimatroid structures which are provided by people from different mo- tivations by generalizing the structure of finite antimatroids, and mean- while, many results relative to infinite antimatroids are provided (cf. [1, 6, 8, 9]). In this paper, we present a new construction of infinite antimatroid– antimatroid of infinite character and discuss some relations among finite antimatroids, antimatroids of finite character and antimatroids of infinite character. -
Good Characterizations and Linear Time Recognition for 2-Probe Block
Good characterizations and linear time recognition for 2-probe block graphs✩ Van Bang Lea, Sheng-Lung Peng∗,b aUniversit¨at Rostock, Institut fur¨ Informatik, Germany bDepartment of Computer Science and Information Engineering, National Dong Hwa University, Shoufeng, Hualien 97401, Taiwan Abstract Block graphs are graphs in which every block (biconnected component) is a clique. A graph G = (V, E) is said to be an (unpartitioned) k-probe block graph if there exist k independent sets Ni ⊆ V , 1 ≤ i ≤ k, such that the graph G′ obtained from G by adding certain edges between vertices inside the sets Ni, 1 ≤ i ≤ k, is a block graph; if the independent sets Ni are given, G is called a partitioned k-probe block graph. In this paper we give good characterizations for 2-probe block graphs, in both unpartitioned and partitioned cases. As an algorithmic implication, partitioned and unpartitioned probe block graphs can be recognized in linear time, improving a recognition algorithm of cubic time complexity previously obtained by Chang et al. [Block-graph width, Theoretical Computer Science 412 (2011), 2496–2502]. Key words: Probe graph, block graph, probe block graph, probe complete graph 1. Introduction Given a graph class C, a graph G = (V, E) is called a probe C graph if there N ′ N exists an independent set ⊆ V (of non-probes) and a set E ⊆ 2 such that ′ ′ N the graph G = (V, E ∪ E ) is in the class C, where 2 stands for the set of all 2-element subsets of N. A graph G = (V, E) with a given independent set arXiv:1611.06285v1 [math.CO] 19 Nov 2016 N ′ N ⊆ V is said to be a partitioned probe C graph if there exists a set E ⊆ 2 such that the graph G′ = (V, E ∪E′) is in the class C. -
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IEICE TRANS. INF. & SYST., VOL.E91-D, NO.2 FEBRUARY 2008 170 PAPER Special Section on Foundations of Computer Science Longest Path Problems on Ptolemaic Graphs*** Yoshihiro TAKAHARA•õ*, Sachio TERAMOTO•õ**, Nonmembers, and Ryuhei UEHARA•õa) , Member SUMMARY Longest path problem is a problem for finding a longest problems for restricted graph classes. A linear time al- path in a given graph. While the graph classes in which the Hamiltonian gorithm for finding a longest path in a tree was invented path problem can be solved efficiently are widely investigated, there are by Dijkstra around 1960, and the formal proof is given by few known graph classes such that the longest path problem can be solved Bulterman et al. [7]. Recently, Uehara and Uno propose two efficiently. Polynomial time algorithms for finding a longest cycle and a longest path in a Ptolemaic graph are proposed. Ptolemaic graphs are the approaches [8]; one is an extension of the Dijkstra's algo- graphs that satisfy the Ptolemy inequality, and they are the intersection of rithm to some tree-like graph classes, and the other is dy- chordal graphs and distance-hereditary graphs. The algorithms use the dy- namic programming to some graph classes that have inter- namic programming technique on a laminar structure of cliques, which is a val representations. As the extensions of the Dijkstra's algo- recent characterization of Ptolemaic graphs. key words: dynamic programming, Hamiltonian path/cycle problem, rithm, they show polynomial time algorithms for weighted longest path/cycle problem, Ptolemaic graphs trees, block graphs, Ptolemaic graphs, and cactus. -
An Order Theoretic Point-Of-View on Subgroup Discovery Aimene Belfodil
An Order Theoretic Point-of-view on Subgroup Discovery Aimene Belfodil To cite this version: Aimene Belfodil. An Order Theoretic Point-of-view on Subgroup Discovery. Artificial Intelligence [cs.AI]. Université de Lyon, 2019. English. tel-02355529 HAL Id: tel-02355529 https://hal.archives-ouvertes.fr/tel-02355529 Submitted on 8 Nov 2019 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. THÈSE DE DOCTORAT DE L’UNIVERSITÉ DE LYON opérée au sein de L’INSA DE LYON ECOLE DOCTORALE N°512 MATHÉMATIQUES ET INFORMATIQUE (INFOMATHS) Spécialité/Discipline de doctorat: Informatique An Order Theoretic Point-of-view on Subgroup Discovery Soutenue publiquement le 30/09/2019 par AIMENE BELFODIL Devant le jury composé de: Pr. Bruno Crémilleux Université de Caen Rapporteur Pr. Bernhard Ganter Technische Universitaet Dresden Rapporteur Pr. Céline Robardet INSA Lyon Directrice de thèse Dr. Mehdi Kaytoue Infologic Co-directeur de thèse Dr. Peggy Cellier INSA Rennes Examinatrice Pr. Miguel Couceiro Université de Lorraine Examinateur Pr. Arno Siebes Universiteit Utrecht Examinateur Pr. Sergei O. Kuznetsov Higher School of Economics (Moscow) Invité Mr. Julien Zarka Mobile Devices Ingenierie Invité Département FEDORA – INSA Lyon - Ecoles Doctorales – Quinquennal 2016-2020 SIGLE ECOLE DOCTORALE NOM ET COORDONNEES DU RESPONSABLE CHIMIE CHIMIE DE LYON M.