## Kernels in planar digraphs (2001)

### Cached

### Download Links

- [www.optimization-online.org]
- [www.cs.rhul.ac.uk]
- [www.dcs.rhul.ac.uk]
- [www.cs.rhul.ac.uk]
- [eprints.pascal-network.org]
- DBLP

### Other Repositories/Bibliography

Venue: | In Optimization Online. Mathematical Programming Society |

Citations: | 15 - 1 self |

### BibTeX

@INPROCEEDINGS{Gutin01kernelsin,

author = {Gregory Gutin and Ton Kloks and Chuan Min Lee and Anders Yeo},

title = {Kernels in planar digraphs},

booktitle = {In Optimization Online. Mathematical Programming Society},

year = {2001},

pages = {174--184}

}

### Years of Citing Articles

### OpenURL

### Abstract

A set S of vertices in a digraph D = (V, A) is a kernel if S is independent and every vertex in V − S has an out-neighbor in S. We show that there exist O(n2 19.1 √ k + n 4)time and O(2 19.1 √ k k 9 + n 2)-time algorithms for checking whether a planar digraph D of order n has a kernel with at most k vertices. Moreover, if D has a kernel of size at most k, the algorithms find such a kernel of minimal size. 1

### Citations

268 |
Theory of games and economic behaviour
- Neumann, Morgenstern
- 1944
(Show Context)
Citation Context ... This sufficient condition for a digraph to have a kernel has been generalized by several authors. For short accounts on the topic see [4, 6]. Kernels in digraphs were introduced in different ways in =-=[22, 27]-=-. It seems that von Neumann and Morgenstern [27] were the first to introduce kernels when describing winning positions in 2-person games. For important applications of kernels in game theory see [13, ... |

251 | Digraphs: Theory, Algorithms, and Applications
- Bang-Jensen, Gutin
- 2001
(Show Context)
Citation Context ...rnels. It is easy to see that every acyclic digraph has a kernel. This sufficient condition for a digraph to have a kernel has been generalized by several authors. For short accounts on the topic see =-=[4, 6]-=-. Kernels in digraphs were introduced in different ways in [22, 27]. It seems that Neumann [27] was the first to introduce kernels when describing winning positions in 2-person games. For important ap... |

204 | A linear time algorithm for finding treedecompositions of small treewidth
- Bodlaender
- 1993
(Show Context)
Citation Context ... tree decompositions of G. Computing treewidth of graphs in general is NP-complete, however the problem is FPT. Moreover there exists a linear time algorithm to check if a graph has bounded treewidth =-=[7, 20]-=-. Alber et al. [1] were the first to prove that a planar graph G of domination number k has treewidth O( √ k). They actually showed that the treewidth of G is at most 6 √ 34 √ k+8. This result was imp... |

190 | The Web as a Graph
- Kumar, Raghavan, et al.
- 2000
(Show Context)
Citation Context ...blem of determining whether a digraph has a subgraph consisting of two sets of vertices A and B, each of size k, with an arc from every vertex of A to every vertex of B. The last problem was posed in =-=[21]-=-, a paper on data-mining the internet to identify on-line communities. The problem seems to be an obvious and perhaps easy candidate for W [1]-hardness, but in fact has resisted much effort [15]. 3s2 ... |

107 | R.: Fixed parameter algorithms for dominating set and related problems on planar graphs. Algorithmica 33
- Alber, Bodlaender, et al.
- 2002
(Show Context)
Citation Context ...e existence an O(8 k n)-algorithm. An important breakthrough was discovery of an algorithm with sub-exponential f(k); such a result, O(c √ k n)) with c ≤ 4 6 √ 34 , was first obtained by Alber et al. =-=[1]-=-. Recently, the upper bound on the constant c was improved to 2 27 [19] and further to 2 15.13 [17]. Also, [2], [16] and [17] obtained algorithms of complexity O(k8 k + n 3 ), O(2 16.48√ k + n 3 ) and... |

73 |
Treewidth: Computations and Approximations
- Kloks
- 1994
(Show Context)
Citation Context ... tree decompositions of G. Computing treewidth of graphs in general is NP-complete, however the problem is FPT. Moreover there exists a linear time algorithm to check if a graph has bounded treewidth =-=[7, 20]-=-. Alber et al. [1] were the first to prove that a planar graph G of domination number k has treewidth O( √ k). They actually showed that the treewidth of G is at most 6 √ 34 √ k+8. This result was imp... |

68 | Parameterized complexity: A framework for systematically confronting computational intractability
- Downey, Fellows, et al.
- 1997
(Show Context)
Citation Context ... if D has an (R, B)-kernel of size at most k. 3 Additive FPT algorithms We start by giving a short description of a general approach to obtain an additive FPT algorithm from a multiplicative one (see =-=[8, 11]-=-). Proposition 3.1 Assume we have an algorithm which runs in time O(f(k)n) for a parameterized problem (on digraphs) with parameter k for some function f(k). Then, alternatively, we can obtain an algo... |

67 | Dominating sets in planar graphs: branch-width and exponential speed-up
- Fomin, Thilikos
(Show Context)
Citation Context ...b-exponential f(k); such a result, O(c √ k n)) with c ≤ 4 6 √ 34 , was first obtained by Alber et al. [1]. Recently, the upper bound on the constant c was improved to 2 27 [19] and further to 2 15.13 =-=[17]-=-. Also, [2], [16] and [17] obtained algorithms of complexity O(k8 k + n 3 ), O(2 16.48√ k + n 3 ) and O(k 4 + 2 15.13√ k k + n 3 ), respectively. Notice that the last three ’additive FPT’ algorithms a... |

59 | Parameterized Computational Feasibility
- Downey, Fellows
- 1995
(Show Context)
Citation Context ...lar, in a sense, to that for Dominating Set. Several researchers studied Dominating Set (and some of its variations including Independent Dominating Set) for planar undirected graphs. Already in 1995 =-=[12]-=- claimed an algorithm of complexity O(11 k n) for checking whether a planar graph on n vertices has a dominating set with at most k vertices. The analysis of the algorithm there turned out to be flawe... |

53 |
Theory of Games and Economic Behaviour
- Neumann, O
- 1972
(Show Context)
Citation Context ... This sufficient condition for a digraph to have a kernel has been generalized by several authors. For short accounts on the topic see [4, 6]. Kernels in digraphs were introduced in different ways in =-=[22, 27]-=-. It seems that Neumann [27] was the first to introduce kernels when describing winning positions in 2-person games. For important applications of kernels in game theory see [13, 14, 26]. Applications... |

37 | A refined search tree technique for Dominating Set on planar graphs
- Alber, Fan, et al.
- 2005
(Show Context)
Citation Context ...an independent dominating set in an undirected graph G is a kernel in the digraph obtained from G by replacing every edge {x, y} by the pair xy, yx of arcs and 2ssince Independent Dominating Set is W =-=[2]-=--complete (see [10], p. 464), Kernel is W [2]-hard. This means that Kernel is fixed-parameter intractable unless very unlikely collapses occur of parameterized complexity classes (see [10] for details... |

18 |
Obstructions to Tree Decomposition
- unknown authors
- 1991
(Show Context)
Citation Context ...d is due to Fomin and Thilikos [17] (using branch decomposition algorithms by Seymour and Thomas [29] and a transformation from a branch decomposition to a tree decomposition by Robertson and Seymour =-=[28]-=-): Theorem 2.2 [17] Let G be a planar graph with n vertices. There is an O(n 4 )-time algorithm that either constructs a tree decomposition of G with O(n) nodes and of width at most 9.55 √ k, or deter... |

13 | Polynomial time data reduction for dominating set
- Alber, Fellows, et al.
(Show Context)
Citation Context ...ear in k and k ′ ≤ ck for some constant c. (Unlike in our reduction above, no set S is allowed.) For Planar Dominating Set such a reduction of both theoretical and practical significance was given in =-=[3]-=-. 5 Acknowledgements We thank Hans Bodlaender, Maw-Shang Chang, and Annelies Jacobs for fruitful discussions and informative comments on the first version of the paper and Fedor Fomin for useful comme... |

12 | Locally Semicomplete Digraphs
- Guo
- 1995
(Show Context)
Citation Context ...digraphs D with degree constraints d + (x) ≤ 2, d − (x) ≤ 2 and d(x) ≤ 3 for all vertices x. Finding kernels in special classes of digraphs seems to be a mostly open field of study. It has been shown =-=[5]-=- that the kernel problem is polynomial time solvable for locally semicomplete digraphs, digraphs in which the out-neighbors (in-neighbors) of every vertex are adjacent. Kernels in some classes of plan... |

11 |
Recent problems and results about kernels in directed graphs
- Berge, Duchet
- 1990
(Show Context)
Citation Context ...rnels. It is easy to see that every acyclic digraph has a kernel. This sufficient condition for a digraph to have a kernel has been generalized by several authors. For short accounts on the topic see =-=[4, 6]-=-. Kernels in digraphs were introduced in different ways in [22, 27]. It seems that Neumann [27] was the first to introduce kernels when describing winning positions in 2-person games. For important ap... |

11 |
Call routing and the ratcatcher, Combinatorica 14
- Seymour, Thomas
- 1994
(Show Context)
Citation Context ... This result was improved to 16.5 √ k+50 by Kanj and Perkovic [19]. The current best result of this kind is due to Fomin and Thilikos [17] (using branch decomposition algorithms by Seymour and Thomas =-=[29]-=- and a transformation from a branch decomposition to a tree decomposition by Robertson and Seymour [28]): Theorem 2.2 [17] Let G be a planar graph with n vertices. There is an O(n 4 )-time algorithm t... |

9 |
Private communication
- Fellows
- 2011
(Show Context)
Citation Context ...ed in [21], a paper on data-mining the internet to identify on-line communities. The problem seems to be an obvious and perhaps easy candidate for W [1]-hardness, but in fact has resisted much effort =-=[15]-=-. 3s2 Small kernels in planar digraphs We start from a well-known definition of a tree decomposition of a graph. Definition 2.1 A tree decomposition of an undirected graph G = (V, E) is a pair (T, S),... |

9 |
Counterexamples of the 0-1 law for fragments of existential second-order logic: an overview. The Bulletin of Symbolic Logic
- Bars
(Show Context)
Citation Context ...l complexity, artificial intelligence, graph theory, combinatorics and coding theory. For recent applications to counterexamples to the 0-1 laws in fragments of monadic second order logic, see, e.g., =-=[24, 25]-=-. Chvátal proved (see [18]) that the problem of deciding whether a digraph has a kernel is NP-complete. Fraenkel [13] showed that the problem remains NP-complete even for planar digraphs D with degree... |

8 | Combinatorial game theory foundations applied to digraph kernels
- Fraenkel
- 1997
(Show Context)
Citation Context ...different ways in [22, 27]. It seems that Neumann [27] was the first to introduce kernels when describing winning positions in 2-person games. For important applications of kernels in game theory see =-=[13, 14, 26]-=-. Applications of kernels are widespread and appear in diverse fields such as logic, computational complexity, artificial intelligence, graph theory, combinatorics and coding theory. For recent applic... |

6 |
The 0-1 law fails for frame satisfiability of propositional modal logic
- Bars
- 2002
(Show Context)
Citation Context ...l complexity, artificial intelligence, graph theory, combinatorics and coding theory. For recent applications to counterexamples to the 0-1 laws in fragments of monadic second order logic, see, e.g., =-=[24, 25]-=-. Chvátal proved (see [18]) that the problem of deciding whether a digraph has a kernel is NP-complete. Fraenkel [13] showed that the problem remains NP-complete even for planar digraphs D with degree... |

5 |
A geometric approach to parameterized algorithms for domination problems on planar graphs
- Fernau, Juedes
(Show Context)
Citation Context ...); such a result, O(c √ k n)) with c ≤ 4 6 √ 34 , was first obtained by Alber et al. [1]. Recently, the upper bound on the constant c was improved to 2 27 [19] and further to 2 15.13 [17]. Also, [2], =-=[16]-=- and [17] obtained algorithms of complexity O(k8 k + n 3 ), O(2 16.48√ k + n 3 ) and O(k 4 + 2 15.13√ k k + n 3 ), respectively. Notice that the last three ’additive FPT’ algorithms are of complexity ... |

4 |
On the existence of subexponential-time parameterized algorithms
- Cai, Juedes
- 2003
(Show Context)
Citation Context ... n 3 ), respectively. Notice that the last three ’additive FPT’ algorithms are of complexity cubic in n. The O(c √ k n)) complexity has a good chance to be optimal, in a sense. Indeed, Cai and Juedes =-=[9]-=- showed that there cannot be a 2 o(√ k) n O(1) algorithm for Planar Dominating Set unless 3SAT ∈ DT IME(2 o(n) ), which is considered to be unlikely. Returning to Planar Kernel, recall that it is more... |

3 |
Generating functions for kernels of digraphs
- Banderier, Bars, et al.
(Show Context)
Citation Context ...ynomial time solvable for locally semicomplete digraphs, digraphs in which the out-neighbors (in-neighbors) of every vertex are adjacent. Kernels in some classes of planar graphs were investigated in =-=[23]-=-. In this paper we study the problem of finding a kernel in a digraph using methods of parameterized complexity [10]; see [10] for all undefined parameterized complexity terms. Definition 1.1 Consider... |

1 |
Advice classes of parameterised tractability, Annals of Pure and Applied Logic 84
- Cai, Chen, et al.
- 1997
(Show Context)
Citation Context ... if D has an (R, B)-kernel of size at most k. 3 Additive FPT algorithms We start by giving a short description of a general approach to obtain an additive FPT algorithm from a multiplicative one (see =-=[8, 11]-=-). Proposition 3.1 Assume we have an algorithm which runs in time O(f(k)n) for a parameterized problem (on digraphs) with parameter k for some function f(k). Then, alternatively, we can obtain an algo... |

1 |
Planar kernel and Grundy with d ≤ 3, d
- Fraenkel
- 1981
(Show Context)
Citation Context ...different ways in [22, 27]. It seems that Neumann [27] was the first to introduce kernels when describing winning positions in 2-person games. For important applications of kernels in game theory see =-=[13, 14, 26]-=-. Applications of kernels are widespread and appear in diverse fields such as logic, computational complexity, artificial intelligence, graph theory, combinatorics and coding theory. For recent applic... |

1 |
Improved parametirized algorithms for planar dominating set
- Kanj, Perkovic
- 2002
(Show Context)
Citation Context ...very of an algorithm with sub-exponential f(k); such a result, O(c √ k n)) with c ≤ 4 6 √ 34 , was first obtained by Alber et al. [1]. Recently, the upper bound on the constant c was improved to 2 27 =-=[19]-=- and further to 2 15.13 [17]. Also, [2], [16] and [17] obtained algorithms of complexity O(k8 k + n 3 ), O(2 16.48√ k + n 3 ) and O(k 4 + 2 15.13√ k k + n 3 ), respectively. Notice that the last three... |

1 |
The generalisation of Richardson theorem
- Kwa´snik
- 1981
(Show Context)
Citation Context ... This sufficient condition for a digraph to have a kernel has been generalized by several authors. For short accounts on the topic see [4, 6]. Kernels in digraphs were introduced in different ways in =-=[22, 27]-=-. It seems that Neumann [27] was the first to introduce kernels when describing winning positions in 2-person games. For important applications of kernels in game theory see [13, 14, 26]. Applications... |

1 |
Leeuwen, Having a grundy numbering is NP-complete
- van
- 1976
(Show Context)
Citation Context ...different ways in [22, 27]. It seems that Neumann [27] was the first to introduce kernels when describing winning positions in 2-person games. For important applications of kernels in game theory see =-=[13, 14, 26]-=-. Applications of kernels are widespread and appear in diverse fields such as logic, computational complexity, artificial intelligence, graph theory, combinatorics and coding theory. For recent applic... |