## Approximation Algorithms for Classes of Graphs Excluding Single-Crossing Graphs as Minors (0)

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Citations: | 26 - 16 self |

### BibTeX

@MISC{Demaine_approximationalgorithms,

author = {Erik D. Demaine and Mohammadtaghi Hajiaghayi and Naomi Nishimura and Prabhakar Ragde and Dimitrios M. Thilikos},

title = {Approximation Algorithms for Classes of Graphs Excluding Single-Crossing Graphs as Minors},

year = {}

}

### Years of Citing Articles

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### Abstract

Many problems that are intractable for general graphs allow polynomial-time solutions for structured classes of graphs, such as planar graphs and graphs of bounded treewidth.

### Citations

11605 | Johnson: Computers and Intractability: A guide to theory of NP hardness, Feeman and - Garey, S - 1979 |

1385 |
Graph theory with applications
- R
- 1976
(Show Context)
Citation Context ...minology We assume that the reader is familiar with general concepts of graph theory such as graphs, trees, and planar graphs. The reader is referred to standard references for appropriate background =-=[BM76]-=-. Here we review a few terms used in this paper; others will be defined in the context of their use. ARTICLE IN PRESS E.D. Demaine et al. / Journal of Computer and System Sciences 69 (2004) 166–195 16... |

1105 | Depth-first search and linear graph algorithms - Tarjan - 1972 |

926 | Parameterized Complexity
- Downey, Fellows
- 1999
(Show Context)
Citation Context ...algorithms for variants of dominating set on single-crossing-minor-free ARTICLE IN PRESS E.D. Demaine et al. / Journal of Computer and System Sciences 69 (2004) 166–195 189 graphs. Downey and Fellows =-=[DF99]-=- introduced the notion of fixed-parameter tractability as a way to categorize the complexity of NP-hard problems based on one or more parameters of a problem. A problem is fixed-parameter tractable if... |

378 | Graph minors. II. Algorithmic aspects of tree-width - Robertson, Seymour - 1986 |

340 |
Complexity of finding embeddings in a k-tree
- Arnborg, Corneil, et al.
- 1987
(Show Context)
Citation Context ...dth, which is NP-complete even if we restrict the input graph to graphs of bounded degree [BT97], cocomparability graphs [ACP87,HM94], bipartite graphs [Klo96], or the complements of bipartite graphs =-=[ACP87]-=-. However, treewidth can be computed exactly in polynomial time for chordal graphs, permutation graphs [BKK95], circular-arc graphs [SSR94], circle graphs [Klo96], distance-hereditary graphs [BDK00], ... |

316 |
Approximation algorithms for NP-complete problems on planar graphs
- Baker
- 1994
(Show Context)
Citation Context ...rmance ratio of two for the independent set problem is achievable only for planar graphs of at least 22 400 vertices [CNS82]. Practical approximation schemes for planar graphs were developed by Baker =-=[Bak94]-=-, who formed a decomposition of G into overlapping k-outerplanar subgraphs. For any planar embedding, vertices can be put into layers by iteratively removing vertices on the outer face of the graph: v... |

225 | A tourist guide through treewidth - Bodlaender - 1993 |

202 |
A linear-time algorithm for finding tree-decompositions of small treewidth
- Bodlaender
- 1996
(Show Context)
Citation Context ...s not provide a constructive algorithm to obtain a tree decomposition of this width. Although we can construct a tree decomposition of width 3ð j iÞ þ cH in linear time using Bodlaender’s algorithm =-=[Bod96]-=-, again the hidden constant factor will depend on this entire width. Below we show how to reduce the constant to depend only on cH ; permitting substitution of approximation algorithms as was done at ... |

186 | Dividing a graph into triconnected components - Hopcraft, Tarjan - 1973 |

175 | Graph minors. X. Obstructions to treedecomposition - Robertson, Seymour - 1991 |

154 | Vertex cover: Further observations and further improvements
- Chen, Kanj, et al.
(Show Context)
Citation Context ...the parameter(s), it is polynomial in the problem size. One example is the problem of finding a vertex cover of size k; as k is the parameter and there is an algorithm running in time Oðkn þ 1:2852kÞ =-=[CKJ01]-=-. In this section, we extend to single-crossing-minor-free graphs a solution to k-dominating set on planar graphs, that is, the problem of finding a set S of at most k vertices such that each vertex i... |

140 | Treewidth: Algorithmic techniques and results - Bodlaender |

115 | Subgraph isomorphism in planar graphs and related problems - Eppstein - 1995 |

107 | Fixed parameter algorithms for Dominating Set and related problems on planar graphs - Alber, Bodlaender, et al. |

96 | Graph minors. iii. planar tree-width - Robertson, Seymour - 1984 |

93 | A separator theorem for graphs with excluded minor and its applications
- Alon, Seymour, et al.
- 1990
(Show Context)
Citation Context ...ntee additional structural properties. Early work in the area demonstrated the possibility of using planarity to obtain approximation schemes [LT80], later generalized to graphs without a fixed minor =-=[AST90]-=-. These approaches are impractical; a performance ratio of two for the independent set problem is achievable only for planar graphs of at least 22 400 vertices [CNS82]. Practical approximation schemes... |

93 | Call routing and the ratcatcher - Seymour, Thomas - 1994 |

88 | Diameter and treewidth in minor-closed graph families. Algorithmica 27 - Eppstein - 2000 |

88 | Efficient approximation for triangulation of minimum treewidth
- Amir
(Show Context)
Citation Context ...e heavily to the constant hidden in the order notation. In some contexts, it may make sense to replace Bodlaender’s linear-time algorithm for determining treewidth exactly with an approximation. Amir =-=[Ami01]-=- gives an algorithm running in time Oð24:38cH n2cHÞ which either returns a treedecomposition of width at most 4cH or answers that the treewidth is more than cH : This can be used to prove a version of... |

76 | Deciding first-order properties of locally tree-decomposable structures - Frick, Grohe - 2001 |

75 | Node-and edge-deletion NP-complete problems - Yannakakis - 1978 |

67 | The pathwidth and treewidth of cographs
- Bodlaender, Möhring
- 1993
(Show Context)
Citation Context ...idth twðHÞ: For W the set of vertices of G and H identified during the" operation, W is a clique in G and in H: As vertices in a clique must appear together in a bag in any decomposition of the graph =-=[BM93]-=-, there exist a node a in TDðGÞ such that WDwa and a node b in TDðHÞ such that WDwb: Hence, we can form a tree decomposition of width maxftwðGÞ; twðHÞg of G"H by adding an edge between a in TDðGÞ and ... |

62 | The extremal function for complete minors - Thomason |

62 | Approximating tree-width, pathwidth, frontsize, and shortest elimination tree
- Bodlaender, Gilbert, et al.
- 1995
(Show Context)
Citation Context ...raphs [BKK95], circular-arc graphs [SSR94], circle graphs [Klo96], distance-hereditary graphs [BDK00], and for graphs of a fixed treewidth [Bod96]. From the approximation viewpoint, Bodlaender et al. =-=[BGHK95]-=- gave an Oðlog nÞapproximation algorithm for treewidth on general graphs. A famous open problem is whether treewidth can be approximated within a constant factor. Constant-factor approximations are po... |

61 | Über eine Eigenschaft der ebenen Komplexe, Mathematische Annalen 114 - Wagner - 1937 |

60 | Parameterized complexity: exponential speedup for planar graph problems
- Alber, Fernau, et al.
- 2001
(Show Context)
Citation Context ... the time to construct tree decompositions and the number of Opti’s. The minimum vertex cover problem can be solved in Oð2wsnÞ time over each n-vertex partial w-tree whose tree decomposition is given =-=[AFN01]-=-. Thus, computing Opti; j takes Oð23ksjVðGLi; j ÞjÞ time on graph Gi; j; and hence for fixed i; as each vertex appears in at most two Li; j’s, Opti can be determined in Oð23ksjVðGÞjÞ time, as claimed... |

56 | Local tree-width, excluded minors, and approximation algorithms - Grohe |

56 | Graph minors -- a survey - Robertson, Seymour - 1985 |

55 | Efficient and constructive algorithms for the pathwidth and treewidth of graphs - Bodlaender, Kloks - 1996 |

52 | Graph Algorithms - Leeuwen - 1990 |

51 | Characterization and recognition of partial 3-trees - Arnborg, Proskurowski - 1986 |

45 | Treewidth and pathwidth of permutation graphs
- Bodlaender, Kloks, et al.
- 1995
(Show Context)
Citation Context ...ity graphs [ACP87,HM94], bipartite graphs [Klo96], or the complements of bipartite graphs [ACP87]. However, treewidth can be computed exactly in polynomial time for chordal graphs, permutation graphs =-=[BKK95]-=-, circular-arc graphs [SSR94], circle graphs [Klo96], distance-hereditary graphs [BDK00], and for graphs of a fixed treewidth [Bod96]. From the approximation viewpoint, Bodlaender et al. [BGHK95] gave... |

37 | Andrzej Proskurowski. Complexity of finding embeddings in a k-tree - Arnborg, Corneil - 1987 |

37 | A refined search tree technique for Dominating Set on planar graphs - Alber, Fan, et al. - 2005 |

34 | Fixed parameter algorithms for planar dom-inating set and related problems, in Algorithm theory-- Scandinavian Workshop on Algorithm Theory 2000 (Bergen,2000 - ALBER, BODLAENDER, et al. - 2000 |

33 | Dynamic programming algorithms on graphs with bounded tree-width - Bodlaender - 1988 |

33 | On simple characterization of k-trees - Rose - 1974 |

32 | Depth-first search and Kuratowski subgraphs - Williamson - 1984 |

32 | Treewidth and minimum fill-in: grouping the minimal separators - Bouchitté, Todinca |

25 | and Robert Endre Tarjan. Applications of a planar separator theorem - Lipton - 1980 |

24 | On linear recognition of tree-width at most four - Sanders - 1996 |

23 |
An approach to the subgraph homeomorphism problem
- Asano
- 1985
(Show Context)
Citation Context ... of planar graphs and graphs of small treewidth (defined formally in Section 3). Our result generalizes previous decomposition results for graphs excluding special single-crossing graphs such as K3;3 =-=[Asa85]-=- and K5 [KM92]. A second structural property is that of locally bounded treewidth which allows a layer decomposition, enabling us to represent a graph as a collection of subgraphs, each of bounded tre... |

23 | On treewidth approximations - Bouchitté, Kratsch, et al. - 2001 |

23 | A tourist guide through treewidth. Acta Cybernetica 11:1–21 - Bodlaender - 1993 |

22 | Über eine Eigenschaft der ebene Komplexe - Wagner - 1937 |

21 | Mohammadtaghi Hajiaghayi, and Dimitrios M. Thilikos. Subexponential parameterized algorithms on graphs of bounded genus and H-minor-free graphs - Demaine, Fomin |

21 | Treewidth of circle graphs - Kloks - 1996 |

20 | Arnborg and Andrzej Proskurowski. Linear time algorithms for NP-hard problems restricted to partial k-trees - Stefan |

20 |
Thilikos. Treewidth for graphs with small chordality
- Bodlaender, M
- 1997
(Show Context)
Citation Context ...complete the proof. & 5. Approximating treewidth A large amount of effort has been put into determining treewidth, which is NP-complete even if we restrict the input graph to graphs of bounded degree =-=[BT97]-=-, cocomparability graphs [ACP87,HM94], bipartite graphs [Klo96], or the complements of bipartite graphs [ACP87]. However, treewidth can be computed exactly in polynomial time for chordal graphs, permu... |