## Dominating Sets in Planar Graphs: Branch-Width and Exponential Speed-up (2002)

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### BibTeX

@INPROCEEDINGS{Fomin02dominatingsets,

author = {Fedor V. Fomin and Dimitrios M. Thilikos},

title = {Dominating Sets in Planar Graphs: Branch-Width and Exponential Speed-up},

booktitle = {},

year = {2002},

pages = {168--177}

}

### Years of Citing Articles

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### Abstract

Graph minors theory, developed by Robertson & Seymour, provides a list of powerful theoretical results and tools. However, the wide spread opinion in Graph Algorithms community about this theory is that it is mainly of theoretical importance.

### Citations

897 | Parameterized complexity
- Downey, Fellows
- 1999
(Show Context)
Citation Context ...cannot be solved in polynomial time unless P=NP. The last five years were the evidence of dramatic improvements of fixed parameter algorithms for the Planar Dominating Set problem. Downey and Fellows =-=[10]-=- suggested an algorithm with running time O(11 k n). Later the running time was reduced to O(8 k n) [2]. The first algorithm with a sublinear exponent for the problem with running time O(4 6 p 34k n) ... |

169 |
Double domination in graphs
- Harary, TW
- 2000
(Show Context)
Citation Context ...antly smaller than those when using tree-width? This paper is partially motivated by this question. Previous results. Dominating Set is one of the NP-complete core problems. The book of Haynes et al. =-=[14] is a nice-=- source for further references on the dominating problem. Planar Dominating Set and related parameters appear to be very "hot" topics at the moment. We counted at least 5 papers on this subj... |

167 | Graph minors. X. Obstructions to treedecomposition
- Robertson, Seymour
- 1991
(Show Context)
Citation Context ... + O(1). This implies that our upper bound 3 p 4:5 p k = 6:364 p k is not far from the optimal. Notice also that combining our bound for branch-width with the well known result of Robertson & Seymour =-=[20]-=- that for any graph G with at least 3 edges tree-width of G is always bounded by 3 2 times its branch-width, we have that the tree-width of a planar graph with dominating set of size k is at most 4:5 ... |

112 |
Quickly excluding a planar graph
- Robertson, Seymour, et al.
- 1994
(Show Context)
Citation Context ...close but surprisingly many Graph Minors theorems are much more easy to prove by using branch-width instead of tree-width. Wonderful examples of using branch-width in proof techniques can be found in =-=[22]-=- and [23]. Another powerful property of branch-width is that it can be naturally generalized for hypergraphs and matroids. A good example of generalization of Robertson & Seymour theory for matroids b... |

106 | Fixed parameter algorithms for dominating set and related problems on planar graphs, Algorithmica 33
- Alber, Bodlaender, et al.
- 2002
(Show Context)
Citation Context ...ning time was reduced to O(8 k n) [2]. The first algorithm with a sublinear exponent for the problem with running time O(4 6√ 34k n) (which is approximately O(2 70 √ k n)) was given by Alber 2set =-=al. [1]. R-=-ecently, Kanj & Perkovuć [17] announced a faster algorithm of running time O(2 27√ k n). The main idea to handle Planar Dominating Set which was used in several papers is that every planar graph wi... |

91 |
Call routing and the ratcatcher
- Seymour, Thomas
- 1994
(Show Context)
Citation Context ... surprisingly many Graph Minors theorems are much more easy to prove by using branch-width instead of tree-width. Wonderful examples of using branch-width in proof techniques can be found in [22] and =-=[23]-=-. Another powerful property of branch-width is that it can be naturally generalized for hypergraphs and matroids. A good example of generalization of Robertson & Seymour theory for matroids by using b... |

62 | Parameterized complexity: exponential speed-up for planar graph problems
- Alber, Fernau, et al.
- 2001
(Show Context)
Citation Context ...tract. Phase 1 can be the common preprocessing step for a wide number of problems related to the Planar Dominating Set problem. Phase 2 is adapted in each problem in the same fashion as it is done in =-=[1, 3, 8, 9, 13, 18, 19]-=-. That way, our upper bound implies the construction of faster algorithms for a series of problems when their inputs are restricted to planar graphs. As a sample we mention the following: Independent ... |

37 | Branch-width and well-quasiordering in matroids and graphs
- Geelen, Gerards, et al.
(Show Context)
Citation Context ...at it can be naturally generalized for hypergraphs and matroids. A good example of generalization of Robertson & Seymour theory for matroids by using branch-width is the recent paper by Geelen et al. =-=[12]-=-. Algorithms for problems expressible in MSOL on matroids of bounded branchwidth are discussed by Hlineny [16]. Alekhnovich & Razborov [5] uses branch-width of hypergraphs to design algorithms for SAT... |

36 | Refined search tree technique for dominating set on planar graphs
- Alber, Fan, et al.
- 2001
(Show Context)
Citation Context ...vements of fixed parameter algorithms for the Planar Dominating Set problem. Downey and Fellows [10] suggested an algorithm with running time O(11 k n). Later the running time was reduced to O(8 k n) =-=[2]-=-. The first algorithm with a sublinear exponent for the problem with running time O(4 6 p 34k n) (which is approximately O(2 70 p k n)) was given by Alber et al. [1]. Recently, Kanj & Perkivu'c [17] a... |

35 | Fixed parameter algorithms for planar dominating set and related problems
- Alber, Bodlaender, et al.
- 2000
(Show Context)
Citation Context ...nning time was reduced to O(8 k n) [2]. The first algorithm with a sublinear exponent for the problem with running time O(4 6 p 34k n) (which is approximately O(2 70 p k n)) was given by Alber et al. =-=[1]-=-. Recently, Kanj & Perkivu'c [17] announced a faster algorithm of running time O(2 27 p k n). In this paper we introduce an algorithm of running time O(2 12:75 p k n + n 4 ), which is a significant st... |

33 | R.: Improved Tree Decomposition Based Algorithms for Domination-like Problems
- Alber, Niedermeier
- 2002
(Show Context)
Citation Context ...g techniques on graphs of bounded tree-width are implemented. The running time of the dynamic programming algorithm for dominating set on graphs of tree-width t is O(2 2t n). (See Alber & Niedermeier =-=[4]-=-.) The main disadvantage of this approach is that the constant c is too large for practical applications. The best known constant c = 6 p 34 = 34:98 due to Alber et al. [1] was very recently beaten by... |

27 |
Planar separators
- Alon, Seymour, et al.
- 1994
(Show Context)
Citation Context ... details). As it was noticed by Robin Thomas (in private communication), a better bound bw(G)sp 4:5 \Delta jV (G)j can be obtained by suitably adapting the arguments from Alon, Seymour & Thomas paper =-=[6]-=-. Another proof of this inequality can be found in [11]. This proof is based on a relation between slopes and majorities, the two notions introduced by Robertson & Seymour in [20] and Alon, Seymour & ... |

25 |
Improved parameterized algorithms for planar dominating set
- Kanj, Perković
(Show Context)
Citation Context ...n) [2]. The first algorithm with a sublinear exponent for the problem with running time O(4 6 p 34k n) (which is approximately O(2 70 p k n)) was given by Alber et al. [1]. Recently, Kanj & Perkivu'c =-=[17]-=- announced a faster algorithm of running time O(2 27 p k n). In this paper we introduce an algorithm of running time O(2 12:75 p k n + n 4 ), which is a significant step towards a practical algorithm.... |

23 | Thilikos, Graphs with branchwidth at most three
- Bodlaender, M
- 1999
(Show Context)
Citation Context ...h is also promising. A comment made by Bill Cook (according to private communication with Hans Bodlaender) is that for some problems branch-width is more suitable for actual implementations (see also =-=[7]-=-). In theory, there is not a big difference between tree-width and branch-width based algorithms. However in practice, branch-width is sometimes easier to use. The question is: Are there examples wher... |

22 | Satisfiability, branch-width and tseitin tautologies
- Alekhnovich, Razborov
- 2002
(Show Context)
Citation Context ...s by using branch-width is the recent paper by Geelen et al. [12]. Algorithms for problems expressible in MSOL on matroids of bounded branchwidth are discussed by Hlineny [16]. Alekhnovich & Razborov =-=[5]-=- uses branch-width of hypergraphs to design algorithms for SAT. From a practical point of view, branch-width is also promising. A comment made by Bill Cook (according to private communication with Han... |

21 | Efficient data reduction for dominating set: A linear problem kernel for the planar case
- Alber, Fellows, et al.
- 2002
(Show Context)
Citation Context ...raph G with a dominating set of size ≤ k, the branch-width of G is at most 3 √ 4.5 √ k. Combining our bound with the Seymour & Thomas algorithm and with recent results of Alber, Fellows & Nieder=-=meier [3] o-=-n a liner problem kernel of Planar Dominating Set and with a dynamic programming approach on graphs of bounded branch-width, we obtain an algorithm of running time O(k 4 + 2 15.13√ k k + n 3 ). Noti... |

20 |
New algorithms for k-face cover, kfeedback vertex set, and k-disjoint cycles on plane and planar graphs
- Kloks, Lee, et al.
- 2002
(Show Context)
Citation Context ...tract. Phase 1 can be the common preprocessing step for a wide number of problems related to the Planar Dominating Set problem. Phase 2 is adapted in each problem in the same fashion as it is done in =-=[1, 3, 8, 9, 13, 18, 19]-=-. That way, our upper bound implies the construction of faster algorithms for a series of problems when their inputs are restricted to planar graphs. As a sample we mention the following: Independent ... |

19 |
Maximum clique transversals
- Chang, Kloks, et al.
(Show Context)
Citation Context ...tract. Phase 1 can be the common preprocessing step for a wide number of problems related to the Planar Dominating Set problem. Phase 2 is adapted in each problem in the same fashion as it is done in =-=[1, 3, 8, 9, 13, 18, 19]-=-. That way, our upper bound implies the construction of faster algorithms for a series of problems when their inputs are restricted to planar graphs. As a sample we mention the following: Independent ... |

13 | Thilikos, Exponential speedup of fixed parameter algorithms on K3;3minor-free or K5-minor-free graphs
- Demaine, Hajiaghayi, et al.
(Show Context)
Citation Context |

13 |
Branch Decompositions and their Applications
- Hicks
- 2000
(Show Context)
Citation Context ...Thomas computing such a branchdecomposition of a planar graph in O(n 4 ) time. This algorithm has not the so-called "enormous hidden constants" and is really practical. (We refer to the work=-= of Hicks [15]-=- on implementations of Seymour & Thomas algorithm.) Our main combinatorial result is that for every planar graph G with dominating set k, the branch-width of G is at most 3 p 4:5 p k. Combining our bo... |

12 |
Thilikos. New upper bounds on the decomposability of planar graphs
- Fomin, M
- 2005
(Show Context)
Citation Context ...ivate communication), a better bound bw(G) ≤ � 4.5 · |V (G)| can be obtained by suitably adapting the arguments from Alon, Seymour & Thomas paper [6]. Another proof of this inequality can be foun=-=d in [12]-=-. This proof is based on a relation between slopes and majorities, the two notions introduced by Robertson & Seymour in [19] and Alon, Seymour & Thomas in [6] respectively. Theorem 2.4 ([12]). For any... |

6 |
P.D.: An algorithm for the ring-routing problem. Bellcore technical memorandum, Bellcore
- Cook, Seymour
- 1994
(Show Context)
Citation Context ...of hypergraphs to design algorithms for SAT. From a practical point of view, branch-width is also promising. For some problems branch-width is more suitable for actual implementations. Cook & Seymour =-=[9]-=- used branch decompositions to solve ring routing problem arising in the design of reliable cost effective SONET networks. (See also [7].) In theory, there is not a big difference between tree-width a... |

5 |
Feedback vertex sets and disjoint cycles in planar (di)graphs
- Kloks, Lee, et al.
- 2001
(Show Context)
Citation Context |

4 |
Parameterized tractability of some (efficient) Y -domination variants for planar graphs and t-degenerate graphs
- Kloks, Cai
- 2000
(Show Context)
Citation Context |

3 |
Parameterized tractability of some (ecient) Y -domination variants for planar graphs and t-degenerate graphs
- Kloks, Cai
- 2000
(Show Context)
Citation Context ...e that similar approach can be applied for a wide number of problems related to the Planar Dominating Set problem. Phase 3 in Section 4 is adapted in each problem in the same fashion as it is done in =-=[1, 4, 8, 10, 18]-=-. When a reduction to a linear kernel is not possible (Phase 1) our approach provides algorithms of running time O(2 c # k n + n 4 ). (Here constant c depends from the type of the problem.) That way, ... |

3 |
Fernau Niedermeier, Refined search tree technique for dominating set on planar graphs
- Alber, Fan, et al.
- 2000
(Show Context)
Citation Context ...ovements of fixed parameter algorithms for the Planar Dominating Set problem. Downey and Fellows [10] suggested an algorithm with running time O(11 k n). Later, a O(8 k n) time algorithm was given in =-=[2]-=-. The first algorithm with a sublinear exponent for the problem with running time O(4 6√ 34k n) (which is approximately O(2 70√ k n)) was given by Alber et al. [1]. Recently, Kanj & Perković [16] anno... |

2 |
parse trees, and second-order monadic logic for matroids over finite fields
- Hlineny, Branch-width
- 2003
(Show Context)
Citation Context ... & Seymour theory for matroids by using branch-width is the recent paper by Geelen et al. [12]. Algorithms for problems expressible in MSOL on matroids of bounded branchwidth are discussed by Hlineny =-=[16]-=-. Alekhnovich & Razborov [5] uses branch-width of hypergraphs to design algorithms for SAT. From a practical point of view, branch-width is also promising. A comment made by Bill Cook (according to pr... |

1 |
Thilikos, An upper bound on the decomposability of planar graphs, Techical Report, Departament de Llenguatges i Sistemes Inform`atics, Universitat Polit`ecnica de Catalunya
- Fomin, M
- 2002
(Show Context)
Citation Context ...a planar graph in terms of its size. The best published bound for the branch-width we were able to find in the literature, is bw(G)s4 p jV (G)j \Gamma 3 which follows directly from 5 Theorem 2.1 (see =-=[11]-=- for the details). As it was noticed by Robin Thomas (in private communication), a better bound bw(G)sp 4:5 \Delta jV (G)j can be obtained by suitably adapting the arguments from Alon, Seymour & Thoma... |