Sub-Coloring and Hypo-Coloring Interval Graphs⋆
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Proper Interval Graphs and the Guard Problem 1
DISCRETE N~THEMATICS ELSEVIER Discrete Mathematics 170 (1997) 223-230 Note Proper interval graphs and the guard problem1 Chiuyuan Chen*, Chin-Chen Chang, Gerard J. Chang Department of Applied Mathematics, National Chiao Tung University, Hsinchu 30050, Taiwan Received 26 September 1995; revised 27 August 1996 Abstract This paper is a study of the hamiltonicity of proper interval graphs with applications to the guard problem in spiral polygons. We prove that proper interval graphs with ~> 2 vertices have hamiltonian paths, those with ~>3 vertices have hamiltonian cycles, and those with />4 vertices are hamiltonian-connected if and only if they are, respectively, 1-, 2-, or 3-connected. We also study the guard problem in spiral polygons by connecting the class of nontrivial connected proper interval graphs with the class of stick-intersection graphs of spiral polygons. Keywords." Proper interval graph; Hamiltonian path (cycle); Hamiltonian-connected; Guard; Visibility; Spiral polygon I. Introduction The main purpose of this paper is to study the hamiltonicity of proper interval graphs with applications to the guard problem in spiral polygons. Our terminology and graph notation are standard, see [2], except as indicated. The intersection graph of a family ~ of nonempty sets is derived by representing each set in ~ with a vertex and connecting two vertices with an edge if and only if their corresponding sets intersect. An interval graph is the intersection graph G of a family J of intervals on a real line. J is usually called the interval model for G. A proper interval graph is an interval graph with an interval model ~¢ such that no interval in J properly contains another. -
Classes of Graphs with Restricted Interval Models Andrzej Proskurowski, Jan Arne Telle
Classes of graphs with restricted interval models Andrzej Proskurowski, Jan Arne Telle To cite this version: Andrzej Proskurowski, Jan Arne Telle. Classes of graphs with restricted interval models. Discrete Mathematics and Theoretical Computer Science, DMTCS, 1999, 3 (4), pp.167-176. hal-00958935 HAL Id: hal-00958935 https://hal.inria.fr/hal-00958935 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 3, 1999, 167–176 Classes of graphs with restricted interval models ✁ Andrzej Proskurowski and Jan Arne Telle ✂ University of Oregon, Eugene, Oregon, USA ✄ University of Bergen, Bergen, Norway received June 30, 1997, revised Feb 18, 1999, accepted Apr 20, 1999. We introduce ☎ -proper interval graphs as interval graphs with interval models in which no interval is properly con- tained in more than ☎ other intervals, and also provide a forbidden induced subgraph characterization of this class of ✆✞✝✠✟ graphs. We initiate a graph-theoretic study of subgraphs of ☎ -proper interval graphs with maximum clique size and give an equivalent characterization of these graphs by restricted path-decomposition. By allowing the parameter ✆ ✆ ☎ to vary from 0 to , we obtain a nested hierarchy of graph families, from graphs of bandwidth at most to graphs of pathwidth at most ✆ . -
On Computing Longest Paths in Small Graph Classes
On Computing Longest Paths in Small Graph Classes Ryuhei Uehara∗ Yushi Uno† July 28, 2005 Abstract The longest path problem is to find a longest path in a given graph. While the graph classes in which the Hamiltonian path problem can be solved efficiently are widely investigated, few graph classes are known to be solved efficiently for the longest path problem. For a tree, a simple linear time algorithm for the longest path problem is known. We first generalize the algorithm, and show that the longest path problem can be solved efficiently for weighted trees, block graphs, and cacti. We next show that the longest path problem can be solved efficiently on some graph classes that have natural interval representations. Keywords: efficient algorithms, graph classes, longest path problem. 1 Introduction The Hamiltonian path problem is one of the most well known NP-hard problem, and there are numerous applications of the problems [17]. For such an intractable problem, there are two major approaches; approximation algorithms [20, 2, 35] and algorithms with parameterized complexity analyses [15]. In both approaches, we have to change the decision problem to the optimization problem. Therefore the longest path problem is one of the basic problems from the viewpoint of combinatorial optimization. From the practical point of view, it is also very natural approach to try to find a longest path in a given graph, even if it does not have a Hamiltonian path. However, finding a longest path seems to be more difficult than determining whether the given graph has a Hamiltonian path or not. -
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. -
Complexity and Exact Algorithms for Vertex Multicut in Interval and Bounded Treewidth Graphs 1
Complexity and Exact Algorithms for Vertex Multicut in Interval and Bounded Treewidth Graphs 1 Jiong Guo a,∗ Falk H¨uffner a Erhan Kenar b Rolf Niedermeier a Johannes Uhlmann a aInstitut f¨ur Informatik, Friedrich-Schiller-Universit¨at Jena, Ernst-Abbe-Platz 2, D-07743 Jena, Germany bWilhelm-Schickard-Institut f¨ur Informatik, Universit¨at T¨ubingen, Sand 13, D-72076 T¨ubingen, Germany Abstract Multicut is a fundamental network communication and connectivity problem. It is defined as: given an undirected graph and a collection of pairs of terminal vertices, find a minimum set of edges or vertices whose removal disconnects each pair. We mainly focus on the case of removing vertices, where we distinguish between allowing or disallowing the removal of terminal vertices. Complementing and refining previous results from the literature, we provide several NP-completeness and (fixed- parameter) tractability results for restricted classes of graphs such as trees, interval graphs, and graphs of bounded treewidth. Key words: Combinatorial optimization, Complexity theory, NP-completeness, Dynamic programming, Graph theory, Parameterized complexity 1 Introduction Motivation and previous results. Multicut in graphs is a fundamental network design problem. It models questions concerning the reliability and ∗ Corresponding author. Tel.: +49 3641 9 46325; fax: +49 3641 9 46002 E-mail address: [email protected]. 1 An extended abstract of this work appears in the proceedings of the 32nd Interna- tional Conference on Current Trends in Theory and Practice of Computer Science (SOFSEM 2006) [19]. Now, in particular the proof of Theorem 5 has been much simplified. Preprint submitted to Elsevier Science 2 February 2007 robustness of computer and communication networks. -
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. -
Parameterized Algorithms for Recognizing Monopolar and 2
Parameterized Algorithms for Recognizing Monopolar and 2-Subcolorable Graphs∗ Iyad Kanj1, Christian Komusiewicz2, Manuel Sorge3, and Erik Jan van Leeuwen4 1School of Computing, DePaul University, Chicago, USA, [email protected] 2Institut für Informatik, Friedrich-Schiller-Universität Jena, Germany, [email protected] 3Institut für Softwaretechnik und Theoretische Informatik, TU Berlin, Germany, [email protected] 4Max-Planck-Institut für Informatik, Saarland Informatics Campus, Saarbrücken, Germany, [email protected] October 16, 2018 Abstract A graph G is a (ΠA, ΠB)-graph if V (G) can be bipartitioned into A and B such that G[A] satisfies property ΠA and G[B] satisfies property ΠB. The (ΠA, ΠB)-Recognition problem is to recognize whether a given graph is a (ΠA, ΠB )-graph. There are many (ΠA, ΠB)-Recognition problems, in- cluding the recognition problems for bipartite, split, and unipolar graphs. We present efficient algorithms for many cases of (ΠA, ΠB)-Recognition based on a technique which we dub inductive recognition. In particular, we give fixed-parameter algorithms for two NP-hard (ΠA, ΠB)-Recognition problems, Monopolar Recognition and 2-Subcoloring, parameter- ized by the number of maximal cliques in G[A]. We complement our algo- rithmic results with several hardness results for (ΠA, ΠB )-Recognition. arXiv:1702.04322v2 [cs.CC] 4 Jan 2018 1 Introduction A (ΠA, ΠB)-graph, for graph properties ΠA, ΠB , is a graph G = (V, E) for which V admits a partition into two sets A, B such that G[A] satisfies ΠA and G[B] satisfies ΠB. There is an abundance of (ΠA, ΠB )-graph classes, and important ones include bipartite graphs (which admit a partition into two independent ∗A preliminary version of this paper appeared in SWAT 2016, volume 53 of LIPIcs, pages 14:1–14:14. -
Shrub-Depth a Successful Depth Measure for Dense Graphs Graphs Petr Hlinˇen´Y Faculty of Informatics, Masaryk University Brno, Czech Republic
page.19 Shrub-Depth a successful depth measure for dense graphs graphs Petr Hlinˇen´y Faculty of Informatics, Masaryk University Brno, Czech Republic Petr Hlinˇen´y, Sparsity, Logic . , Warwick, 2018 1 / 19 Shrub-depth measure for dense graphs page.19 Shrub-Depth a successful depth measure for dense graphs graphs Petr Hlinˇen´y Faculty of Informatics, Masaryk University Brno, Czech Republic Ingredients: joint results with J. Gajarsk´y,R. Ganian, O. Kwon, J. Neˇsetˇril,J. Obdrˇz´alek, S. Ordyniak, P. Ossona de Mendez Petr Hlinˇen´y, Sparsity, Logic . , Warwick, 2018 1 / 19 Shrub-depth measure for dense graphs page.19 Measuring Width or Depth? • Being close to a TREE { \•-width" sparse dense tree-width / branch-width { showing a structure clique-width / rank-width { showing a construction Petr Hlinˇen´y, Sparsity, Logic . , Warwick, 2018 2 / 19 Shrub-depth measure for dense graphs page.19 Measuring Width or Depth? • Being close to a TREE { \•-width" sparse dense tree-width / branch-width { showing a structure clique-width / rank-width { showing a construction • Being close to a STAR { \•-depth" sparse dense tree-depth { containment in a structure ??? (will show) Petr Hlinˇen´y, Sparsity, Logic . , Warwick, 2018 2 / 19 Shrub-depth measure for dense graphs page.19 1 Recall: Width Measures Tree-width tw(G) ≤ k if whole G can be covered by bags of size ≤ k + 1, arranged in a \tree-like fashion". Petr Hlinˇen´y, Sparsity, Logic . , Warwick, 2018 3 / 19 Shrub-depth measure for dense graphs page.19 1 Recall: Width Measures Tree-width tw(G) ≤ k if whole G can be covered by bags of size ≤ k + 1, arranged in a \tree-like fashion". -