On planar graphs with large tree-width and small grid minors

Grigoriev, Alexander; Marchal, Bert; Usotskaya, Natalya

doi:10.1016/j.endm.2009.02.006

Cited by 6 publications

(2 citation statements)

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“…Theorem 1.1 gives a better constant 4.5. Grigoriev et al [13] study the tightness of these bounds for treewidth and conjecture that the best constant in this bound is 2. Theorem 1.3 verifies one side of their conjecture that the constant in the bound cannot be smaller than 2, since tw(C 2h,h ) = 2h = 2gm (C 2h,h ).…”

Section: Theorem 13 For Every Integer H ≥ 2 We Have Gm(c 2hh ) = Hmentioning

confidence: 99%

Improved Bounds on the Planar Branchwidth with Respect to the Largest Grid Minor Size

Tamaki

2012

Algorithmica

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For graph G, let bw(G) denote the branchwidth of G and gm(G) the largest integer g such that G contains a g × g grid as a minor. We show that bw(G) ≤ 3gm(G) for every planar graph G. This is an improvement over the bound bw(G) ≤ 4gm(G) due to Robertson, Seymour and Thomas. Our proof is constructive and implies quadratic time constant-factor approximation algorithms for planar graphs for both problems of finding a largest grid minor and of finding an optimal branch-decomposition: (3 + )-approximation for the former and (2 + )-approximation for the latter, where is an arbitrary positive constant. We also study the tightness of the above bound. We show that for any constant c < 2, the bound of bw(G) ≤ c gm(G) + o(gm(G)) does not hold in general for a planar graph G.

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Section: Theorem 13 For Every Integer H ≥ 2 We Have Gm(c 2hh ) = Hmentioning

confidence: 99%

Improved Bounds on the Planar Branchwidth with Respect to the Largest Grid Minor Size

Tamaki

2012

Algorithmica

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“…This topic started with one well-known result on planar graph, independently proved by Kuratowski and Wagner, which says that a graph is planar if and only if it do not include as a minor neither the complete graph K 5 nor the complete bipartite graph K 3,3 (see [64,92]). There are previous works relating minor graphs with tree-length and tree-width, which are parameters closely related to hyperbolicity (see [16,51,79,80]).…”

Section: Introductionmentioning

confidence: 99%

Small values of the hyperbolicity constant in graphs

Bermudo

Rodrı́guez

Rosario

et al. 2016

Discrete Mathematics

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to my familyGracias a mis padres y hermanos por todo el apoyo y confianza desde que decidí emprender esta aventura, pero sobre todo por el infinito cariño y amor. A todos mis amigos que siempre han estado presentes de una u otra manera, gracias por ese gran apoyo moral y por sus palabras deánimo. La vida me ha dado tantos amigos que no acabaría de mencionarlos. Infinitas gracias. Conclusions Contributions of this work Open problems BibliographyNote that, if we consider a graph G whose edges have length equal to one and a graph G k obtained from G stretching out their edges until length k, then δ(G k ) = kδ(G). Therefore, all the results in this work can be generalized when the edges of the graph have length equal to k.

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Improved Bounds on the Planar Branchwidth with Respect to the Largest Grid Minor Size

Tamaki

2010

Algorithms and Computation

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On planar graphs with large tree-width and small grid minors

Cited by 6 publications

References 5 publications

Improved Bounds on the Planar Branchwidth with Respect to the Largest Grid Minor Size

Improved Bounds on the Planar Branchwidth with Respect to the Largest Grid Minor Size

Small values of the hyperbolicity constant in graphs

Improved Bounds on the Planar Branchwidth with Respect to the Largest Grid Minor Size

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