Results 1  10
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20
Efficient and Constructive Algorithms for the Pathwidth and Treewidth of Graphs
, 1993
"... In this paper we give, for all constants k, l, explicit algorithms, that given a graph G = (V; E) with a treedecomposition of G with treewidth at most l, decide whether the treewidth (or pathwidth) of G is at most k, and if so, find a treedecomposition or (pathdecomposition) of G of width at most ..."
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Cited by 49 (11 self)
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In this paper we give, for all constants k, l, explicit algorithms, that given a graph G = (V; E) with a treedecomposition of G with treewidth at most l, decide whether the treewidth (or pathwidth) of G is at most k, and if so, find a treedecomposition or (pathdecomposition) of G of width at most k, and that use O(V) time. In contrast with previous solutions, our algorithms do not rely on nonconstructive reasoning, and are single exponential in k and l. This result can be combined with a result of Reed [37], yielding explicit O(n log n) algorithms for the problem, given a graph G, to determine whether the treewidth (or pathwidth) of G is at most k, and if so, to find a tree (or path)decomposition of width at most k (k constant). Also, Bodlaender [13] has used the result of this paper to obtain linear time algorithms for these problems. We also show that for all constants k, there exists a polynomial time algorithm, that, when given a graph G = (V; E) with treewidth k, computes the pathwidth of G and a minimum path decomposition of G.
The Structure of the Models of Decidable Monadic Theories of Graphs
, 1991
"... In this article the structure of the models of decidable (weak) monadic theories of planar graphs is investigated. It is shown that if the (weak) monadic theory of a class K of planar graphs is decidable, then the treewidth in the sense of Robertson and Seymour (1984) of the elements of K is univer ..."
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Cited by 47 (2 self)
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In this article the structure of the models of decidable (weak) monadic theories of planar graphs is investigated. It is shown that if the (weak) monadic theory of a class K of planar graphs is decidable, then the treewidth in the sense of Robertson and Seymour (1984) of the elements of K is universally bounded and there is a class T of trees such that the (weak) monadic theory of K is interpretable in the (weak) monadic theory of T.
On the Treewidth and Pathwidth of Permutation Graphs
, 1992
"... In this paper we discuss the problem of finding the treewidth and pathwidth of permutation graphs. If G[r] is a permutation graph with treewidth k, then we show that the pathwidth of G[r] is at most 2k, and we give an algo rithm which constructs a pathdecomposition with width at most 2k in time ..."
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Cited by 41 (11 self)
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In this paper we discuss the problem of finding the treewidth and pathwidth of permutation graphs. If G[r] is a permutation graph with treewidth k, then we show that the pathwidth of G[r] is at most 2k, and we give an algo rithm which constructs a pathdecomposition with width at most 2k in time O(nk). We assume that the permutation r is given. For permutation graphs of which the treewidth is bounded by some constant, this result, together with previous results ([9], [15]), shows that the treewidth and pathwidth can be computed in linear time.
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 19 (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.
Parameterized Algorithms for Directed Maximum Leaf Problems
 Proc. ICALP 2007, LNCS 4596
, 2007
"... Abstract. We prove that finding a rooted subtree with at least k leaves in a digraph is a fixed parameter tractable problem. A similar result holds for finding rooted spanning trees with many leaves in digraphs from a wide family L that includes all strong and acyclic digraphs. This settles complete ..."
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Cited by 12 (7 self)
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Abstract. We prove that finding a rooted subtree with at least k leaves in a digraph is a fixed parameter tractable problem. A similar result holds for finding rooted spanning trees with many leaves in digraphs from a wide family L that includes all strong and acyclic digraphs. This settles completely an open question of Fellows and solves another one for digraphs in L. Our algorithms are based on the following combinatorial result which can be viewed as a generalization of many results for a ‘spanning tree with many leaves ’ in the undirected case, and which is interesting on its own: If a digraph D ∈ L of order n with minimum indegree at least 3 contains a rooted spanning tree, then D contains one with at least (n/2) 1/5 − 1 leaves. 1
A new algorithm for finding trees with many leaves
, 2001
"... We present an algorithm that finds trees with at least k leaves in undirected and directed graphs. These problems are known as Maximum Leaf Spanning Tree for undirected graphs, and, respectively, Directed Maximum Leaf OutTree and Directed Maximum Leaf Spanning OutTree in the case of directed grap ..."
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Cited by 12 (1 self)
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We present an algorithm that finds trees with at least k leaves in undirected and directed graphs. These problems are known as Maximum Leaf Spanning Tree for undirected graphs, and, respectively, Directed Maximum Leaf OutTree and Directed Maximum Leaf Spanning OutTree in the case of directed graphs. The run time of our algorithm is O(poly(V ) + 4 k k 2) on undirected graphs, and O(4 k V ·E) on directed graphs. Currently, the fastest algorithms for these problems have run times of O(poly(n) + 6.75 k poly(k)) and 2 O(k log k) poly(n), respectively.
Two Results on a Partial Ordering of Finite Sequences
 Comment. Math. Univ. Carolinae
, 1993
"... In the first part of the paper we are concerned about finite sequences (over arbitrary symbols) u for which Ex(u; n) = O(n). The function Ex(u; n) measures the maximum length of finite sequences over n symbols which contain no subsequence of the type u. It follows from the result of Hart and Sharir ..."
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Cited by 10 (5 self)
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In the first part of the paper we are concerned about finite sequences (over arbitrary symbols) u for which Ex(u; n) = O(n). The function Ex(u; n) measures the maximum length of finite sequences over n symbols which contain no subsequence of the type u. It follows from the result of Hart and Sharir that the containment ababa OE u is a (minimal) obstacle to Ex(u; n) = O(n). We show by means of a construction due to Sharir and Wiernik that there is another obstacle to the linear growth.
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 10 (5 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 ...