DOCSLIB.ORG

Partitioning Cographs Into Cliques and Stable Sets

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Partitioning Cographs Into Cliques and Stable Sets View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Optimization 2 (2005) 145–153 www.elsevier.com/locate/disopt Partitioning cographs into cliques and stable sets Marc Demangea, Tınaz Ekimb,∗,1, Dominique de Werrab aESSEC, Département SID, France bEPFL, Recherche Opérationnelle Sud Est (ROSE), 1015Lausanne, Switzerland Received 10 May 2004; received in revised form 30 November 2004; accepted 8 March2005 Available online 5 July 2005 Abstract We consider the problem of partitioning the node set of a graph into p cliques and k stable sets, namely the (p, k)-coloring problem. Results have been obtained for general graphs [Feder et al., SIAM J. Discrete Math. 16 (3) (2003) 449–478], chordal graphs [Hell et al., Discrete Appl. Math. 141 (2004) 185–194] and cacti for the case where k = p in [Ekim and de Werra, On split-coloring problems, submitted for publication] where some upper and lower bounds on the optimal value minimizing k are also presented. We focus on cographs and devise some efficient algorithms for solving (p, k)-coloring problems and cocoloring / problems in O(n2 + nm) time and O(n3 2) time, respectively. We also give an algorithm for finding the maximum induced (p, k)-colorable subgraph. In addition to this, we present characterizations of (2, 1)- and (2, 2)-colorable cographs by forbidden configurations. © 2005 Elsevier B.V. All rights reserved. Keywords: (p, k)-coloring; Cocoloring; Split-coloring; Cographs 1. Introduction Partitioning the node set of a graph into two kinds of subsets (like for instance cliques and stable sets) is a natural extension of the classical coloring problem which has therefore been a topic of extensive research in the recent years (see [1,2,5,6,10,11]). Choosing subsets of nodes that induce cliques and stable sets has in addition the nice property of complementarity: the problem is basically the same whether we consider the graph G or its complement. In a more general context, Feder et al. [6] have considered so-called dense graphs and sparse graphs (instead of collection of cliques and collection of stable sets) and they have studied the problem of partitioning the node set of a graph G into a dense graphand a sparse graph.Complexity results are also given in [6]. Along the same line, Hell et al. [10] consider the problem of partitioning the node set into p cliques and k stable sets and examine in particular the case of chordal graphs. These are the graphs in which every cycle of length at least 4 contains a chord (an edge linking two nonadjacent nodes of a cycle). They devise polynomial time algorithms using a perfect elimination order defined on a chordal graph for solving the above mentioned partitioning problems. As a special case when k = p, the problem consists in partitioning the node set into k split graphs (these are graphs in which the node set is the union of a clique and a stable set). It has been studied in [5]. ∗ Corresponding author. Tel.: +41 21 693 25 68; fax: +41 21 693 58 40. E-mail address: tinaz.ekim@epfl.ch (T. Ekim). 1 The research of Tınaz Ekim was supported by Grant 200021-101455/1 of the Swiss National Science Foundation whose support is greatly acknowledged. 1572-5286/$ - see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.disopt.2005.03.003 146 M. Demange et al. / Discrete Optimization 2 (2005) 145–153 In this paper we intend to discuss the problem of partitioning the node set of a graph into p cliques and k stable sets; we call (p, k)-coloring problems all different versions of this partitioning problem. We shall first review some basic results concerning suchpartitioning in general graphs [6] and after that we shall derive and exploit some consequences. We will consider cographs also called P4-free graphs since they are defined as graphs without induced chain on four nodes, i.e., without P4. These are special cases of permutation graphs (graphs G for which G and its complement have a transitive orientation). We shall give a new characterization of cographs which lead to efficient algorithms for all (p, k)-coloring problems. General remarks on these (p, k)-coloring problems will also be given as well as some avenues for further research. Given a graph G = (V, E), |V |=n and |E|=m. We denote by G the complement of G.ForV ⊂ V , G[V ] denotes the subgraphof G induced by V . A graph G=(V, E) is (p, k)-colorable if V can be partitioned into p cliques and k stable sets of G: p k i j V = (K1,...,K ,S1,...,S ) where K , i = 1,...,pis a clique and S , j = 1,...,kis a stable set. Sucha partition is called a (p, k)-coloring of G.Ifp = k, then it is called a k-split-coloring of G [5]. Another problem, called cocoloring [9,7], consists in minimizing the total number of cliques and/or stable sets covering all the nodes of a graph. In more general (p, k)-coloring problems, we minimize k (resp. p)forafixedp (resp. k). We denote by KpSk the class of (p, k)-colorable graphs. Finally, given a class C of graphs, C ∩ KpSk is said to be polynomially determined if there exists a polynomial-time algorithm computing, for every graph G ∈ C,a(p, k)-coloring of G whenever G ∈ KpSk or recognizing that G/∈ KpSk. Given a graph G, we denote by p,k(G) the maximum cardinality (number of nodes) of a (p, k)-colorable subgraphof G.By convention, 0,0(G) = 0. Note that 1,1(G) is called the split-independence number of G and denoted by S(G) in [5]. A class G of graphs is called hereditary if every node-induced subgraphof a graph G ∈ G also belongs to G. 2. (p, k)-coloring general graphs In this section, we focus on (p, k)-colorings of general graphs. K0S1 (stable sets) and K1S0 (cliques) are trivially poly- nomially determined as well as K0S2 (bipartite graphs) and K2S0 (co-bipartite graphs). Let us also note that a graph G is (p, k)-colorable if and only if its complement G is (k, p)-colorable; consequently for every class of graphs C closed under complementation, KpSk ∩ C is polynomially determined if and only if KkSp ∩ C is polynomially determined. The following theorem is proved in [6]: Theorem 1 (Feder et al. [6]). Let G be a hereditary class of graphs, let (p, k) ∈ N×N, k 1. If K0Sk ∩G and KpS0 ∩G are polynomially determined, then KpSk ∩G is polynomially determined. Moreover, if KpS0 ∩G (resp. K0Sk ∩G) is determined kp+ in time Tp(n) (resp. Tk(n)), then KpSk ∩ G can be determined in time O((n2 2) max(Tp(n), Tk(n))). Corollary 2. If G is a subclass of perfect graphs, then KpSk ∩ G is polynomially determined. This follows also from the fact that hereditary families of perfect (p, k)-colorable graphs have a finite forbidden subgraph characterization (and hence a polynomial time recognition algorithm) [11]. For instance, KpSk ∩ G can be determined in the kp+ class of cographs in time O(mn2 2), where m denotes the number of edges and n the number of nodes. A special case of Theorem 1 is the test of (i, j)-colorability with i, j 2; note that the cases (0, 2) and (2, 0) reduce to testing for the existence of a node 2-coloring. Corollary 3 (Brandstädt [1] and Brandstädt et al. [2]). Since KiS0 and K0Sj are polynomially determined for i, j 2, K2S2 is also polynomially determined. Let us note that the condition “G is hereditary” in Theorem 1 is needed unless NP = P. In fact, let us consider the class G of graphs obtained by adding to any graph G a 4-clique K4 (a clique of size 4) without any edge between G and this clique. G∩K0S3 is trivially polynomially determined since suchgraphsare never (0, 3)-colorable. On the other hand, G∩K1S0 is also polynomially determined (recognition of a clique). But it is NP-complete to decide whether a graph of G is (1, 3)-colorable. In fact a graph G is (0, 3)-colorable if and only if the graph obtained by adding to G a 4-clique not connected to G is (1, 3)-colorable. Let us also point out that recognizing a (1, 3)-coloring remains difficult even if a maximum induced (p, k)-colorable subgraph with (p, k) equal to (1, 0), (0, 1), (0, 2) and (0, 3) can be polynomially computed: Proposition 4. There exists a class of graphs G such that a maximum clique, a maximum stable set, a maximum induced bipartite and also a maximum 3-colorable induced subgraph can be polynomially computed on G while it is NP-complete to decide whether a graph in G is (1, 3)-colorable. M. Demange et al. / Discrete Optimization 2 (2005) 145–153 147 Proof. Let us revisit the proof of NP-completeness of graph 3-colorability by reduction to 3-SAT [8]. Given an instance of 3-SAT with p clauses (C1,...,Cp) and n boolean variables (x1,...,xn), one constructs a graph G = (V, E) defined by V = V1 ∪ V2 ∪ V3, V ={R,T,F} V = n {x , x¯ } V = p {yj ,yj ,yj ,yj ,yj } where 1 , 2 i=1 i i and 3 j=1 1 2 3 4 5 . V contains two nodes per literal, five nodes per clause and three color nodes R,T,F (standing for Red, True and False).
Load more

Recommended publications
  • A Fast and Scalable Graph Coloring Algorithm for Multi-Core and Many-Core Architectures
    A Fast and Scalable Graph Coloring Algorithm for Multi-core and Many-core Architectures Georgios Rokos1, Gerard Gorman2, and Paul H J Kelly1 1 Software Peroformance Optimisation Group, Department of Computing 2 Applied Modelling and Computation Group, Department of Earth Science and Engineering Imperial College London, South Kensington Campus, London SW7 2AZ, United Kingdom, fgeorgios.rokos09,g.gorman,[email protected] Abstract. Irregular computations on unstructured data are an impor- tant class of problems for parallel programming. Graph coloring is often an important preprocessing step, e.g. as a way to perform dependency analysis for safe parallel execution. The total run time of a coloring algo- rithm adds to the overall parallel overhead of the application whereas the number of colors used determines the amount of exposed parallelism. A fast and scalable coloring algorithm using as few colors as possible is vi- tal for the overall parallel performance and scalability of many irregular applications that depend upon runtime dependency analysis. C¸ataly¨ureket al. have proposed a graph coloring algorithm which relies on speculative, local assignment of colors. In this paper we present an improved version which runs even more optimistically with less thread synchronization and reduced number of conflicts compared to C¸ataly¨urek et al.'s algorithm. We show that the new technique scales better on multi- core and many-core systems and performs up to 1.5x faster than its pre- decessor on graphs with high-degree vertices, while keeping the number of colors at the same near-optimal levels. Keywords: Graph Coloring; Greedy Coloring; First-Fit Coloring; Ir- regular Data; Parallel Graph Algorithms; Shared-Memory Parallelism; Optimistic Execution; Many-core Architectures; Intel®Xeon Phi™ 1 Introduction Many modern applications are built around algorithms which operate on irreg- ular data structures, usually in form of graphs.
    [Show full text]
  • I?'!'-Comparability Graphs
    Discrete Mathematics 74 (1989) 173-200 173 North-Holland I?‘!‘-COMPARABILITYGRAPHS C.T. HOANG Department of Computer Science, Rutgers University, New Brunswick, NJ 08903, U.S.A. and B.A. REED Discrete Mathematics Group, Bell Communications Research, Morristown, NJ 07950, U.S.A. In 1981, Chv&tal defined the class of perfectly orderable graphs. This class of perfect graphs contains the comparability graphs. In this paper, we introduce a new class of perfectly orderable graphs, the &comparability graphs. This class generalizes comparability graphs in a natural way. We also prove a decomposition theorem which leads to a structural characteriza- tion of &comparability graphs. Using this characterization, we develop a polynomial-time recognition algorithm and polynomial-time algorithms for the clique and colouring problems for &comparability graphs. 1. Introduction A natural way to color the vertices of a graph is to put them in some order v*cv~<” l < v, and then assign colors in the following manner: (1) Consider the vertxes sequentially (in the sequence given by the order) (2) Assign to each Vi the smallest color (positive integer) not used on any neighbor vi of Vi with j c i. We shall call this the greedy coloring algorithm. An order < of a graph G is perfect if for each subgraph H of G, the greedy algorithm applied to H, gives an optimal coloring of H. An obstruction in a ordered graph (G, <) is a set of four vertices {a, b, c, d} with edges ab, bc, cd (and no other edge) and a c b, d CC. Clearly, if < is a perfect order on G, then (G, C) contains no obstruction (because the chromatic number of an obstruction is twu, but the greedy coloring algorithm uses three colors).
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Finding Articulation Points and Bridges Articulation Points Articulation Point
    Finding Articulation Points and Bridges Articulation Points Articulation Point Articulation Point A vertex v is an articulation point (also called cut vertex) if removing v increases the number of connected components. A graph with two articulation points. 3 / 1 Articulation Points Given I An undirected, connected graph G = (V; E) I A DFS-tree T with the root r Lemma A DFS on an undirected graph does not produce any cross edges. Conclusion I If a descendant u of a vertex v is adjacent to a vertex w, then w is a descendant or ancestor of v. 4 / 1 Removing a Vertex v Assume, we remove a vertex v 6= r from the graph. Case 1: v is an articulation point. I There is a descendant u of v which is no longer reachable from r. I Thus, there is no edge from the tree containing u to the tree containing r. Case 2: v is not an articulation point. I All descendants of v are still reachable from r. I Thus, for each descendant u, there is an edge connecting the tree containing u with the tree containing r. 5 / 1 Removing a Vertex v Problem I v might have multiple subtrees, some adjacent to ancestors of v, and some not adjacent. Observation I A subtree is not split further (we only remove v). Theorem A vertex v is articulation point if and only if v has a child u such that neither u nor any of u's descendants are adjacent to an ancestor of v. Question I How do we determine this efficiently for all vertices? 6 / 1 Detecting Descendant-Ancestor Adjacency Lowpoint The lowpoint low(v) of a vertex v is the lowest depth of a vertex which is adjacent to v or a descendant of v.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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).
    [Show full text]
  • 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.
    [Show full text]
  • COLORING and DEGENERACY for DETERMINING VERY LARGE and SPARSE DERIVATIVE MATRICES ASHRAFUL HUQ SUNY Bachelor of Science, Chittag
    COLORING AND DEGENERACY FOR DETERMINING VERY LARGE AND SPARSE DERIVATIVE MATRICES ASHRAFUL HUQ SUNY Bachelor of Science, Chittagong University of Engineering & Technology, 2013 A Thesis Submitted to the School of Graduate Studies of the University of Lethbridge in Partial Fulfillment of the Requirements for the Degree MASTER OF SCIENCE Department of Mathematics and Computer Science University of Lethbridge LETHBRIDGE, ALBERTA, CANADA c Ashraful Huq Suny, 2016 COLORING AND DEGENERACY FOR DETERMINING VERY LARGE AND SPARSE DERIVATIVE MATRICES ASHRAFUL HUQ SUNY Date of Defence: December 19, 2016 Dr. Shahadat Hossain Supervisor Professor Ph.D. Dr. Daya Gaur Committee Member Professor Ph.D. Dr. Robert Benkoczi Committee Member Associate Professor Ph.D. Dr. Howard Cheng Chair, Thesis Examination Com- Associate Professor Ph.D. mittee Dedication To My Parents. iii Abstract Estimation of large sparse Jacobian matrix is a prerequisite for many scientific and engi- neering problems. It is known that determining the nonzero entries of a sparse matrix can be modeled as a graph coloring problem. To find out the optimal partitioning, we have proposed a new algorithm that combines existing exact and heuristic algorithms. We have introduced degeneracy and maximum k-core for sparse matrices to solve the problem in stages. Our combined approach produce better results in terms of partitioning than DSJM and for some test instances, we report optimal partitioning for the first time. iv Acknowledgments I would like to express my deep acknowledgment and profound sense of gratitude to my supervisor Dr. Shahadat Hossain, for his continuous guidance, support, cooperation, and persistent encouragement throughout the journey of my MSc program.
    [Show full text]
  • Graph Coloring Major Themes Vertex Colorings Χ(G) =
    CSC 1300 – Discrete Structures 17: Graph Coloring Major Themes • Vertex coloring • Chromac number χ(G) Graph Coloring • Map coloring • Greedy coloring algorithm • CSC 1300 – Discrete Structures Applicaons Villanova University Villanova CSC 1300 - Dr Papalaskari 1 Villanova CSC 1300 - Dr Papalaskari 2 What is the least number of colors needed for the verFces of Vertex Colorings this graph so that no two adjacent verFces have the same color? Adjacent ver,ces cannot have the same color χ(G) = Chromac number χ(G) = least number of colors needed to color the verFces of a graph so that no two adjacent verFces are assigned the same color? Source: “Discrete Mathemacs” by Chartrand & Zhang, 2011, Waveland Press. Source: “Discrete Mathemacs with Ducks” by Sara-Marie Belcastro, 2012, CRC Press, Fig 13.1. 4 5 Dr Papalaskari 1 CSC 1300 – Discrete Structures 17: Graph Coloring Map Coloring Map Coloring What is the least number of colors needed to color a map? Region è vertex Common border è edge G A D C E B F G H I Villanova CSC 1300 - Dr Papalaskari 8 Villanova CSC 1300 - Dr Papalaskari 9 Coloring the USA Four color theorem Every planar graph is 4-colorable The proof of this theorem is one of the most famous and controversial proofs in mathemacs, because it relies on a computer program. It was first presented in 1976. A more recent reformulaon can be found in this arFcle: Formal Proof – The Four Color Theorem, Georges Gonthier, NoFces of the American Mathemacal Society, December 2008. hp://www.printco.com/pages/State%20Map%20Requirements/USA-colored-12-x-8.gif
    [Show full text]