### Citations

441 |
Graph minors II. Algorithmic aspects of tree-width.
- Robertson, Seymour
- 1986
(Show Context)
Citation Context ...most k? We prove that the problem is NP-complete for any fixed k ≥ 4 and polynomial for k ≤ 2. On going work also suggests it is polynomial for k = 3. 1 Introduction A tree-decomposition of a graph G =-=[11]-=- is a way to represent G by a family of subsets of its vertex-set organized in a tree-like manner and satisfying some connectivity property. The treewidth of G measures the proximity of G with a tree.... |

386 | Complexity of finding embeddings in a k-tree
- Arnborg, Corneil, et al.
- 1987
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Citation Context ...Work. The problem of computing “good” tree-decompositions has been extensively studied. Computing optimal tree-decomposition - i.e., with width tw(G) - is NP-complete in the class of general graphs G =-=[1]-=-. For any fixed k ≥ 1, Bodlaender designed an algorithm that computes, in time O(kk3n), a treedecomposition of width k of any n-node graph with treewidth at most k [3]. Very recently, a single-exponen... |

301 | The monadic second-order logic of graphs I: recognizable sets of finite graphs
- Courcelle
- 1990
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Citation Context ... algorithms for solving graph problems. As an example, the famous Courcelle’s Theorem states that any problem expressible in MSOL can be solved in linear-time in the class of bounded treewidth graphs =-=[5]-=-. Another framework based on graph decompositions is the bi-dimensionality theory that allowed the design of sub-exponential-time algorithms for many problems in the class of graphs excluding some fix... |

288 |
A linear-time algorithm for finding tree-decompositions of small treewidth.
- Bodlaender
- 1996
(Show Context)
Citation Context ...in the class of general graphs G [1]. For any fixed k ≥ 1, Bodlaender designed an algorithm that computes, in time O(kk3n), a treedecomposition of width k of any n-node graph with treewidth at most k =-=[3]-=-. Very recently, a single-exponential (in k) algorithm has been proposed that computes a tree-decomposition with width at most 5k in the class of graphs with treewidth at most k [4]. As far as we know... |

52 |
Characterization and recognition of partial 3-trees.
- Arnborg, Proskurowski
- 1986
(Show Context)
Citation Context ...ractical algorithms for computing optimal tree-decompositions hold for graphs with treewidth at most 1 (trivial since tw(G) = 1 if and only if G is a tree), 2 (graphs excluding K4 as a minor) [13], 3 =-=[2, 9, 10]-=- and 4 [12]. We are not aware of any work dealing with the computation of tree-decompositions with minimum size. In [7], Dereniowski et al. consider the problem of size-constrained path-decompositions... |

47 | The bidimensionality theory and its algorithmic applications.
- Demaine, Hajiaghayi
- 2008
(Show Context)
Citation Context ...n graph decompositions is the bi-dimensionality theory that allowed the design of sub-exponential-time algorithms for many problems in the class of graphs excluding some fixed graph as a minor (e.g., =-=[6]-=-). Given a tree-decomposition with width w and size n, the time-complexity of most of such dynamic programming algorithms can be expressed as O(2wn) (or O(2w logwn) in the case of global problems). Th... |

29 | Parallel recognition of series-parallel graphs.
- Eppstein
- 1992
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Citation Context ...e theorem is to consider 2-connected graphs with treewidth 2. It is known that any 2-connected graph has treewidth 2 if and only if it has an open nested ear decomposition starting from a single edge =-=[8]-=-. In particular, this implies that any such a graph contains a node with degree 2. Given a 2-connected graph G with treewidth 2, let v be a node with degree 2 and u and w its neighbors. Let G′ obtaine... |

26 |
On linear recognition of tree-width at most four
- Sanders
- 1996
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Citation Context ...ms for computing optimal tree-decompositions hold for graphs with treewidth at most 1 (trivial since tw(G) = 1 if and only if G is a tree), 2 (graphs excluding K4 as a minor) [13], 3 [2, 9, 10] and 4 =-=[12]-=-. We are not aware of any work dealing with the computation of tree-decompositions with minimum size. In [7], Dereniowski et al. consider the problem of size-constrained path-decompositions. Given any... |

25 | Algorithms Finding Tree-Decompositions of Graphs
- Matousek, Thomas
- 1991
(Show Context)
Citation Context ...ractical algorithms for computing optimal tree-decompositions hold for graphs with treewidth at most 1 (trivial since tw(G) = 1 if and only if G is a tree), 2 (graphs excluding K4 as a minor) [13], 3 =-=[2, 9, 10]-=- and 4 [12]. We are not aware of any work dealing with the computation of tree-decompositions with minimum size. In [7], Dereniowski et al. consider the problem of size-constrained path-decompositions... |

17 |
tree, partial 2-trees, and minimum IFI networks, Networks 13
- Wald, Colboum, et al.
- 1983
(Show Context)
Citation Context ...e only practical algorithms for computing optimal tree-decompositions hold for graphs with treewidth at most 1 (trivial since tw(G) = 1 if and only if G is a tree), 2 (graphs excluding K4 as a minor) =-=[13]-=-, 3 [2, 9, 10] and 4 [12]. We are not aware of any work dealing with the computation of tree-decompositions with minimum size. In [7], Dereniowski et al. consider the problem of size-constrained path-... |

10 | An O(ck · n) 5-Approximation Algorithm for Treewidth
- Bodlaender, Drange, et al.
- 2013
(Show Context)
Citation Context ...reewidth at most k [3]. Very recently, a single-exponential (in k) algorithm has been proposed that computes a tree-decomposition with width at most 5k in the class of graphs with treewidth at most k =-=[4]-=-. As far as we know, the only practical algorithms for computing optimal tree-decompositions hold for graphs with treewidth at most 1 (trivial since tw(G) = 1 if and only if G is a tree), 2 (graphs ex... |

2 | Minimum length path decompositions
- DERENIOWSKI, KUBIAK, et al.
- 2013
(Show Context)
Citation Context ...1 if and only if G is a tree), 2 (graphs excluding K4 as a minor) [13], 3 [2, 9, 10] and 4 [12]. We are not aware of any work dealing with the computation of tree-decompositions with minimum size. In =-=[7]-=-, Dereniowski et al. consider the problem of size-constrained path-decompositions. Given any positive integer k and any graph G with pathwidth at most k. Let lk(G) denote the smallest size (length) of... |

2 |
Characterization of partial 3-trees in terms of three structures
- KAJITANI, ISHIZUKA, et al.
- 1986
(Show Context)
Citation Context ...ractical algorithms for computing optimal tree-decompositions hold for graphs with treewidth at most 1 (trivial since tw(G) = 1 if and only if G is a tree), 2 (graphs excluding K4 as a minor) [13], 3 =-=[2, 9, 10]-=- and 4 [12]. We are not aware of any work dealing with the computation of tree-decompositions with minimum size. In [7], Dereniowski et al. consider the problem of size-constrained path-decompositions... |