Recognizability, hypergraph operations, and logical types

Blumensath, Achim; Courcelle, Bruno

doi:10.1016/j.ic.2005.11.006

Cited by 17 publications

(20 citation statements)

References 25 publications

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“…This is formally defined and proved in [1]. Letting ST R n denote the restriction of ST R to relations (and the corresponding domains and operations) of arity at most n we get the following statement :…”

Section: Theorem 2 [4] : a Set Of Graphs Is Vr-equational Iff It Is Smentioning

confidence: 94%

“…Hence, with respect to sets of hypergraphs of maximal arity n, the full signature ST R is equivalent to ST R n , its restriction to structures of arity at most n. Stronger formulations and their proofs can be found in [1].…”

Section: Theorem 2 [4] : a Set Of Graphs Is Vr-equational Iff It Is Smentioning

confidence: 99%

“…The general framework of this research which associates tightly algebraic notions (equational and recognizable sets) and Monadic Second-Order logic (for defining sets and transformations of graphs, hypergraphs and relational structures) is reviewed. The lecture [3] is based on two submitted but nevertheless available articles [1,4] ; the present text is an informal overview. The numerous definitions and results can be found in the two articles.…”

mentioning

confidence: 99%

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Recognizable Sets of Graphs, Hypergraphs and Relational Structures: A Survey

Courcelle

2004

Developments in Language Theory

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New results on the recognizability of sets of finite graphs, hypergraphs and relational structures are presented. The general framework of this research which associates tightly algebraic notions (equational and recognizable sets) and Monadic Second-Order logic (for defining sets and transformations of graphs, hypergraphs and relational structures) is reviewed. The lecture [3] is based on two submitted but nevertheless available articles [1,4] ; the present text is an informal overview. The numerous definitions and results can be found in the two articles.

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Section: Theorem 2 [4] : a Set Of Graphs Is Vr-equational Iff It Is Smentioning

confidence: 94%

Section: Theorem 2 [4] : a Set Of Graphs Is Vr-equational Iff It Is Smentioning

confidence: 99%

mentioning

confidence: 99%

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Recognizable Sets of Graphs, Hypergraphs and Relational Structures: A Survey

Courcelle

2004

Developments in Language Theory

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“…Quantifier-free transformations modify logical structures by redefining certain relations by quantifier-free formulas (see [9,3] for graph algebras).…”

Section: Introductionmentioning

confidence: 99%

Graph Operations Characterizing Rank-Width and Balanced Graph Expressions

Courcelle

Kanté

2007

Graph-Theoretic Concepts in Computer Science

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Graph complexity measures like tree-width, clique-width, NLC-width and rank-width are important because they yield Fixed Parameter Tractable algorithms. Rank-width is based on ranks of adjacency matrices of graphs over GF(2). We propose here algebraic operations on graphs that characterize rank-width. For algorithmic purposes, it is important to represent graphs by balanced terms. We give a unique theorem that generalizes several "balancing theorems" for tree-width and clique-width. New results are obtained for rank-width and a variant of clique-width, called m-clique-width.

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“…General quantifier-free transformations modify logical structures, by redefining certain relations by quantifier-free formulas (see [1,4,6,13] for graph algebras).…”

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confidence: 99%

Graph operations characterizing rank-width

Courcelle¹,

Kanté²

2009

Discrete Applied Mathematics

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a b s t r a c tGraph complexity measures such as tree-width, clique-width and rank-width are important because they yield Fixed Parameter Tractable algorithms. These algorithms are based on hierarchical decompositions of the considered graphs, and on boundedness conditions on the graph operations that, more or less explicitly, recombine the components of decompositions into larger graphs. Rank-width is defined in a combinatorial way, based on a tree, and not in terms of graph operations. We define operations on graphs that characterize rank-width and help for the construction of Fixed Parameter Tractable algorithms, especially for problems specified in monadic second-order logic.

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Recognizability, hypergraph operations, and logical types

Abstract: International audienc

Cited by 17 publications

References 25 publications

Recognizable Sets of Graphs, Hypergraphs and Relational Structures: A Survey

Recognizable Sets of Graphs, Hypergraphs and Relational Structures: A Survey

Graph Operations Characterizing Rank-Width and Balanced Graph Expressions

Graph operations characterizing rank-width

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