Practical and Efficient Split Decomposition via Graph-Labelled Trees

Gioan, Emeric; Paul, Christophe; Tedder, Marc; Corneil, Derek G.

doi:10.1007/s00453-013-9752-9

Cited by 15 publications

(54 citation statements)

References 39 publications

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“…3). The contents of this subsection are based on the original work of Cunningham [13] as well as on more recent results by several authors [25][26][27].…”

Section: Splits and Graph Labeled Treesmentioning

confidence: 99%

“…When clear from the context, we may use u as a shorthand for G(u) ∈ F; for instance, we use V (u) to denote V (G(u)) and we say that an edge of T incident to u is incident to the vertex of G(u) mapped to it. Graph-labeled trees were introduced by Gioan and Paul [25,26], and in the following paragraphs we recall some useful definitions and facts that also appeared in follow-up work [27].…”

Section: Splits and Graph Labeled Treesmentioning

confidence: 99%

“…A graph-labeled tree is reduced if all its labels are either prime or degenerate, and no node-join of two cliques or two stars S 1 and S 2 where the center of S 1 is adjacent to a leaf of S 2 is not possible. It is known that for every connected graph G, there exists a unique reduced graph-labeled tree (T, F) such that G = Gr(T, F) [13,[25][26][27]; this unique reduced graph-labeled tree is called the split-tree and is denoted ST(G). …”

Section: Definition 1 Let (T F)mentioning

confidence: 99%

“…4 we provide a fixed-parameter algorithm for computing the wsn H along with the corresponding decomposition for any graph class H which can be characterized by a finite set of forbidden induced subgraphs (obstructions). This is achieved by building on the polynomial algorithm for computing split-decompositions [27] in combination with the fixed-parameter algorithm for computing rank-width [29].…”

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confidence: 99%

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Solving Problems on Graphs of High Rank-Width

2017

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A modulator in a graph is a vertex set whose deletion places the considered graph into some specified graph class. The cardinality of a modulator to various graph classes has long been used as a structural parameter which can be exploited to obtain fixed-parameter algorithms for a range of hard problems. Here we investigate what happens when a graph contains a modulator which is large but "well-structured" (in the sense of having bounded rank-width). Can such modulators still be exploited to obtain efficient algorithms? And is it even possible to find such modulators efficiently? We first show that the parameters derived from such well-structured modulators are more powerful for fixed-parameter algorithms than the cardinality of modulators and rank-width itself. Then, we develop a fixed-parameter algorithm for finding such wellstructured modulators to every graph class which can be characterized by a finite set of forbidden induced subgraphs. We proceed by showing how well-structured modulators can be used to obtain efficient parameterized algorithms for Minimum Vertex Cover and Maximum Clique. Finally, we use the concept of well-structured modulators to develop an algorithmic meta-theorem for deciding problems expressible in monadic second order logic, and prove that this result is tight in the sense that it cannot be generalized to LinEMSO problems.

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“…3). The contents of this subsection are based on the original work of Cunningham [13] as well as on more recent results by several authors [25][26][27].…”

Section: Splits and Graph Labeled Treesmentioning

confidence: 99%

Section: Splits and Graph Labeled Treesmentioning

confidence: 99%

Section: Definition 1 Let (T F)mentioning

confidence: 99%

mentioning

confidence: 99%

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Solving Problems on Graphs of High Rank-Width

2017

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“…When clear from the context, we may use u as a shorthard for G(u) ∈ F ; for instance, we use V (u) to denote V (G(u)) and we say that an edge of T incident to u is incident to the vertex of G(u) mapped to it. Graph-labeled trees were introduced in [16,17] and in the following paragraphs we recall some useful definitions and theorems that appear in [18].…”

Section: Splits and Graph Labeled Treesmentioning

confidence: 99%

Solving Problems on Graphs of High Rank-Width

Eiben

Ganian

Szeider

2015

Lecture Notes in Computer Science

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A modulator of a graph G to a specified graph class H is a set of vertices whose deletion puts G into H. The cardinality of a modulator to various graph classes has long been used as a structural parameter which can be exploited to obtain FPT algorithms for a range of hard problems. Here we investigate what happens when a graph contains a modulator which is large but "well-structured" (in the sense of having bounded rank-width). Can such modulators still be exploited to obtain efficient algorithms? And is it even possible to find such modulators efficiently? We first show that the parameters derived from such well-structured modulators are strictly more general than the cardinality of modulators and rank-width itself. Then, we develop an FPT algorithm for finding such well-structured modulators to any graph class which can be characterized by a finite set of forbidden induced subgraphs. We proceed by showing how well-structured modulators can be used to obtain efficient parameterized algorithms for MINIMUM VERTEX COVER and MAXIMUM CLIQUE. Finally, we use the concept of well-structured modulators to develop an algorithmic meta-theorem for efficiently deciding problems expressible in Monadic Second Order (MSO) logic, and prove that this result is tight in the sense that it cannot be generalized to LinEMSO problems. IntroductionMany important graph problems are known to be NP-hard, and yet admit efficient solutions in practice due to the inherent structure of instances. The parameterized complexity paradigm [10,24] allows a more refined analysis of the complexity of various problems and hence enables the design of more efficient algorithms. In particular, given an instance of size n and a numerical parameter k which captures some property of the instance, one asks whether the instance can be solved in time f (k) · n O(1) . Parameterized problems which admit such an algorithm are called fixed parameter tractable (FPT), and the algorithms themselves are often called FPT algorithms.Given the above, it is natural to ask what kind of structure can be exploited to obtain FPT algorithms for a wide range of natural graph problems. There are two very successful, mutually incomparable approaches which tackle this question.A. Width measures. Treewidth has become an extremely successful structural parameter with a wide range of applications in many fields of computer science. However, treewidth is not suitable for use in dense graphs. This led to the development of algorithms that use the parameter clique-width [7], which can be viewed as a relaxation ⋆ Supported by the Austrian Science Fund (FWF), project P26696.1. We introduce a family of "hybrid" parameters that combine approaches A and B.Given a graph G and a fixed graph class H, the new parameters capture (roughly speaking) the minimum rank-width of any modulator of G into H. We call this the wellstructure number of G or wsn H (G). The formal definition of the parameter also relies on the notion of split decompositions [8] and is provided in Section 3, where we also prove ...

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Unified View of Graph Searching and LDFS-Based Certifying Algorithms

Corneil¹,

Habib²

2016

Encyclopedia of Algorithms

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Practical and Efficient Split Decomposition via Graph-Labelled Trees

Cited by 15 publications

References 39 publications

Solving Problems on Graphs of High Rank-Width

Solving Problems on Graphs of High Rank-Width

Solving Problems on Graphs of High Rank-Width

Unified View of Graph Searching and LDFS-Based Certifying Algorithms

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