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## Bidimensional parameters and local treewidth (2004)

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Venue: | SIAM Journal on Discrete Mathematics |

Citations: | 31 - 17 self |

### Citations

1211 | Parameterized Complexity
- Downey, Fellows
- 1999
(Show Context)
Citation Context ...he last author was supported by EC contract IST-1999-14186: Project ALCOMFT (Algorithms and Complexity) - Future Technologies and by the Spanish CICYT project TIC-2002-04498-C05-03 (TRACER)s& Fellows =-=[14]-=-. Roughly speaking, a parameterized problem with parameter k is fixed-parameter tractable (FPT) if it admits an algorithm with running time f(k)|I| O(1) . (Here f is a function depending only on k and... |

1198 | Graph Theory
- Diestel
- 1997
(Show Context)
Citation Context ...e need the following facts about treewidth. The first fact is trivial. • For any complete graph Kn on n vertices, tw(Kn) =n − 1. The second fact is well known but its proof is not trivial. (See e.=-=g., [17].) -=-3s• The treewidth of the m × m grid is m. The next fact we need is the improved version of the Robertson & Seymour theorem on excluded grid minors [32] due to Diestel et al. [18]. (See also the tex... |

439 |
Graph minors II. Algorithmic aspects of treewidth,
- Robertson, Seymour
- 1986
(Show Context)
Citation Context ...he m × m grid with i ∈ {1, m} or j ∈ {1, m} are called boundary vertices and the rest of the vertices are called non-boundary vertices. The notion of treewidth was introduced by Robertson and Seymour =-=[25]-=-. A tree decomposition of a graph G is a pair ({Xi | i ∈ I}, T = (I, F )), with {Xi | i ∈ I} a family of subsets of V (G) and T a tree, such that 1. � i∈I Xi = V (G); 2. for all {v, w} ∈ E(G), there i... |

375 |
Approximation algorithms for NP-complete problems on planar graphs.
- BAKER
- 1994
(Show Context)
Citation Context ...roving that some problems like subgraph isomorphism and induced subgraph isomorphism can be solved in linear time on minor-closed graphs of bounded local treewidth. Also the classic Baker’s technique =-=[4]-=- for obtaining approximation schemes on planar graphs for different NP-hard problems can be generalized to minor-closed families of bounded local treewidth. (See [16] for a generalization of these tec... |

292 |
Graph rewriting: An algebraic and logic approach.
- COURCELLE
- 1990
(Show Context)
Citation Context ...th property for P if and only if F is of bounded local treewidth. The function f(p) in the parameter-treewidth property is 2 O(√ p log p) . 4 Algorithmic consequences and concluding remarks Courcelle =-=[6]-=- proved a meta-theorem on graphs of bounded treewidth; he showed that, if φ is a property of graphs that is definable in monadic second-order logic, then φ can be decided in linear time on graphs of b... |

268 | A tourist guide through treewidth
- Bodlaender
- 1993
(Show Context)
Citation Context ...ounded genus, and single-crossing-minor-free graphs. Many difficult graph problems can be solved efficiently when the input is restricted to graphs of bounded treewidth (see e.g., Bodlaender’s survey =-=[5]-=-). Eppstein [16] made a step forward by proving that some problems like subgraph isomorphism and induced subgraph isomorphism can be solved in linear time on minor-closed graphs of bounded local treew... |

185 | Vertex cover: Further observations and further improvements
- Chen, Kanj, et al.
(Show Context)
Citation Context ...ertex Cover, asking whether an input graph has at most k vertices that are incident to all its edges. When parameterized by k, the k-Vertex Cover problem admits a solution as fast as O(kn + 1.285 k ) =-=[7]-=-. Moreover, if we restrict k-Vertex Cover to planar graphs then it is possible to design FPT-algorithms where the contribution of k in the non-polynomial part of their complexity is subexponential. Th... |

146 |
On Local and Non-Local Properties
- Gaifman
- 1982
(Show Context)
Citation Context ... on 1/ε, φ, and the local treewidth bound. However, the running time of Frick and Grohe’s algorithm remains unanalyzed in terms of φ: their algorithm transforms φ into so-called “Gaifman norma=-=l form” [28]-=- and the complexity of this transformation is unknown. Using Theorems 2.3 and 4.1, we obtain a result along similar lines of Frick and Grohe. Specifically, consider any property that is solvable in po... |

111 | Fixed parameter algorithms for dominating set and related problems on planar graphs.
- ALBER, BODLAENDER, et al.
- 2002
(Show Context)
Citation Context ...rithm with running time O(11 d n). Later the running time was reduced to O(8 d n) [2]. An algorithm with a sublinear exponent for the problem with running time O(4 6√ 34d n) was given by Alber et al. =-=[1]-=-. Recently, Kanj & Perkovuć [23] improved the running time to O(2 27√ d n) and Fomin & Thilikos to O(2 15.13√ d d+n 3 +d 4 ) [18]. The fixed-parameter algorithms for extensions of planar graphs like b... |

110 | Diameter and treewidth in minor-closed graph families.
- EPPSTEIN
- 2000
(Show Context)
Citation Context ...ating set, and q-dominating set (for fixed q) (see also [11] for more examples). The proof of the main result uses the characterization of Eppstein for minorclosed families of bounded local treewidth =-=[16]-=- and Diestel et al.’s modification of the Robertson & Seymour excluded-grid-minor theorem [13]. In addition, the proof is constructive and can be used for constructing fixed-parameter algorithms to de... |

98 | Deciding first-order properties of locally treedecomposable graphs,
- Frick, Grohe
- 2001
(Show Context)
Citation Context ...g fixed-parameter algorithms to decide bidimensional parameters on minor-closed families of bounded local treewidth. In this sense, we extend to fixed-parameter algorithms the result of Frick & Grohe =-=[21]-=- that, for each property φ definable in first-order logic, and for each class of minor-closed graphs of bounded local treewidth, there is a (nonfixed-parameter) O(n 1+ɛ )-time algorithm deciding wheth... |

89 | Efficient approximation for triangulation of minimum treewidth
- Amir
- 2001
(Show Context)
Citation Context ...follows. First we check whether tw(G) is in 2 O(g−1 (k) log g −1 (k)) . By Theorems 2 and 4, if it is not, parameter P has value more than k on graph G. This step can be performed by Amir’s algorithm =-=[3]-=-, which for a given graph G and integer ω, either reports that the treewidth of G is at least ω, or producessa tree decomposition of width at most (3 + 2 3 )ω in time O(23.698ω n 3 ω 3 log 4 n). Thus ... |

83 |
Excluding a planar graph
- unknown authors
- 1986
(Show Context)
Citation Context ... known but its proof is not trivial. (See e.g., [12].) – The treewidth of the m × m grid is m. The next fact we need is the improved version of the Robertson & Seymour theorem on excluded grid minors =-=[26]-=- due to Diestel et al. [13]. (See also the textbook [12].) Theorem 1 ([13]). Let r, m be integers, and let G be a graph of treewidth at least m 4r2 (m+2) . Then G contains either Kr or the m × m grid ... |

68 | Dominating sets in planar graphs: branch-width and exponential speed-up,
- Fomin, Thilikos
- 2006
(Show Context)
Citation Context ...r the problem with running time O(4 6√ 34d n) was given by Alber et al. [1]. Recently, Kanj & Perkovuć [23] improved the running time to O(2 27√ d n) and Fomin & Thilikos to O(2 15.13√ d d+n 3 +d 4 ) =-=[18]-=-. The fixed-parameter algorithms for extensions of planar graphs like bounded-genus graphs and graphs excluding single-crossing graphs as minors are introduced in [11, 9, 15]. In the majority of these... |

68 | Local tree-width, excluded minors, and approximation algorithms.
- GROHE
- 2003
(Show Context)
Citation Context ...ain the desired bound. Theorem 2 can be applied for minor-bidimensional parameters such as vertex cover or feedback vertex set. The notion of local treewidth was introduced by Eppstein [16] (see also =-=[22]-=-). The local treewidth of a graph G is ltw(G, r) = max{tw(G[N r G[v]]): v ∈ V (G)}. For a function f : N → N we define the minor-closed class of graphs of bounded local treewidth L(f) = {G: ∀H � G ∀r ... |

66 | Parameterized complexity: Exponential speed-up for planar graph problems.
- ALBER, FERNAU, et al.
- 2004
(Show Context)
Citation Context ...n it is possible to design FPT-algorithms where the contribution of k in the non-polynomial part of their complexity is subexponential. The first algorithm of this type was given by Alber et al. (see =-=[4]). R-=-ecently, Fomin and Thilikos reported a O(k 4 +2 4.5√ k + kn) algorithm for planar k-Vertex Cover [25]. However, not all parameterized problems are fixed-parameter tractable. A typical ∗ The result... |

64 | Polynomial-time data reduction for Dominating Set.
- Alber, Fellows, et al.
- 2004
(Show Context)
Citation Context ...families of bounded local treewidth for any such graph parameter. 510 DEMAINE, FOMIN, HAJIAGHAYI, AND THILIKOS It is known that the dominating set problem admits a linear size kernel on planar graphs =-=[3]-=-. Recently, this result was extended to graphs of bounded genus [26]. It is tempting to ask whether such a kernel exists for any minor-closed graph class of bounded local treewidth, i.e., any minor-cl... |

63 | Subexponential parameterized algorithms on graphs of bounded genus and Hminor-free graphs.
- Demaine, Fomin, et al.
- 2005
(Show Context)
Citation Context ...kos to O(2 15.13√ d d+n 3 +d 4 ) [18]. The fixed-parameter algorithms for extensions of planar graphs like bounded-genus graphs and graphs excluding single-crossing graphs as minors are introduced in =-=[11, 9, 15]-=-. In the majority of these results, the design of FPT algorithms for solving problems such as k-Vertex Cover or k-Dominating Set in a sparse graph class F is based on the following lemma: every graph ... |

40 |
Thilikos. Subexponential parameterized algorithms on graphs of bounded genus and H-minor-free graphs
- Demaine, Fomin, et al.
(Show Context)
Citation Context ...to O(2 15.13 √ k k+n 3 +k 4 ) [23, 25]. The fixedparameter algorithms for extensions of planar graphs like bounded-genus graphs and graphs excluding single-crossing graphs as minors are introduced i=-=n [13, 15, 20]-=-. In the majority of these results, the design of FPT algorithms for solving problems such as k-Vertex Cover or k-Dominating Set in a sparse graph class F is based on the following lemma: every graph ... |

39 | Refined search tree technique for dominating sets on planar graphs.
- Alber, Fan, et al.
- 2004
(Show Context)
Citation Context ...n it is possible to design FPT-algorithms where the contribution of k in the non-polynomial part of their complexity is subexponential. The first algorithm of this type was given by Alber et al. (see =-=[2]-=-). Recently, Fomin and Thilikos reported a O(k 4 +2 4.5√ k +kn) algorithm for planar k-Vertex Cover [19]. However, not all parameterized problems are fixed-parameter tractable. A typical example of su... |

37 | Fixed-parameter tractability, definability and model checking. - Flum, Grohe - 2001 |

35 | Fixed parameter algorithms for planar dominating set and related problems.
- Alber, Bodlaender, et al.
- 2000
(Show Context)
Citation Context ...ven tree decomposition of width at most ω can be computed in time O(qO(ω) n) [12]. Also, algorithms on graphs of bounded treewidth for clique-transversal set, and Πdomination set appeared in [26] and =-=[3]-=- respectively. Using these algorithms, and the fact that all these 8sparameters are lower bounded by the domination number, the methodology of the proof of Theorem 11 can give algorithmic results for ... |

34 | Highly connected sets and the excluded grid theorem,
- Diestel, Jensen, et al.
- 1999
(Show Context)
Citation Context ...the main result uses the characterization of Eppstein for minorclosed families of bounded local treewidth [16] and Diestel et al.’s modification of the Robertson & Seymour excluded-grid-minor theorem =-=[13]-=-. In addition, the proof is constructive and can be used for constructing fixed-parameter algorithms to decide bidimensional parameters on minor-closed families of bounded local treewidth. In this sen... |

31 | Thilikos, Fixed-parameter algorithms for the (k, r)-center in planar graphs and map graphs
- Demaine, Fomin, et al.
(Show Context)
Citation Context ...the minimum size of a q-dominating set. Indeed, Theorem 5 can be applied because the q-dominating set of a graph with a given tree decomposition of width at most ω can be computed in time O(q O(ω) n) =-=[8]-=-. Also, algorithms on graphs of bounded treewidth for clique-transversal set, and Π-domination set appeared in [24] and [1] respectively. Using these algorithms, and the fact that all these parameters... |

31 | Equivalence of local treewidth and linear local treewidth and its algorithmic applications
- Demaine, Hajiaghayi
- 2004
(Show Context)
Citation Context ...any other graph parameters [1]. The same bound has also been proved for more general graph classes like graphs of bounded genus [13, 26, 16] and minor-closed graph families of bounded local treewidth =-=[14]-=-. It is natural to ask whether such a smaller bound holds in the case of any bidimensional parameter. This would provide subexponential fixed-parameter algorithms on minor-closed graph families of bou... |

23 |
Improved parameterized algorithms for planar dominating set.
- KANJ, PERKOVIC
- 2002
(Show Context)
Citation Context ...n). Later the running time was reduced to O(8 d n) [2]. An algorithm with a sublinear exponent for the problem with running time O(4 6√ 34d n) was given by Alber et al. [1]. Recently, Kanj & Perkovuć =-=[23]-=- improved the running time to O(2 27√ d n) and Fomin & Thilikos to O(2 15.13√ d d+n 3 +d 4 ) [18]. The fixed-parameter algorithms for extensions of planar graphs like bounded-genus graphs and graphs e... |

21 |
Graph theory, vol
- Diestel
- 2010
(Show Context)
Citation Context ... need the following facts about treewidth. The first fact is trivial. – For any complete graph Kn on n vertices, tw(Kn) = n − 1. The second fact is well known but its proof is not trivial. (See e.g., =-=[12]-=-.) – The treewidth of the m × m grid is m. The next fact we need is the improved version of the Robertson & Seymour theorem on excluded grid minors [26] due to Diestel et al. [13]. (See also the textb... |

21 | Efficient data reduction for dominating set: A linear problem kernel for the planar case
- Alber, Fellows, et al.
- 2002
(Show Context)
Citation Context ...d-parameter algorithms on minor-closed graph families of bounded local treewidth for any such graph parameter. It is known that the dominating set problem admits a linear size kernel on planar graphs =-=[3]-=-. Recently, this result was extended to graphs of bounded genus [26]. It 9sis tempting to ask whether such a kernel exists for any minor-closed graph class of bounded local treewidth, i.e., any minor-... |

19 |
The dominating set problem is fixed parameter tractable for graphs of bounded genus
- Ellis, Fan, et al.
(Show Context)
Citation Context ...kos to O(2 15.13√ d d+n 3 +d 4 ) [18]. The fixed-parameter algorithms for extensions of planar graphs like bounded-genus graphs and graphs excluding single-crossing graphs as minors are introduced in =-=[11, 9, 15]-=-. In the majority of these results, the design of FPT algorithms for solving problems such as k-Vertex Cover or k-Dominating Set in a sparse graph class F is based on the following lemma: every graph ... |

17 |
Maximum clique transversals
- Chang, Kloks, et al.
- 2001
(Show Context)
Citation Context ...ith a given tree decomposition of width at most ω can be computed in time O(q O(ω) n) [8]. Also, algorithms on graphs of bounded treewidth for clique-transversal set, and Π-domination set appeared in =-=[24]-=- and [1] respectively. Using these algorithms, and the fact that all these parameters are lower bounded by the domination number, the methodology of the proof of Theorem 5 can give algorithmic results... |

16 | A simple and fast approach for solving problems on planar graphs,
- Fomin, Thilikos
- 2004
(Show Context)
Citation Context ...ir complexity is subexponential. The first algorithm of this type was given by Alber et al. (see [2]). Recently, Fomin and Thilikos reported a O(k 4 +2 4.5√ k +kn) algorithm for planar k-Vertex Cover =-=[19]-=-. However, not all parameterized problems are fixed-parameter tractable. A typical example of such a problem is Dominating Set, asking whether an input graph has at most k vertices that are adjacent t... |

15 | The Bidimensional Theory of Bounded-Genus Graphs
- Demaine, Hajiaghayi, et al.
(Show Context)
Citation Context ...nating set of size at most k, wehavetw(G) =O( √ k). The same holds for many other graph parameters [1]. The same bound has also been proved for more general graph classes like graphs of bounded genu=-=s [13, 26, 16]-=- and minor-closed graph families of bounded local treewidth [14]. It is natural to ask whether such a smaller bound holds in the case of any bidimensional parameter. This would provide subexponential ... |

15 |
Thilikos. New upper bounds on the decomposability of planar graphs
- Fomin, M
(Show Context)
Citation Context ... complexity is subexponential. The first algorithm of this type was given by Alber et al. (see [4]). Recently, Fomin and Thilikos reported a O(k 4 + 2 4.5√ k + kn) algorithm for planar k-Vertex Cover =-=[21]-=-. However, not all parameterized problems are fixed-parameter tractable. A typical example of such a problem is Dominating Set, asking whether an input graph has at most k vertices that are adjacent t... |

14 | Exponential speedup of fixed parameter algorithms on K3,3-minor-free or K5-minor-free graphs.
- DEMAINE, HAJIAGHAYI, et al.
- 2003
(Show Context)
Citation Context ...kos to O(2 15.13√ d d+n 3 +d 4 ) [18]. The fixed-parameter algorithms for extensions of planar graphs like bounded-genus graphs and graphs excluding single-crossing graphs as minors are introduced in =-=[11, 9, 15]-=-. In the majority of these results, the design of FPT algorithms for solving problems such as k-Vertex Cover or k-Dominating Set in a sparse graph class F is based on the following lemma: every graph ... |

8 | Deciding properties of locally tree-decomposable structures - Frick, Grohe |

6 | Dominating sets and local treewidth, in:
- Fomin, Thilikos
- 2003
(Show Context)
Citation Context ... a (nonfixed-parameter) O(n 1+ɛ )-time algorithm deciding whether a given graph has property φ. A preliminary and special case of our result, concerning only the dominating set parameter, appeared in =-=[20]-=- with a different and more complicated proof. Also, another proof of the same result appeared in [10]. In this paper we present shorter and more elegant proofs of the combinatorial results of [20] and... |

2 | Thilikos, Exponential speedup of parameter algorithms on K3;3 -minor-free or K5 -minor-free graphs - Demaine, Hajiaghayi, et al. |

2 | The dominating set problem is parameter tractable for graphs of bounded genus - Ellis, Fan, et al. - 2002 |

2 | Fixed-parameter tractability, de and model-checking - Flum, Grohe - 2001 |

1 |
Fixed Parameter Algorithms for Minor-Closed Graphs (of Locally Bounded Treewidth
- Demaine, Hajiaghayi
- 2003
(Show Context)
Citation Context ...liminary and special case of our result, concerning only the dominating set parameter, appeared in [20] with a different and more complicated proof. Also, another proof of the same result appeared in =-=[10]-=-. In this paper we present shorter and more elegant proofs of the combinatorial results of [20] and [10] while we extend their applicability to general families of parameters. 2 Definitions and prelim... |

1 |
Thilikos, Parameterized algorithms for Dominating Set in graphs on surfaces: Linear kernel and exponential speed-up
- Fomin, M
(Show Context)
Citation Context ...nating set of size at most k, wehavetw(G) =O( √ k). The same holds for many other graph parameters [1]. The same bound has also been proved for more general graph classes like graphs of bounded genu=-=s [13, 26, 16]-=- and minor-closed graph families of bounded local treewidth [14]. It is natural to ask whether such a smaller bound holds in the case of any bidimensional parameter. This would provide subexponential ... |