Upper bounds to the clique width of graphs

Courcelle, Bruno; Olariu, Stephan

doi:10.1016/s0166-218x(99)00184-5

Cited by 558 publications

(529 citation statements)

References 24 publications

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“…Since treewidth is lowerbounded by degeneracy, we know that there is an O(ω · m)-time algorithm for △-Enum, where ω is the treewidth of the input graph. A parameter lower bounding treewidth in the parameter hierarchy is clique-width k (it holds that k ≤ 2 ω+1 + 1 and k can be arbitrarily small compared to ω [15]). Moreover, clique-width also lower-bounds distance to cograph.…”

Section: Parameters Incomparable With Degeneracymentioning

confidence: 99%

Parameterized Aspects of Triangle Enumeration

Bentert

Fluschnik

Nichterlein

et al. 2017

Fundamentals of Computation Theory

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Listing all triangles in an undirected graph is a fundamental graph primitive with numerous applications. It is trivially solvable in time cubic in the number of vertices. It has seen a significant body of work contributing to both theoretical aspects (e.g., lower and upper bounds on running time, adaption to new computational models) as well as practical aspects (e.g. algorithms tuned for large graphs). Motivated by the fact that the worst-case running time is cubic, we perform a systematic parameterized complexity study of triangle enumeration, providing both positive results (new enumerative kernelizations, "subcubic" parameterized solving algorithms) as well as negative results (likely uselessness in terms of "faster" parameterized algorithms of certain parameters such as graph diameter).

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Section: Parameters Incomparable With Degeneracymentioning

confidence: 99%

Parameterized Aspects of Triangle Enumeration

Bentert

Fluschnik

Nichterlein

et al. 2017

Fundamentals of Computation Theory

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“…The notion of clique-width was first introduced by Courcelle and Olariu [2000]. Let k be a positive integer.…”

Section: Rank-widthmentioning

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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“…Hence we can determine the hardness of a given instance with respect to our algorithm in advance. This is not possible for treewidth and related parameters: computation of tree-width or branch-width is NP-hard [3,26], and it is not known whether graphs with fixed cliquewidth ≥ 4 can be recognized in polynomial time [5]. (3) Franco, et al [13] show that satisfiability of certain propositional formulas whose only connective is the implication is fixed-parameter tractable with respect to the number of occurrences of the always-false constant f (this result is listed in the appendix of [10] as pure implicational satisfiability of fixed f-depth); an improved algorithm is presented in [16].…”

Section: Fixed-parameter Tractable Parameterizations Of Satmentioning

confidence: 99%

Minimal Unsatisfiable Formulas with Bounded Clause-Variable Difference are Fixed-Parameter Tractable

Szeider

2003

Lecture Notes in Computer Science

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Upper bounds to the clique width of graphs

Cited by 558 publications

References 24 publications

Parameterized Aspects of Triangle Enumeration

Parameterized Aspects of Triangle Enumeration

Approximating Rank-Width and Clique-Width Quickly

Minimal Unsatisfiable Formulas with Bounded Clause-Variable Difference are Fixed-Parameter Tractable

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