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Width parameters beyond treewidth and their applications
 Computer Journal
, 2007
"... Besides the very successful concept of treewidth (see [Bodlaender, H. and Koster, A. (2007) Combinatorial optimisation on graphs of bounded treewidth. These are special issues on Parameterized Complexity]), many concepts and parameters measuring the similarity or dissimilarity of structures compare ..."
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Cited by 40 (0 self)
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Besides the very successful concept of treewidth (see [Bodlaender, H. and Koster, A. (2007) Combinatorial optimisation on graphs of bounded treewidth. These are special issues on Parameterized Complexity]), many concepts and parameters measuring the similarity or dissimilarity of structures compared to trees have been born and studied over the past years. These concepts and parameters have proved to be useful tools in many applications, especially in the design of efficient algorithms. Our presented novel look at the contemporary developments of these ‘width ’ parameters in combinatorial structures delivers—besides traditional treewidth and derived dynamic programming schemes—also a number of other useful parameters like branchwidth, rankwidth (cliquewidth) or hypertreewidth. In this contribution, we demonstrate how ‘width ’ parameters of graphs and generalized structures (such as matroids or hypergraphs), can be used to improve the design of parameterized algorithms and the structural analysis in other applications on an abstract level.
Graphs with Branchwidth at most Three
 J. Algorithms
, 1997
"... In this paper we investigate both the structure of graphs with branchwidth at most three, as well as algorithms to recognise such graphs. We show that a graph has branchwidth at most three, if and only if it has treewidth at most three and does not contain the threedimensional binary cube graph as ..."
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Cited by 23 (2 self)
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In this paper we investigate both the structure of graphs with branchwidth at most three, as well as algorithms to recognise such graphs. We show that a graph has branchwidth at most three, if and only if it has treewidth at most three and does not contain the threedimensional binary cube graph as a minor. A set of four graphs is shown to be the obstruction set of graphs with branchwidth at most three. We give a safe and complete set of reduction rules for the graphs with branchwidth at most three. Using this set, a linear time algorithm is given that checks if a given graph has branchwidth at most three, and, if so, outputs a minimum width branch decomposition. Keywords: graph algorithms, branchwidth, obstruction set, graph minor, reduction rule. 1 Introduction This paper considers the study of the graphs with branchwidth at most three. The notion of branchwidth has a close relationship to the more wellknown notion of treewidth, a notion that has come to play a large role in many ...
Computing Excluded Minors
, 2008
"... By Robertson and Seymour’s graph minor theorem, every minor ideal can be characterised by a finite family of excluded minors. (A minor ideal is a class of graphs closed under taking minors.) We study algorithms for computing excluded minor characterisations of minor ideals. We propose a general meth ..."
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Cited by 23 (4 self)
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By Robertson and Seymour’s graph minor theorem, every minor ideal can be characterised by a finite family of excluded minors. (A minor ideal is a class of graphs closed under taking minors.) We study algorithms for computing excluded minor characterisations of minor ideals. We propose a general method for obtaining such algorithms, which is based on definability in monadic secondorder logic and the decidability of the monadic secondorder theory of trees. A straightforward application of our method yields algorithms that, for a given k, compute excluded minor characterisations for the minor ideal Tk of all graphs of tree width at most k, the minor ideal Bk of all graphs of branch width at most k, and the minor ideal Gk of all graphs of genus at most k. Our main results are concerned with constructions of new minor ideals from given ones. Answering a question that goes back to Fellows and Langston [11], we prove that there is an algorithm that, given excluded minor characterisations of two minor ideals C and D, computes such a characterisation for the ideal C ∪ D. Furthermore, we obtain an algorithm for computing an excluded minor characterisation for the class of all apex graphs over a minor ideal C, given an excluded minor characterisation for C. (An apex graph over C is a graph G from which one vertex can be removed to obtain a graph in C.) A corollary of this result is a uniform ftpalgorithm for the “distance k from planarity” problem.
Algorithmic MetaTheorems
 In M. Grohe and R. Neidermeier eds, International Workshop on Parameterized and Exact Computation (IWPEC), volume 5018 of LNCS
, 2008
"... Algorithmic metatheorems are algorithmic results that apply to a whole range of problems, instead of addressing just one specific problem. This kind of theorems are often stated relative to a certain class of graphs, so the general form of a meta theorem reads “every problem in a certain class C of ..."
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Cited by 22 (6 self)
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Algorithmic metatheorems are algorithmic results that apply to a whole range of problems, instead of addressing just one specific problem. This kind of theorems are often stated relative to a certain class of graphs, so the general form of a meta theorem reads “every problem in a certain class C of problems can be solved efficiently on every graph satisfying a certain property P”. A particularly well known example of a metatheorem is Courcelle’s theorem that every decision problem definable in monadic secondorder logic (MSO) can be decided in linear time on any class of graphs of bounded treewidth [1]. The class C of problems can be defined in a number of different ways. One option is to state combinatorial or algorithmic criteria of problems in C. For instance, Demaine, Hajiaghayi and Kawarabayashi [5] showed that every minimisation problem that can be solved efficiently on graph classes of bounded treewidth and for which approximate solutions can be computed efficiently from solutions of certain subinstances, have a PTAS on any class of graphs excluding a fixed minor. While this gives a strong unifying explanation for PTAS of many
Branch and Tree Decomposition Techniques for Discrete Optimization
, 2005
"... This chapter gives a general overview of two emerging techniques for discrete optimization that have footholds in mathematics, computer science, and operations research: branch decompositions and tree decompositions. Branch decompositions and tree decompositions along with their respective connectiv ..."
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Cited by 21 (3 self)
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This chapter gives a general overview of two emerging techniques for discrete optimization that have footholds in mathematics, computer science, and operations research: branch decompositions and tree decompositions. Branch decompositions and tree decompositions along with their respective connectivity invariants, branchwidth and treewidth, were first introduced to aid in proving the Graph Minors Theorem, a wellknown conjecture (Wagner’s conjecture) in graph theory. The algorithmic importance of branch decompositions and tree decompositions for solving NPhard problems modelled on graphs was first realized by computer scientists in relation to formulating graph problems in monadic second order logic. The dynamic programming techniques utilizing branch decompositions and tree decompositions, called branch decomposition and tree decomposition based algorithms, fall into a class of algorithms known as fixedparameter tractable algorithms and have been shown to be effective in a practical setting for NPhard problems such as minimum domination, the travelling salesman problem, general minor containment, and frequency assignment problems.
A Parametrized Algorithm for Matroid BranchWidth
, 2005
"... Branchwidth is a structural parameter very closely related to treewidth, but branchwidth has an immediate generalization from graphs to matroids. We present an algorithm that, for a given matroid M of bounded branchwidth t which is represented over a finite field, finds a branch decomposition o ..."
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Cited by 18 (1 self)
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Branchwidth is a structural parameter very closely related to treewidth, but branchwidth has an immediate generalization from graphs to matroids. We present an algorithm that, for a given matroid M of bounded branchwidth t which is represented over a finite field, finds a branch decomposition of M of width at most 3t in cubic time. Then we show that the branchwidth of M is a uniformly fixedparameter tractable problem. Other applications include recognition of matroid properties definable in the monadic secondorder logic for bounded branchwidth, or [Oum] a cubic approximation algorithm for graph rankwidth and cliquewidth.
Algorithms and Obstructions for LinearWidth and Related Search Parameters
 Discrete Applied Mathematics
, 1997
"... The linearwidth of a graph G is defined to be the smallest integer k such that the edges of G can be arranged in a linear ordering (e 1 ; : : : ; e r ) in such a way that for every i = 1; : : : ; r \Gamma 1, there are at most k vertices incident to edges that belong both to fe 1 ; : : : ; e i g an ..."
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Cited by 15 (8 self)
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The linearwidth of a graph G is defined to be the smallest integer k such that the edges of G can be arranged in a linear ordering (e 1 ; : : : ; e r ) in such a way that for every i = 1; : : : ; r \Gamma 1, there are at most k vertices incident to edges that belong both to fe 1 ; : : : ; e i g and to fe i+1 ; : : : ; e r g. In this paper, we give a set of 57 graphs and prove that it is the set of the minimal forbidden minors for the class of graphs with linearwidth at most two. Our proof also gives a linear time algorithm that either reports that a given graph has linearwidth more than two or outputs an edge ordering of minimum linearwidth. We further prove a structural connection between linearwidth and the mixed search number which enables us to determine, for any k 1, the set acyclic forbidden minors for the class of graphs with linearwidth k. Moreover, due to this connection, our algorithm can be transfered to two linear time algorithms that check whether a graph has mixe...
A Constructive Linear Time Algorithm for Small Cutwidth
 PROC. 11TH INTERNATINAL CONFERENCE ISAAC 2000, NUMBER 1969
, 2000
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Logic, Graphs, and Algorithms
, 2007
"... Algorithmic meta theorems are algorithmic results that apply to whole families of combinatorial problems, instead of just specific problems. These families are usually defined in terms of logic and graph theory. An archetypal algorithmic meta theorem is Courcelle’s Theorem [9], which states that all ..."
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Cited by 11 (0 self)
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Algorithmic meta theorems are algorithmic results that apply to whole families of combinatorial problems, instead of just specific problems. These families are usually defined in terms of logic and graph theory. An archetypal algorithmic meta theorem is Courcelle’s Theorem [9], which states that all graph properties definable in monadic secondorder logic can be decided in linear time on graphs of bounded tree width. This article is an introduction into the theory underlying such meta theorems and a survey of the most important results in this area.
Computing Small Search Numbers in Linear Time
, 1998
"... Let G = (V; E) be a graph. The linearwidth of G is defined as the smallest integer k such that E can be arranged in a linear ordering (e 1 ; : : : ; e r ) such that for every i = 1; : : : ; r \Gamma 1, there are at most k vertices both incident to an edge that belongs to fe 1 ; : : : ; e i g as to ..."
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Cited by 11 (8 self)
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Let G = (V; E) be a graph. The linearwidth of G is defined as the smallest integer k such that E can be arranged in a linear ordering (e 1 ; : : : ; e r ) such that for every i = 1; : : : ; r \Gamma 1, there are at most k vertices both incident to an edge that belongs to fe 1 ; : : : ; e i g as to an edge that belongs to fe i+1 ; : : : ; e r g. For each fixed constant k, a linear time algorithm is given, that decides for any graph G = (V; E) whether the linearwidth of G is at most k, and if so, finds the corresponding ordering of E. Linearwidth has been proven to be related with the following graph searching parameters: mixed search number, node search number, and edge search number. A consequence of this is that we obtain for fixed k, linear time algorithms that check whether a given graph can be mixed, node, or edge searched with at most k searchers, and if so, output the corresponding search strategies. 1 Introduction In this paper, we study algorithmic aspects of a relatively ...