On Finite Groups Whose Power Graph Is a Cograph
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Induced Minors and Well-Quasi-Ordering Jaroslaw Blasiok, Marcin Kamiński, Jean-Florent Raymond, Théophile Trunck
Induced minors and well-quasi-ordering Jaroslaw Blasiok, Marcin Kamiński, Jean-Florent Raymond, Théophile Trunck To cite this version: Jaroslaw Blasiok, Marcin Kamiński, Jean-Florent Raymond, Théophile Trunck. Induced minors and well-quasi-ordering. EuroComb: European Conference on Combinatorics, Graph Theory and Appli- cations, Aug 2015, Bergen, Norway. pp.197-201, 10.1016/j.endm.2015.06.029. lirmm-01349277 HAL Id: lirmm-01349277 https://hal-lirmm.ccsd.cnrs.fr/lirmm-01349277 Submitted on 27 Jul 2016 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. Available online at www.sciencedirect.com www.elsevier.com/locate/endm Induced minors and well-quasi-ordering Jaroslaw Blasiok 1,2 School of Engineering and Applied Sciences, Harvard University, United States Marcin Kami´nski 2 Institute of Computer Science, University of Warsaw, Poland Jean-Florent Raymond 2 Institute of Computer Science, University of Warsaw, Poland, and LIRMM, University of Montpellier, France Th´eophile Trunck 2 LIP, ENS´ de Lyon, France Abstract AgraphH is an induced minor of a graph G if it can be obtained from an induced subgraph of G by contracting edges. Otherwise, G is said to be H-induced minor- free. -
Arxiv:1907.08517V3 [Math.CO] 12 Mar 2021
RANDOM COGRAPHS: BROWNIAN GRAPHON LIMIT AND ASYMPTOTIC DEGREE DISTRIBUTION FRÉDÉRIQUE BASSINO, MATHILDE BOUVEL, VALENTIN FÉRAY, LUCAS GERIN, MICKAËL MAAZOUN, AND ADELINE PIERROT ABSTRACT. We consider uniform random cographs (either labeled or unlabeled) of large size. Our first main result is the convergence towards a Brownian limiting object in the space of graphons. We then show that the degree of a uniform random vertex in a uniform cograph is of order n, and converges after normalization to the Lebesgue measure on [0; 1]. We finally an- alyze the vertex connectivity (i.e. the minimal number of vertices whose removal disconnects the graph) of random connected cographs, and show that this statistics converges in distribution without renormalization. Unlike for the graphon limit and for the degree of a random vertex, the limiting distribution of the vertex connectivity is different in the labeled and unlabeled settings. Our proofs rely on the classical encoding of cographs via cotrees. We then use mainly combi- natorial arguments, including the symbolic method and singularity analysis. 1. INTRODUCTION 1.1. Motivation. Random graphs are arguably the most studied objects at the interface of com- binatorics and probability theory. One aspect of their study consists in analyzing a uniform random graph of large size n in a prescribed family, e.g. perfect graphs [MY19], planar graphs [Noy14], graphs embeddable in a surface of given genus [DKS17], graphs in subcritical classes [PSW16], hereditary classes [HJS18] or addable classes [MSW06, CP19]. The present paper focuses on uniform random cographs (both in the labeled and unlabeled settings). Cographs were introduced in the seventies by several authors independently, see e.g. -
ELEMENTARY ABELIAN P-SUBGROUPS of ALGEBRAIC GROUPS
ROBERT L. GRIESS, JR ELEMENTARY ABELIAN p-SUBGROUPS OF ALGEBRAIC GROUPS Dedicated to Jacques Tits for his sixtieth birthday ABSTRACT. Let ~ be an algebraically closed field and let G be a finite-dimensional algebraic group over N which is nearly simple, i.e. the connected component of the identity G O is perfect, C6(G °) = Z(G °) and G°/Z(G °) is simple. We classify maximal elementary abelian p-subgroups of G which consist of semisimple elements, i.e. for all primes p ~ char K. Call a group quasisimple if it is perfect and is simple modulo the center. Call a subset of an algebraic group total if it is in a toms; otherwise nontoral. For several quasisimple algebraic groups and p = 2, we define complexity, and give local criteria for whether an elementary abelian 2-subgroup of G is total. For all primes, we analyze the nontoral examples, include a classification of all the maximal elementary abelian p-groups, many of the nonmaximal ones, discuss their normalizers and fusion (i.e. how conjugacy classes of the ambient algebraic group meet the subgroup). For some cases, we give a very detailed discussion, e.g. p = 3 and G of type E6, E 7 and E 8. We explain how the presence of spin up and spin down elements influences the structure of projectively elementary abelian 2-groups in Spin(2n, C). Examples of an elementary abelian group which is nontoral in one algebraic group but toral in a larger one are noted. Two subsets of a maximal torus are conjugate in G iff they are conjugate in the normalizer of the torus; this observation, with our discussion of the nontoral cases, gives a detailed guide to the possibilities for the embedding of an elementary abelian p-group in G. -
Partitioning a Graph Into Disjoint Cliques and a Triangle-Free Graph
This is a repository copy of Partitioning a graph into disjoint cliques and a triangle-free graph. White Rose Research Online URL for this paper: http://eprints.whiterose.ac.uk/85292/ Version: Accepted Version Article: Abu-Khzam, FN, Feghali, C and Muller, H (2015) Partitioning a graph into disjoint cliques and a triangle-free graph. Discrete Applied Mathematics, 190-19. 1 - 12. ISSN 0166-218X https://doi.org/10.1016/j.dam.2015.03.015 © 2015, Elsevier. Licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International http://creativecommons.org/licenses/by-nc-nd/4.0/ Reuse Unless indicated otherwise, fulltext items are protected by copyright with all rights reserved. The copyright exception in section 29 of the Copyright, Designs and Patents Act 1988 allows the making of a single copy solely for the purpose of non-commercial research or private study within the limits of fair dealing. The publisher or other rights-holder may allow further reproduction and re-use of this version - refer to the White Rose Research Online record for this item. Where records identify the publisher as the copyright holder, users can verify any specific terms of use on the publisher’s website. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing [email protected] including the URL of the record and the reason for the withdrawal request. [email protected] https://eprints.whiterose.ac.uk/ Partitioning a Graph into Disjoint Cliques and a Triangle-free Graph Faisal N. Abu-Khzam, Carl Feghali, Haiko M¨uller Abstract A graph G =(V, E) is partitionable if there exists a partition {A,B} of V such that A induces a disjoint union of cliques and B induces a triangle- free graph. -
Arxiv:1906.05510V2 [Math.AC]
BINOMIAL EDGE IDEALS OF COGRAPHS THOMAS KAHLE AND JONAS KRUSEMANN¨ Abstract. We determine the Castelnuovo–Mumford regularity of binomial edge ideals of complement reducible graphs (cographs). For cographs with n vertices the maximum regularity grows as 2n/3. We also bound the regularity by graph theoretic invariants and construct a family of counterexamples to a conjecture of Hibi and Matsuda. 1. Introduction Let G = ([n], E) be a simple undirected graph on the vertex set [n]= {1,...,n}. x1 ··· xn x1 ··· xn Let X = ( y1 ··· yn ) be a generic 2 × n matrix and S = k[ y1 ··· yn ] the polynomial ring whose indeterminates are the entries of X and with coefficients in a field k. The binomial edge ideal of G is JG = hxiyj −yixj : {i, j} ∈ Ei⊆ S, the ideal of 2×2 mi- nors indexed by the edges of the graph. Since their inception in [5, 15], connecting combinatorial properties of G with algebraic properties of JG or S/ JG has been a popular activity. Particular attention has been paid to the minimal free resolution of S/ JG as a standard N-graded S-module [3, 11]. The data of a minimal free res- olution is encoded in its graded Betti numbers βi,j (S/ JG) = dimk Tori(S/ JG, k)j . An interesting invariant is the highest degree appearing in the resolution, the Castelnuovo–Mumford regularity reg(S/ JG) = max{j − i : βij (S/ JG) 6= 0}. It is a complexity measure as low regularity implies favorable properties like vanish- ing of local cohomology. Binomial edge ideals have square-free initial ideals by arXiv:1906.05510v2 [math.AC] 10 Mar 2021 [5, Theorem 2.1] and, using [1], this implies that the extremal Betti numbers and regularity can also be derived from those initial ideals. -
P 4-Colorings and P 4-Bipartite Graphs Chinh T
P_4-Colorings and P_4-Bipartite Graphs Chinh T. Hoàng, van Bang Le To cite this version: Chinh T. Hoàng, van Bang Le. P_4-Colorings and P_4-Bipartite Graphs. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2001, 4 (2), pp.109-122. hal-00958951 HAL Id: hal-00958951 https://hal.inria.fr/hal-00958951 Submitted on 13 Mar 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 4, 2001, 109–122 P4-Free Colorings and P4-Bipartite Graphs Ch´ınh T. Hoang` 1† and Van Bang Le2‡ 1Department of Physics and Computing, Wilfrid Laurier University, 75 University Ave. W., Waterloo, Ontario N2L 3C5, Canada 2Fachbereich Informatik, Universitat¨ Rostock, Albert-Einstein-Straße 21, D-18051 Rostock, Germany received May 19, 1999, revised November 25, 2000, accepted December 15, 2000. A vertex partition of a graph into disjoint subsets Vis is said to be a P4-free coloring if each color class Vi induces a subgraph without a chordless path on four vertices (denoted by P4). Examples of P4-free 2-colorable graphs (also called P4-bipartite graphs) include parity graphs and graphs with “few” P4s like P4-reducible and P4-sparse graphs. -
Characterization and Recognition of P4-Sparse Graphs Partitionable Into K Independent Sets and ℓ Cliques✩
Discrete Applied Mathematics 159 (2011) 165–173 Contents lists available at ScienceDirect Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam Characterization and recognition of P4-sparse graphs partitionable into k independent sets and ` cliquesI Raquel S.F. Bravo a, Sulamita Klein a, Loana Tito Nogueira b, Fábio Protti b,∗ a COPPE-PESC, Universidade Federal do Rio de Janeiro, Brazil b IC, Universidade Federal Fluminense, Brazil article info a b s t r a c t Article history: In this work, we focus on the class of P4-sparse graphs, which generalizes the well-known Received 28 December 2009 class of cographs. We consider the problem of verifying whether a P4-sparse graph is a Received in revised form 22 October 2010 .k; `/-graph, that is, a graph that can be partitioned into k independent sets and ` cliques. Accepted 28 October 2010 First, we describe in detail the family of forbidden induced subgraphs for a cograph to be Available online 27 November 2010 a .k; `/-graph. Next, we show that the same forbidden structures suffice to characterize P -sparse graphs which are .k; `/-graphs. Finally, we describe how to recognize .k; `/-P - Keywords: 4 4 sparse graphs in linear time by using special auxiliary cographs. .k; `/-graphs Cographs ' 2010 Elsevier B.V. All rights reserved. P4-sparse graphs 1. Introduction The class of P4-sparse graphs was introduced by Hoàng [14] as the class of graphs for which every set of five vertices induces at most one P4. Hoàng also gave a number of characterizations for these graphs, and showed that P4-sparse graphs are perfect (a graph G is perfect if for every induced subgraph H of G, the chromatic number of H equals the largest number of pairwise adjacent vertices in H). -
Generalizing Cographs to 2-Cographs
GENERALIZING COGRAPHS TO 2-COGRAPHS JAMES OXLEY AND JAGDEEP SINGH Abstract. In this paper, we consider 2-cographs, a natural general- ization of the well-known class of cographs. We show that, as with cographs, 2-cographs can be recursively defined. However, unlike cographs, 2-cographs are closed under induced minors. We consider the class of non-2-cographs for which every proper induced minor is a 2-cograph and show that this class is infinite. Our main result finds the finitely many members of this class whose complements are also induced-minor-minimal non-2-cographs. 1. Introduction In this paper, we only consider simple graphs. Except where indicated otherwise, our notation and terminology will follow [5]. A cograph is a graph in which every connected induced subgraph has a disconnected com- plement. By convention, the graph K1 is assumed to be a cograph. Re- placing connectedness by 2-connectedness, we define a graph G to be a 2-cograph if G has no induced subgraph H such that both H and its com- plement, H, are 2-connected. Note that K1 is a 2-cograph. Cographs have been extensively studied over the last fifty years (see, for example, [6, 12, 4]). They are also called P4-free graphs due to following characterization [3]. Theorem 1.1. A graph G is a cograph if and only if G does not contain the path P4 on four vertices as an induced subgraph. In other words, P4 is the unique non-cograph with the property that every proper induced subgraph of P4 is a cograph. -
Combinatorial Optimization and Recognition of Graph Classes with Applications to Related Models
Combinatorial Optimization and Recognition of Graph Classes with Applications to Related Models Von der Fakult¨at fur¨ Mathematik, Informatik und Naturwissenschaften der RWTH Aachen University zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften genehmigte Dissertation vorgelegt von Diplom-Mathematiker George B. Mertzios aus Thessaloniki, Griechenland Berichter: Privat Dozent Dr. Walter Unger (Betreuer) Professor Dr. Berthold V¨ocking (Zweitbetreuer) Professor Dr. Dieter Rautenbach Tag der mundlichen¨ Prufung:¨ Montag, den 30. November 2009 Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online verfugbar.¨ Abstract This thesis mainly deals with the structure of some classes of perfect graphs that have been widely investigated, due to both their interesting structure and their numerous applications. By exploiting the structure of these graph classes, we provide solutions to some open problems on them (in both the affirmative and negative), along with some new representation models that enable the design of new efficient algorithms. In particular, we first investigate the classes of interval and proper interval graphs, and especially, path problems on them. These classes of graphs have been extensively studied and they find many applications in several fields and disciplines such as genetics, molecular biology, scheduling, VLSI design, archaeology, and psychology, among others. Although the Hamiltonian path problem is well known to be linearly solvable on interval graphs, the complexity status of the longest path problem, which is the most natural optimization version of the Hamiltonian path problem, was an open question. We present the first polynomial algorithm for this problem with running time O(n4). Furthermore, we introduce a matrix representation for both interval and proper interval graphs, called the Normal Interval Representation (NIR) and the Stair Normal Interval Representation (SNIR) matrix, respectively. -
Arxiv:1305.5974V1 [Math-Ph]
INTRODUCTION TO SPORADIC GROUPS for physicists Luis J. Boya∗ Departamento de F´ısica Te´orica Universidad de Zaragoza E-50009 Zaragoza, SPAIN MSC: 20D08, 20D05, 11F22 PACS numbers: 02.20.a, 02.20.Bb, 11.24.Yb Key words: Finite simple groups, sporadic groups, the Monster group. Juan SANCHO GUIMERA´ In Memoriam Abstract We describe the collection of finite simple groups, with a view on physical applications. We recall first the prime cyclic groups Zp, and the alternating groups Altn>4. After a quick revision of finite fields Fq, q = pf , with p prime, we consider the 16 families of finite simple groups of Lie type. There are also 26 extra “sporadic” groups, which gather in three interconnected “generations” (with 5+7+8 groups) plus the Pariah groups (6). We point out a couple of physical applications, in- cluding constructing the biggest sporadic group, the “Monster” group, with close to 1054 elements from arguments of physics, and also the relation of some Mathieu groups with compactification in string and M-theory. ∗[email protected] arXiv:1305.5974v1 [math-ph] 25 May 2013 1 Contents 1 Introduction 3 1.1 Generaldescriptionofthework . 3 1.2 Initialmathematics............................ 7 2 Generalities about groups 14 2.1 Elementarynotions............................ 14 2.2 Theframeworkorbox .......................... 16 2.3 Subgroups................................. 18 2.4 Morphisms ................................ 22 2.5 Extensions................................. 23 2.6 Familiesoffinitegroups ......................... 24 2.7 Abeliangroups .............................. 27 2.8 Symmetricgroup ............................. 28 3 More advanced group theory 30 3.1 Groupsoperationginspaces. 30 3.2 Representations.............................. 32 3.3 Characters.Fourierseries . 35 3.4 Homological algebra and extension theory . 37 3.5 Groupsuptoorder16.......................... -
Arxiv:2011.00522V2 [Math.CO] 17 May 2021 Correcting the Algorithm
Correcting the algorithm for the secure domination number of cographs by Jha, Pradhan, and Banerjee Anja Kiˇseka∗ Sandi Klavˇzar a,b,c† May 18, 2021 a Faculty of Mathematics and Physics, University of Ljubljana, Slovenia b Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia c Faculty of Natural Sciences and Mathematics, University of Maribor, Slovenia Abstract Jha, Pradhan, and Banerjee devised a linear algorithm to compute the se- cure domination number of a cograph. Here it is shown that their Lemma 2, which is crucial for the computational complexity of the algorithm, is incom- plete. An accordingly modified lemma is proved and it is demonstrated that the complexity of the modified algorithm remains linear. Keywords: domination number; secure domination number; cograph; algorithm; computational complexity AMS Subj. Class. (2020): 05C57, 05C69, 05C76, 05C85 arXiv:2011.00522v2 [math.CO] 17 May 2021 1 Introduction and preliminaries Cographs have been independently introduced in many different contexts and un- der many different names (including D*-graphs, hereditary Dacey graphs, 2-parity graphs), clearly indicating the intrinsic role of this class of graphs in graph theory and elsewhere. Cographs can be characterized in several different ways, see [10]. ∗Email: [email protected] †Email: [email protected] 1 Moreover, their research is still very active, the list [1, 5, 11, 12, 13, 17] presents just a sample of recent studies and applications of cographs. Numerous problems, including the independence number, the clique number, and the existence of a Hamiltonian cycle, that are difficult in general, are polynomial (in most cases linear) on cographs. -
Construction of the Rudvalis Group of Order 145,926,144,000*
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA 27, 538-548 (1973) Construction of the Rudvalis Group of Order 145,926,144,000* J. H. CONWAY Department of Pure Mathematics and Mathematical Statistics, Cambridge, England AND D. B. WALES Department of Mathematics, California Institute of Technology, Pasadena, California 91109 Communicated by Marshall Hall, Jr. Received August 4, 1972 Recently, Arunas Rudvalis [l] provided evidence for the existence of a new simple group R of order 145,926,144,000 = 2r4 * 33 * 5a . 7 * 13 . 29. He describes the group as a rank 3 permutation group on 4060 letters, in which the stabilizer of a point is the (nonsimple) Ree group F = V’,(2), which has orbits of sizes 1,1755,2304, corresponding to subgroups of F of orders 20480 and 15600. The first of these is the centralizer of an involution of F, while the second has the known subgroup L = PSLa(25) of P’ (see Ref. [4]) as a subgroup of index 2. Since one of the involutions in R has no fixed point and so just 2030 2-cycles, an argument of Griess and Schur [2] shows that R has a proper double cover 2R. Rudvalis and Frame gave evidence for supposing that this group had a 28-dimensional complex representation not splitting over F. (Feit and Lyons have proved this under further assumptions on R.) We sup- pose that this unitary representation of 2R exists while constructing the group, but then conclude independently of this supposition that the con- struction defines a group 2R whose central quotient is R.