Edge dominating set and colorings on graphs with fixed clique-width

Kobler, Daniel; Rotics, Udi

doi:10.1016/s0166-218x(02)00198-1

Cited by 144 publications

(128 citation statements)

References 18 publications

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“…This algorithm significantly improves over a previous algorithm of Kobler and Rotics [18] which runs in time O n 4 k on graphs of clique-width k. When comparing these two algorithms, the readers should keep in mind that our parameter t is the rankwidth of the input graph, and the clique-width k can reach up to 2 t/2−1 by [3]. We, moreover, straightforwardly extend our algorithm (Theorem 4.8) to compute the chromatic polynomial, again improving runtime over previous [1].…”

Section: Introductionmentioning

confidence: 77%

“…Many examples using the clique-width parameter can be found in [1,7,13,18,20]. Our goal in this paper is to design and use a mathematically precise and sound formalism for solving problems on graphs of bounded rank-width in pseudopolynomial time.…”

Section: Definitions and Basicsmentioning

confidence: 99%

“…We illustrate our formalism on the graph chromatic number problem, for which we strongly improve the previous algorithm of [18] running on graphs of bounded clique-width. For the purposes of this section, it is useful to think about colouring not as a function from vertices to colours but rather as a vertex-partition of G. Formally, a colour partition of G is a partition N of V (G) into pairwise disjoint nonempty(!)…”

Section: Computing the Chromatic Numbermentioning

confidence: 99%

“…of subspaces of GF(2) t . The crucial finding, inspired by the colouring algorithm in [18], reads: (4.6) For any t-labeled graphsḠ 1 ,Ḡ 2 and any…”

Section: Computing the Chromatic Numbermentioning

confidence: 99%

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Better Polynomial Algorithms on Graphs of Bounded Rank-Width

Ganian

Hliněný

2009

Lecture Notes in Computer Science

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Abstract. Although there exist many polynomial algorithms for N Phard problems running on a clique-width expression of the input graph, there exists only little comparable work on such algorithms for rankwidth. We believe that one reason for this is the somewhat obscure and hard-to-grasp nature of rank-decompositions. Nevertheless, strong arguments for using the rank-width parameter have been given by recent formalisms independently developed by Courcelle and Kanté, by the authors, and by Bui-Xuan et al. This article focuses on designing formally clean and understandable pseudopolynomial algorithms solving hard problems (non-FPT) on graphs of bounded rank-width. Those include computing the chromatic number and polynomial or testing the Hamiltonicity of a graph and are extendable to many other problems.

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Section: Introductionmentioning

confidence: 77%

Section: Definitions and Basicsmentioning

confidence: 99%

Section: Computing the Chromatic Numbermentioning

confidence: 99%

“…of subspaces of GF(2) t . The crucial finding, inspired by the colouring algorithm in [18], reads: (4.6) For any t-labeled graphsḠ 1 ,Ḡ 2 and any…”

Section: Computing the Chromatic Numbermentioning

confidence: 99%

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Better Polynomial Algorithms on Graphs of Bounded Rank-Width

Ganian

Hliněný

2009

Lecture Notes in Computer Science

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“…(In fact, the clique-width is the minimum k so that there is a k-expression of G.) In general, every graph problem expressible in monadic second-order logic with quantifications over vertices and vertex sets (MS 1 -logic) can be solved in linear time if the input graph is given with a k-expression . However, problems such as deciding whether the graph is Hamiltonian [Wanke 1994] and finding the chromatic number [Kobler and Rotics 2003] are not expressible in monadic second-order logic but have nevertheless polynomial-time algorithms on graphs of bounded clique-width if the input graph is given with a k-expression. Therefore we hope to have, for each fixed k, a polynomial-time algorithm to find a k-expression of an input graph if the input graph has clique-width at most k. This problem is still open when k > 3.…”

Section: Papermentioning

confidence: 99%

Approximating Rank-Width and Clique-Width Quickly

Oum

2005

Graph-Theoretic Concepts in Computer Science

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Rank-width was defined by Oum and Seymour [2006. Approximating clique-width and branchwidth. J. Combin. Theory Ser. B 96, 4, 514-528] to investigate clique-width. They constructed an algorithm that either outputs a rank-decomposition of width at most f (k) for some function f or confirms that rank-width is larger than k in time O(|V | 9 log |V |) for an input graph G = (V, E) and a fixed k. We develop three separate algorithms of this kind with faster running time. We construct an O(|V | 4 )-time algorithm with f (k) = 3k + 1 by constructing a subroutine for the previous algorithm; we avoid generic algorithms minimizing submodular functions used by Oum and Seymour. Another one is an O(|V | 3 )-time algorithm with f (k) = 24k by giving a reduction from graphs to binary matroids; then we use an approximation algorithm for matroid branch-width by Hliněný [2005 INTRODUCTIONGraph complexity measures such as tree-width and branch-width are important for algorithmic purposes and for understanding the structure of families of graphs. One of them is the clique-width, defined by Courcelle and Olariu [2000]. We discuss its definition in the next section. Many NP-hard graph problems are solvable in polynomial time if a tree-like decomposition corresponding to clique-width, called Permission to make digital/hard copy of all or part of this material without fee for personal or classroom use provided that the copies are not made or distributed for profit or commercial advantage, the ACM copyright/server notice, the title of the publication, and its date appear, and notice is given that copying is by permission of the ACM, Inc. To copy otherwise, to republish, to post on servers, or to redistribute to lists requires prior specific permission and/or a fee. c 202008 ACM 0000-0000/202008/0000-0001 $5.00 2003] are not expressible in monadic second-order logic but have nevertheless polynomial-time algorithms on graphs of bounded clique-width if the input graph is given with a k-expression. Therefore we hope to have, for each fixed k, a polynomial-time algorithm to find a k-expression of an input graph if the input graph has clique-width at most k. This problem is still open when k > 3. Corneil et al. [2000] solved this problem when k = 3.Instead, Oum and Seymour [2006] found an "approximation" algorithm that either outputs a (2 3k+2 − 1)-expression or confirms that the clique-width of G is larger than k. This can be combined with algorithms requiring a k-expression and therefore those algorithms no longer have to require a k-expression as an input to be polynomial-time algorithms. To obtain this approximation algorithm, they defined another graph width parameter, called the rank-width and showed that rank-width is at most clique-width and clique-width is at most 2 1+rank-width − 1. In addition, they showed a polynomial-time algorithm to find a rank-decomposition of width 3k + 1 or to confirm that the rank-width is larger than k. (Rank-width is defined as the minimum possible width of all rank-decompositions. We will discuss its...

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Graph Coloring Problems

Werra

Kobler

2013

Paradigms of Combinatorial Optimization

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Edge dominating set and colorings on graphs with fixed clique-width

Cited by 144 publications

References 18 publications

Better Polynomial Algorithms on Graphs of Bounded Rank-Width

Better Polynomial Algorithms on Graphs of Bounded Rank-Width

Approximating Rank-Width and Clique-Width Quickly

Graph Coloring Problems

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