K-Degenerate Graphs
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Linear K-Arboricities on Trees
Discrete Applied Mathematics 103 (2000) 281–287 View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Note Linear k-arboricities on trees Gerard J. Changa; 1, Bor-Liang Chenb, Hung-Lin Fua, Kuo-Ching Huangc; ∗;2 aDepartment of Applied Mathematics, National Chiao Tung University, Hsinchu 300, Taiwan bDepartment of Business Administration, National Taichung Institue of Commerce, Taichung 404, Taiwan cDepartment of Applied Mathematics, Providence University, Shalu 433, Taichung, Taiwan Received 31 October 1997; revised 15 November 1999; accepted 22 November 1999 Abstract For a ÿxed positive integer k, the linear k-arboricity lak (G) of a graph G is the minimum number ‘ such that the edge set E(G) can be partitioned into ‘ disjoint sets and that each induces a subgraph whose components are paths of lengths at most k. This paper studies linear k-arboricity from an algorithmic point of view. In particular, we present a linear-time algo- rithm to determine whether a tree T has lak (T)6m. ? 2000 Elsevier Science B.V. All rights reserved. Keywords: Linear forest; Linear k-forest; Linear arboricity; Linear k-arboricity; Tree; Leaf; Penultimate vertex; Algorithm; NP-complete 1. Introduction All graphs in this paper are simple, i.e., ÿnite, undirected, loopless, and without multiple edges. A linear k-forest is a graph whose components are paths of length at most k.Alinear k-forest partition of G is a partition of the edge set E(G) into linear k-forests. The linear k-arboricity of G, denoted by lak (G), is the minimum size of a linear k-forest partition of G. -
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. -
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. -
Orienting Fully Dynamic Graphs with Worst-Case Time Bounds⋆
Orienting Fully Dynamic Graphs with Worst-Case Time Bounds? Tsvi Kopelowitz1??, Robert Krauthgamer2 ???, Ely Porat3, and Shay Solomon4y 1 University of Michigan, [email protected] 2 Weizmann Institute of Science, [email protected] 3 Bar-Ilan University, [email protected] 4 Weizmann Institute of Science, [email protected] Abstract. In edge orientations, the goal is usually to orient (direct) the edges of an undirected network (modeled by a graph) such that all out- degrees are bounded. When the network is fully dynamic, i.e., admits edge insertions and deletions, we wish to maintain such an orientation while keeping a tab on the update time. Low out-degree orientations turned out to be a surprisingly useful tool for managing networks. Brodal and Fagerberg (1999) initiated the study of the edge orientation problem in terms of the graph's arboricity, which is very natural in this context. Their solution achieves a constant out-degree and a logarithmic amortized update time for all graphs with constant arboricity, which include all planar and excluded-minor graphs. It remained an open ques- tion { first proposed by Brodal and Fagerberg, later by Erickson and others { to obtain similar bounds with worst-case update time. We address this 15 year old question by providing a simple algorithm with worst-case bounds that nearly match the previous amortized bounds. Our algorithm is based on a new approach of maintaining a combina- torial invariant, and achieves a logarithmic out-degree with logarithmic worst-case update times. This result has applications to various dynamic network problems such as maintaining a maximal matching, where we obtain logarithmic worst-case update time compared to a similar amor- tized update time of Neiman and Solomon (2013). -
Faster Sublinear Approximation of the Number of K-Cliques in Low-Arboricity Graphs
Faster sublinear approximation of the number of k-cliques in low-arboricity graphs Talya Eden ✯ Dana Ron ❸ C. Seshadhri ❹ Abstract science [10, 49, 40, 58, 7], with a wide variety of applica- Given query access to an undirected graph G, we consider tions [37, 13, 53, 18, 44, 8, 6, 31, 54, 38, 27, 57, 30, 39]. the problem of computing a (1 ε)-approximation of the This problem has seen a resurgence of interest because number of k-cliques in G. The± standard query model for general graphs allows for degree queries, neighbor queries, of its importance in analyzing massive real-world graphs and pair queries. Let n be the number of vertices, m be (like social networks and biological networks). There the number of edges, and nk be the number of k-cliques. are a number of clever algorithms for exactly counting Previous work by Eden, Ron and Seshadhri (STOC 2018) ∗ n mk/2 k-cliques using matrix multiplications [49, 26] or combi- gives an O ( 1 + )-time algorithm for this problem n /k nk k natorial methods [58]. However, the complexity of these (we use O∗( ) to suppress poly(log n, 1/ε,kk) dependencies). algorithms grows with mΘ(k), where m is the number of · Moreover, this bound is nearly optimal when the expression edges in the graph. is sublinear in the size of the graph. Our motivation is to circumvent this lower bound, by A line of recent work has considered this question parameterizing the complexity in terms of graph arboricity. from a sublinear approximation perspective [20, 24]. -
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. -
Sub-Coloring and Hypo-Coloring Interval Graphs⋆
Sub-coloring and Hypo-coloring Interval Graphs? Rajiv Gandhi1, Bradford Greening, Jr.1, Sriram Pemmaraju2, and Rajiv Raman3 1 Department of Computer Science, Rutgers University-Camden, Camden, NJ 08102. E-mail: [email protected]. 2 Department of Computer Science, University of Iowa, Iowa City, Iowa 52242. E-mail: [email protected]. 3 Max-Planck Institute for Informatik, Saarbr¨ucken, Germany. E-mail: [email protected]. Abstract. In this paper, we study the sub-coloring and hypo-coloring problems on interval graphs. These problems have applications in job scheduling and distributed computing and can be used as “subroutines” for other combinatorial optimization problems. In the sub-coloring problem, given a graph G, we want to partition the vertices of G into minimum number of sub-color classes, where each sub-color class induces a union of disjoint cliques in G. In the hypo-coloring problem, given a graph G, and integral weights on vertices, we want to find a partition of the vertices of G into sub-color classes such that the sum of the weights of the heaviest cliques in each sub-color class is minimized. We present a “forbidden subgraph” characterization of graphs with sub-chromatic number k and use this to derive a a 3-approximation algorithm for sub-coloring interval graphs. For the hypo-coloring problem on interval graphs, we first show that it is NP-complete and then via reduction to the max-coloring problem, show how to obtain an O(log n)-approximation algorithm for it. 1 Introduction Given a graph G = (V, E), a k-sub-coloring of G is a partition of V into sub-color classes V1,V2,...,Vk; a subset Vi ⊆ V is called a sub-color class if it induces a union of disjoint cliques in G. -
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". -