Highly Connected Sets and the Excluded Grid Theorem

Diestel, Reinhard; Jensen, Tommy R.; Gorbunov, K. Yu.; Thomassen, Carsten

doi:10.1006/jctb.1998.1862

Cited by 114 publications

(118 citation statements)

References 6 publications

(11 reference statements)

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“…This result has been re-proved by Robertson, Seymour, and Thomas [RST94], Reed [Ree97], and Diestel, Jensen, Gorbunov, and Thomassen [DJGT99]. Among these proofs, the best known bound on w in terms of r is that every H-minor-free graph of treewidth larger than 20…”

Section: Theorem 4 ([Dh05b]) For Any Fixed Graph H Every H-minor-frmentioning

confidence: 74%

Fast Algorithms for Hard Graph Problems: Bidimensionality, Minors, and Local Treewidth

Demaine¹,

Hajiaghayi²

2005

Graph Drawing

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Section: Theorem 4 ([Dh05b]) For Any Fixed Graph H Every H-minor-frmentioning

confidence: 74%

Fast Algorithms for Hard Graph Problems: Bidimensionality, Minors, and Local Treewidth

Demaine¹,

Hajiaghayi²

2005

Graph Drawing

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“…It is well known that planar graphs having a large treewidth must have also a large grid as a minor; see e.g. [17,19,36]. The following theorem and corollary present the algorithmic consequences of this fact.…”

Section: Theorem 31 the Lower Bound Lb 2 On The Treewidth Of A Planamentioning

confidence: 92%

Treewidth Lower Bounds with Brambles

2007

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In this paper we present a new technique for computing lower bounds for graph treewidth. Our technique is based on the fact that the treewidth of a graph G is the maximum order of a bramble of G minus one. We give two algorithms: one for general graphs, and one for planar graphs. The algorithm for planar graphs is shown to give a lower bound for both the treewidth and branchwidth that is at most a constant factor away from the optimum. For both algorithms, we report on extensive computational experiments that show that the algorithms often give excellent lower bounds, in particular when applied to (close to) planar graphs.

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“…The best known upper bound for f 1 (t) is 20 2t 5 , see [4,20,25]. The best known lower bound is Θ(t 2 log t), see [25].…”

Section: Theorem 7 For Any T There Exists a Constant F 1 (T) Such Tmentioning

confidence: 99%

Edge-disjoint odd cycles in 4-edge-connected graphs

Kawarabayashi

Kobayashi

2016

Journal of Combinatorial Theory, Series B

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Finding edge-disjoint odd cycles is one of the most important problems in graph theory, graph algorithm and combinatorial optimization. In fact, it is closely related to the well-known max-cut problem. One of the difficulties of this problem is that the Erdős-Pósa property does not hold for odd cycles in general. Motivated by this fact, we prove that for any positive integer k, there exists an integer f (k) satisfying the following: For any 4-edge-connected graph G = (V, E), either G has edge-disjoint k odd cycles or there exists an edge set F ⊆ E with |F | ≤ f (k) such that G − F is bipartite. We note that the 4-edge-connectivity is best possible in this statement. Similar approach can be applied to an algorithmic question. Suppose that the input graph G is a 4-edgeconnected graph with n vertices. We show that, for any ε > 0, if k = O((log log log n) 1/2−ε ), then the edge-disjoint k odd cycle packing in G can be solved in polynomial time of n. ACM Subject Classification G.2.2 Graph Theory

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Highly Connected Sets and the Excluded Grid Theorem

Cited by 114 publications

References 6 publications

Fast Algorithms for Hard Graph Problems: Bidimensionality, Minors, and Local Treewidth

Fast Algorithms for Hard Graph Problems: Bidimensionality, Minors, and Local Treewidth

Treewidth Lower Bounds with Brambles

Edge-disjoint odd cycles in 4-edge-connected graphs

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