DOCSLIB.ORG

The Clique Operator on Cographs and Serial Graphs

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Clique Operator on Cographs and Serial Graphs View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Mathematics 282 (2004) 183–191 www.elsevier.com/locate/disc The clique operator on cographs and serial graphs F. LarriÃona, C.P. de Mellob;1, A. Morganac, V. Neumann-Laraa, M.A. Piza˜nad aInstituto de Matematicas,à U.N.A.M., Mexico bInstituto de Computacao,˜ UNICAMP, Brazil cDipartimento di Matematica, Universita di Roma “La Sapienza”, Italy dDepartamento de IngenierÃ(a Electrica,à UAM-I, Mexico Received 20 September 2001; received in revised form 24 March 2003; accepted 23 October 2003 Dedicated to the memory of our beloved friend VÃ6ctor Neumann-Lara who passed away on 26 February 2004. Abstract The clique graph of a graph G is the intersection graph K(G) of the (maximal) cliques of G. The iterated clique graphs K n(G) are deÿned by K 0(G)=G and K i(G)=K(K i−1(G)), i¿0 and K is the clique operator. A cograph is a graph with no induced subgraph isomorphic to P4. In this article we use the modular decomposition technique to characterize the K-behaviour of cographs and to give some partial results for the larger class of serial (i.e. complement-disconnected) graphs. We prove that a cograph is K-convergent if and only if it is clique-Helly. This characterization leads to a polynomial time algorithm for deciding the K-convergence or K-divergence of any cograph. c 2003 Elsevier B.V. All rights reserved. Keywords: Iterated clique graphs; Clique divergence; Modular decompositions; Cographs 1. Introduction The clique graph of a graph G is the intersection graph K(G) of the (maximal) cliques of G. The iterated clique graphs K n(G) are deÿned by K 0(G)=G and K i(G)=K(K i−1(G)), i¿0. We refer to [27,29] for the literature on iterated clique graphs. Graphs behave in a variety of ways under the iterates of the clique operator K, the main distinction being n between K-convergence and K-divergence. A graph G is said to be K-null if K (G) is the trivial graph K1 for some n. We say that G is K-periodic if K n(G) ∼= G for some n ¿ 1; the smallest such n is the period of G. More generally, a graph G is said to be K-convergent if K n(G) ∼= K m(G) for some pair of non-negative integers n¡m.Ifn and m are the smallest such integers, we say that n is the transition index and m − n is the period of G. A graph G is K-convergent if and only if G is, in the obvious sense, eventually K-periodic. This last is equivalent to the boundedness of the sequence n n of the orders |V (K (G))|. A graph G is said to be K-divergent if limn→∞ |V (K (G))| = ∞. The graph G is K-divergent if and only if it is not K-convergent. A graph is clique-Helly if its (maximal) cliques satisfy the Helly property: each family of mutually intersecting cliques has non-trivial intersection. For instance, all triangleless graphs are clique-Helly. Clique-Helly graphs have been introduced in [9,11] and studied in [20,25,26], among others. Clique-Helly graphs are always K-convergent: indeed, they are all eventually K-periodic of period 1 or 2 [9]. Clique-Helly graphs can be recognized in polynomial time [30]. So far, most of the results on convergence of iterated clique graphs are on the domain of clique-Helly graphs. In particular, several classes of clique-Helly graphs have been shown to be K-null, as is the case of interval graphs [13]. More generally, 1 Partially supported by CNPq, FAEP and PRONEX/CNPq (664107/1997-4). E-mail addresses: [email protected] (F. LarriÃon), [email protected] (C.P. de Mello), [email protected] (A. Morgana), [email protected] (V. Neumann-Lara), [email protected] (M.A. Piza˜na). 0012-365X/$ - see front matter c 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.disc.2003.10.023 184 F. Larrionà et al. / Discrete Mathematics 282 (2004) 183–191 Bandelt and Prisner characterized the clique-Helly graphs which are K-null [2]. In general, much less is known about clique convergence, when non-clique-Helly graphs are considered. Some results on convergence of graphs which are not clique-Helly can be found in [2,3,6]. All the graphs which will be shown to be K-convergent in this paper are clique-Helly. The ÿrst examples of K-divergent graphs were given by Neumann–Lara (see [9,23]). For n ¿ 2, deÿne the n-dimensional octahedron On as the complement of a perfect matching on 2n vertices. Then On is the complete multipartite graph K2;2;:::;2. ∼ Neumann–Lara showed that K(On) = O2n−1 and hence, for n ¿ 3, On is K-divergent with superexponential growth. Moreover, he showed that all complete multipartite graphs Kp1;:::;pn , with n ¿ 3 and pi ¿ 2; 1 6 i 6 n, are K-divergent with superexponential growth. This last result completed the determination of the behaviour under K (or K-behaviour) of the class CM of complete multipartite graphs. In fact, the rest of the graphs in CM were previously known to be K-convergent: the complete graphs Kn are in CM and K(Kn) is trivial. The other graphs in CM that have a universal 2 vertex are those of the form G = K1;p2;:::;pn = Kn and here K(G) is non-trivial but K (G) is trivial. The remaining graphs in CM are those which have exactly two parts and at least two vertices in each part; they are triangleless and without degree one vertices, so by Hedetniemi and Stater [12] they are K-periodic of period one (only for the square K2;2 = C4) or two (all the rest). In [24] it has been asked whether there are divergent graphs with polynomial growth. Recently, aLrmative answers have been given in [16–18]. The graphs that will be shown to be K-divergent in this work have all superexponential growth. Given a class C of graphs, denote by K(C) the class whose members are the clique graphs of the members of C.A class C is closed when its clique class K(C) satisÿes K(C) ⊆ C and ÿxed when K(C)=C. Escalante [9] showed that the class of clique-Helly graphs is ÿxed. After this, some other ÿxed classes have been identiÿed: indiMerence graphs [13], dismantable graphs [2], strongly chordal graphs [2,4], among others. The clique graph of an interval graph is an indiMerence graph [13], hence the class of interval graphs is closed. The class of chordal graphs is not closed, but its clique class is known [4,10,31]. The class of cographs is not closed. In this article, we characterize the cographs whose clique graph is also a cograph. Modular decompositions (which will be reviewed in Section 3) play an important role in this paper. The class of cographs properly contains the class of complete multipartite graphs, and it is known to be characterized by the absence of neighbourhood nodes in the modular decomposition tree (MDT). The wider class of serial graphs is deÿned by the fact that they are connected, non-trivial, and the root of the MDT is not neighbourhood, i.e. the graphs whose complement is disconnected. In Section 4 we will characterize the serial graphs which are clique-Helly (a suLcient condition for K-convergence). An important class of serial graphs is that of parallel-decomposable serial graphs: for these, the MDT does not contain neither leaves nor neighbourhood modules in the root and the ÿrst level. In Section 5 we will give some suLcient conditions for K-divergence for such graphs. In Section 6 we shall determine completely the K-behaviour for the class of cographs. An interesting result is that a cograph is K-convergent if and only if it is clique-Helly. Finally, in Section 7 we characterize the cographs whose clique graph is also a cograph and, in Section 8, we provide some conclusions. 2. Preliminaries and deÿnitions We consider simple, undirected, ÿnite graphs. The sets V (G) and E(G) are the vertex and edge sets of a graph G. A trivial graph is a graph with a single vertex. A complete is a set of pairwise adjacent vertices in G.Aclique of G is a complete not properly contained in any other complete. A subgraph of G is a graph H with V (H) ⊆ V (G) and E(H) ⊆ E(G). For X ⊆ V (G), we denote by G[X ] the subgraph induced by X , that is, V (G[X ]) = X and E(G[X ]) consists of those edges of E(G) having both ends in X . We often identify induced subgraphs with their vertex sets. If v is a vertex of a subgraph H of G adjacent to every other vertex of H, then we say that v is universal in H, or that H is a cone with apex v. Let X be a subset of V (G) and x any vertex of X . The quotient graph G=X is deÿned as V (G=X )=(V (G) − X ) ∪{x} and E(G=X )=E(G[V (G) − X ]) ∪{{x; v}|{u; v}∈E(G);u∈ X; v ∈ V (G) − X }. Let H and H be vertex-disjoint graphs.
Load more

Recommended publications
  • On Treewidth and Graph Minors
    On Treewidth and Graph Minors Daniel John Harvey Submitted in total fulfilment of the requirements of the degree of Doctor of Philosophy February 2014 Department of Mathematics and Statistics The University of Melbourne Produced on archival quality paper ii Abstract Both treewidth and the Hadwiger number are key graph parameters in structural and al- gorithmic graph theory, especially in the theory of graph minors. For example, treewidth demarcates the two major cases of the Robertson and Seymour proof of Wagner's Con- jecture. Also, the Hadwiger number is the key measure of the structural complexity of a graph. In this thesis, we shall investigate these parameters on some interesting classes of graphs. The treewidth of a graph defines, in some sense, how \tree-like" the graph is. Treewidth is a key parameter in the algorithmic field of fixed-parameter tractability. In particular, on classes of bounded treewidth, certain NP-Hard problems can be solved in polynomial time. In structural graph theory, treewidth is of key interest due to its part in the stronger form of Robertson and Seymour's Graph Minor Structure Theorem. A key fact is that the treewidth of a graph is tied to the size of its largest grid minor. In fact, treewidth is tied to a large number of other graph structural parameters, which this thesis thoroughly investigates. In doing so, some of the tying functions between these results are improved. This thesis also determines exactly the treewidth of the line graph of a complete graph. This is a critical example in a recent paper of Marx, and improves on a recent result by Grohe and Marx.
    [Show full text]
  • The AVD-Edge-Coloring Conjecture for Some Split Graphs
    Matem´atica Contempor^anea, Vol. 44,1{10 c 2015, Sociedade Brasileira de Matem´atica The AVD-edge-coloring conjecture for some split graphs Alo´ısiode Menezes Vilas-B^oas C´eliaPicinin de Mello Abstract Let G be a simple graph. An adjacent vertex distinguishing edge- coloring (AVD-edge-coloring) of G is an edge-coloring of G such that for each pair of adjacent vertices u; v of G, the set of colors assigned to the edges incident with u differs from the set of colors assigned to the edges incident with v. The adjacent vertex distinguishing 0 chromatic index of G, denoted χa(G), is the minimum number of colors required to produce an AVD-edge-coloring for G. The AVD- edge-coloring conjecture states that every simple connected graph G ∼ 0 with at least three vertices and G =6 C5 has χa(G) ≤ ∆(G) + 2. The conjecture is open for arbitrary graphs, but it holds for some classes of graphs. In this note we focus on split graphs. We prove this AVD-edge- coloring conjecture for split-complete graphs and split-indifference graphs. 1 Introduction In this paper, G denotes a simple, undirected, finite, connected graph. The sets V (G) and E(G) are the vertex and edge sets of G. Let u; v 2 2000 AMS Subject Classification: 05C15. Key Words and Phrases: edge-coloring, adjacent strong edge-coloring, split graph. Supported by CNPq (132194/2010-4 and 308314/2013-1). The AVD-edge-coloring conjecture for some split graphs 2 V (G). We denote an edge by uv.A clique is a set of vertices pairwise adjacent in G and a stable set is a set of vertices such that no two of which are adjacent.
    [Show full text]
  • Graph Varieties Axiomatized by Semimedial, Medial, and Some Other Groupoid Identities
    Discussiones Mathematicae General Algebra and Applications 40 (2020) 143–157 doi:10.7151/dmgaa.1344 GRAPH VARIETIES AXIOMATIZED BY SEMIMEDIAL, MEDIAL, AND SOME OTHER GROUPOID IDENTITIES Erkko Lehtonen Technische Universit¨at Dresden Institut f¨ur Algebra 01062 Dresden, Germany e-mail: [email protected] and Chaowat Manyuen Department of Mathematics, Faculty of Science Khon Kaen University Khon Kaen 40002, Thailand e-mail: [email protected] Abstract Directed graphs without multiple edges can be represented as algebras of type (2, 0), so-called graph algebras. A graph is said to satisfy an identity if the corresponding graph algebra does, and the set of all graphs satisfying a set of identities is called a graph variety. We describe the graph varieties axiomatized by certain groupoid identities (medial, semimedial, autodis- tributive, commutative, idempotent, unipotent, zeropotent, alternative). Keywords: graph algebra, groupoid, identities, semimediality, mediality. 2010 Mathematics Subject Classification: 05C25, 03C05. 1. Introduction Graph algebras were introduced by Shallon [10] in 1979 with the purpose of providing examples of nonfinitely based finite algebras. Let us briefly recall this concept. Given a directed graph G = (V, E) without multiple edges, the graph algebra associated with G is the algebra A(G) = (V ∪ {∞}, ◦, ∞) of type (2, 0), 144 E. Lehtonen and C. Manyuen where ∞ is an element not belonging to V and the binary operation ◦ is defined by the rule u, if (u, v) ∈ E, u ◦ v := (∞, otherwise, for all u, v ∈ V ∪ {∞}. We will denote the product u ◦ v simply by juxtaposition uv. Using this representation, we may view any algebraic property of a graph algebra as a property of the graph with which it is associated.
    [Show full text]
  • Counting Independent Sets in Graphs with Bounded Bipartite Pathwidth∗
    Counting independent sets in graphs with bounded bipartite pathwidth∗ Martin Dyery Catherine Greenhillz School of Computing School of Mathematics and Statistics University of Leeds UNSW Sydney, NSW 2052 Leeds LS2 9JT, UK Australia [email protected] [email protected] Haiko M¨uller∗ School of Computing University of Leeds Leeds LS2 9JT, UK [email protected] 7 August 2019 Abstract We show that a simple Markov chain, the Glauber dynamics, can efficiently sample independent sets almost uniformly at random in polynomial time for graphs in a certain class. The class is determined by boundedness of a new graph parameter called bipartite pathwidth. This result, which we prove for the more general hardcore distribution with fugacity λ, can be viewed as a strong generalisation of Jerrum and Sinclair's work on approximately counting matchings, that is, independent sets in line graphs. The class of graphs with bounded bipartite pathwidth includes claw-free graphs, which generalise line graphs. We consider two further generalisations of claw-free graphs and prove that these classes have bounded bipartite pathwidth. We also show how to extend all our results to polynomially-bounded vertex weights. 1 Introduction There is a well-known bijection between matchings of a graph G and independent sets in the line graph of G. We will show that we can approximate the number of independent sets ∗A preliminary version of this paper appeared as [19]. yResearch supported by EPSRC grant EP/S016562/1 \Sampling in hereditary classes". zResearch supported by Australian Research Council grant DP190100977. 1 in graphs for which all bipartite induced subgraphs are well structured, in a sense that we will define precisely.
    [Show full text]
  • Forbidding Subgraphs
    Graph Theory and Additive Combinatorics Lecturer: Prof. Yufei Zhao 2 Forbidding subgraphs 2.1 Mantel’s theorem: forbidding a triangle We begin our discussion of extremal graph theory with the following basic question. Question 2.1. What is the maximum number of edges in an n-vertex graph that does not contain a triangle? Bipartite graphs are always triangle-free. A complete bipartite graph, where the vertex set is split equally into two parts (or differing by one vertex, in case n is odd), has n2/4 edges. Mantel’s theorem states that we cannot obtain a better bound: Theorem 2.2 (Mantel). Every triangle-free graph on n vertices has at W. Mantel, "Problem 28 (Solution by H. most bn2/4c edges. Gouwentak, W. Mantel, J. Teixeira de Mattes, F. Schuh and W. A. Wythoff). Wiskundige Opgaven 10, 60 —61, 1907. We will give two proofs of Theorem 2.2. Proof 1. G = (V E) n m Let , a triangle-free graph with vertices and x edges. Observe that for distinct x, y 2 V such that xy 2 E, x and y N(x) must not share neighbors by triangle-freeness. Therefore, d(x) + d(y) ≤ n, which implies that d(x)2 = (d(x) + d(y)) ≤ mn. ∑ ∑ N(y) x2V xy2E y On the other hand, by the handshake lemma, ∑x2V d(x) = 2m. Now by the Cauchy–Schwarz inequality and the equation above, Adjacent vertices have disjoint neigh- borhoods in a triangle-free graph. !2 ! 4m2 = ∑ d(x) ≤ n ∑ d(x)2 ≤ mn2; x2V x2V hence m ≤ n2/4.
    [Show full text]
  • The Labeled Perfect Matching in Bipartite Graphs Jérôme Monnot
    The Labeled perfect matching in bipartite graphs Jérôme Monnot To cite this version: Jérôme Monnot. The Labeled perfect matching in bipartite graphs. Information Processing Letters, Elsevier, 2005, to appear, pp.1-9. hal-00007798 HAL Id: hal-00007798 https://hal.archives-ouvertes.fr/hal-00007798 Submitted on 5 Aug 2005 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. The Labeled perfect matching in bipartite graphs J´erˆome Monnot∗ July 4, 2005 Abstract In this paper, we deal with both the complexity and the approximability of the labeled perfect matching problem in bipartite graphs. Given a simple graph G = (V, E) with |V | = 2n vertices such that E contains a perfect matching (of size n), together with a color (or label) function L : E → {c1, . , cq}, the labeled perfect matching problem consists in finding a perfect matching on G that uses a minimum or a maximum number of colors. Keywords: labeled matching; bipartite graphs; NP-complete; approximate algorithms. 1 Introduction Let Π be a NPO problem accepting simple graphs G = (V,E) as instances, edge-subsets E′ ⊆ E verifying a given polynomial-time decidable property Pred as solutions, and the solutions cardinality as objective function; the labeled problem associated to Π, denoted by Labeled Π, seeks, given an instance I = (G, L) where G = (V,E) is a simple graph ′ and L is a mapping from E to {c1,...,cq}, in finding a subset E verifying Pred that optimizes the size of the set L(E′)= {L(e): e ∈ E′}.
    [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]
  • Arxiv:2006.06067V2 [Math.CO] 4 Jul 2021
    Treewidth versus clique number. I. Graph classes with a forbidden structure∗† Cl´ement Dallard1 Martin Milaniˇc1 Kenny Storgelˇ 2 1 FAMNIT and IAM, University of Primorska, Koper, Slovenia 2 Faculty of Information Studies, Novo mesto, Slovenia [email protected] [email protected] [email protected] Treewidth is an important graph invariant, relevant for both structural and algo- rithmic reasons. A necessary condition for a graph class to have bounded treewidth is the absence of large cliques. We study graph classes closed under taking induced subgraphs in which this condition is also sufficient, which we call (tw,ω)-bounded. Such graph classes are known to have useful algorithmic applications related to variants of the clique and k-coloring problems. We consider six well-known graph containment relations: the minor, topological minor, subgraph, induced minor, in- duced topological minor, and induced subgraph relations. For each of them, we give a complete characterization of the graphs H for which the class of graphs excluding H is (tw,ω)-bounded. Our results yield an infinite family of χ-bounded induced-minor-closed graph classes and imply that the class of 1-perfectly orientable graphs is (tw,ω)-bounded, leading to linear-time algorithms for k-coloring 1-perfectly orientable graphs for every fixed k. This answers a question of Breˇsar, Hartinger, Kos, and Milaniˇc from 2018, and one of Beisegel, Chudnovsky, Gurvich, Milaniˇc, and Servatius from 2019, respectively. We also reveal some further algorithmic implications of (tw,ω)- boundedness related to list k-coloring and clique problems. In addition, we propose a question about the complexity of the Maximum Weight Independent Set prob- lem in (tw,ω)-bounded graph classes and prove that the problem is polynomial-time solvable in every class of graphs excluding a fixed star as an induced minor.
    [Show full text]
  • Structural Parameterizations of Clique Coloring
    Structural Parameterizations of Clique Coloring Lars Jaffke University of Bergen, Norway lars.jaff[email protected] Paloma T. Lima University of Bergen, Norway [email protected] Geevarghese Philip Chennai Mathematical Institute, India UMI ReLaX, Chennai, India [email protected] Abstract A clique coloring of a graph is an assignment of colors to its vertices such that no maximal clique is monochromatic. We initiate the study of structural parameterizations of the Clique Coloring problem which asks whether a given graph has a clique coloring with q colors. For fixed q ≥ 2, we give an O?(qtw)-time algorithm when the input graph is given together with one of its tree decompositions of width tw. We complement this result with a matching lower bound under the Strong Exponential Time Hypothesis. We furthermore show that (when the number of colors is unbounded) Clique Coloring is XP parameterized by clique-width. 2012 ACM Subject Classification Mathematics of computing → Graph coloring Keywords and phrases clique coloring, treewidth, clique-width, structural parameterization, Strong Exponential Time Hypothesis Digital Object Identifier 10.4230/LIPIcs.MFCS.2020.49 Related Version A full version of this paper is available at https://arxiv.org/abs/2005.04733. Funding Lars Jaffke: Supported by the Trond Mohn Foundation (TMS). Acknowledgements The work was partially done while L. J. and P. T. L. were visiting Chennai Mathematical Institute. 1 Introduction Vertex coloring problems are central in algorithmic graph theory, and appear in many variants. One of these is Clique Coloring, which given a graph G and an integer k asks whether G has a clique coloring with k colors, i.e.
    [Show full text]
  • Theoretical Computer Science a Polynomial Solution to the K-Fixed
    Theoretical Computer Science 411 (2010) 967–975 Contents lists available at ScienceDirect Theoretical Computer Science journal homepage: www.elsevier.com/locate/tcs A polynomial solution to the k-fixed-endpoint path cover problem on proper interval graphs Katerina Asdre, Stavros D. Nikolopoulos ∗ Department of Computer Science, University of Ioannina, P.O. Box 1186, GR-45110 Ioannina, Greece article info a b s t r a c t Article history: We study a variant of the path cover problem, namely, the k-fixed-endpoint path cover Received 29 January 2008 problem, or kPC for short. Given a graph G and a subset T of k vertices of V .G/, a k-fixed- Received in revised form 24 December 2008 endpoint path cover of G with respect to T is a set of vertex-disjoint paths P that covers Accepted 7 November 2009 the vertices of G such that the k vertices of T are all endpoints of the paths in P . The kPC Communicated by P. Spirakis problem is to find a k-fixed-endpoint path cover of G of minimum cardinality; note that, if T is empty (or, equivalently, k D 0), the stated problem coincides with the classical path Keywords: cover problem. The kPC problem generalizes some path cover related problems, such as Perfect graphs Proper interval graphs the 1HP and 2HP problems, which have been proved to be NP-complete. Note that the Path cover complexity status for both 1HP and 2HP problems on interval graphs remains an open Fixed-endpoint path cover question (Damaschke (1993) [9]). In this paper, we show that the kPC problem can be solved Linear-time algorithms in linear time on the class of proper interval graphs, that is, in O.n C m/ time on a proper interval graph on n vertices and m edges.
    [Show full text]
  • Clique-Width and Tree-Width of Sparse Graphs
    Clique-width and tree-width of sparse graphs Bruno Courcelle Labri, CNRS and Bordeaux University∗ 33405 Talence, France email: [email protected] June 10, 2015 Abstract Tree-width and clique-width are two important graph complexity mea- sures that serve as parameters in many fixed-parameter tractable (FPT) algorithms. The same classes of sparse graphs, in particular of planar graphs and of graphs of bounded degree have bounded tree-width and bounded clique-width. We prove that, for sparse graphs, clique-width is polynomially bounded in terms of tree-width. For planar and incidence graphs, we establish linear bounds. Our motivation is the construction of FPT algorithms based on automata that take as input the algebraic terms denoting graphs of bounded clique-width. These algorithms can check properties of graphs of bounded tree-width expressed by monadic second- order formulas written with edge set quantifications. We give an algorithm that transforms tree-decompositions into clique-width terms that match the proved upper-bounds. keywords: tree-width; clique-width; graph de- composition; sparse graph; incidence graph Introduction Tree-width and clique-width are important graph complexity measures that oc- cur as parameters in many fixed-parameter tractable (FPT) algorithms [7, 9, 11, 12, 14]. They are also important for the theory of graph structuring and for the description of graph classes by forbidden subgraphs, minors and vertex-minors. Both notions are based on certain hierarchical graph decompositions, and the associated FPT algorithms use dynamic programming on these decompositions. To be usable, they need input graphs of "small" tree-width or clique-width, that furthermore are given with the relevant decompositions.
    [Show full text]