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

Gu, Qian-Ping; Tamaki, Hisao

doi:10.1007/978-3-642-17514-5_8

Cited by 10 publications

(2 citation statements)

References 15 publications

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“…The first variant of Theorem 5 for special graph classes appeared in [113] (proved for the twin parameter of branchwidth) from which it follows that if G is a planar graph, then tw(G) ≤ 6 · gm(G). Actually, with some more careful application of the results of [113] it can also be proven that tw(G) ≤ 5 · gm(G), which can be improved further to tw(G) ≤ 9 2 · gm(G) using the results of [62]. An analogous upper bound holds also for graphs embedded in surfaces.…”

Section: Bidimensionalitymentioning

confidence: 85%

Graph Minors and Parameterized Algorithm Design

Thilikos

2012

The Multivariate Algorithmic Revolution and Beyond

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Abstract. The Graph Minors Theory, developed by Robertson and Seymour, has been one of the most influential mathematical theories in parameterized algorithm design. We present some of the basic algorithmic techniques and methods that emerged from this theory. We discuss its direct meta-algorithmic consequences, we present the algorithmic applications of core theorems such as the grid-exclusion theorem, and we give a brief description of the irrelevant vertex technique.

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Section: Bidimensionalitymentioning

confidence: 85%

Graph Minors and Parameterized Algorithm Design

Thilikos

2012

The Multivariate Algorithmic Revolution and Beyond

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“…We use the variant for planar graphs. Proposition 1 [6,20,31] Given a planar graph G on n vertices with…”

Section: Remark 1 For Every Two Edges E F ∈ E(t ) Withmentioning

confidence: 99%

Fast Minor Testing in Planar Graphs

et al. 2011

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Minor Containment is a fundamental problem in Algorithmic Graph Theory used as a subroutine in numerous graph algorithms. A model of a graph H in a graph G is a set of disjoint connected subgraphs of G indexed by the vertices of H , such that if {u, v} is an edge of H , then there is an edge of G between components C u and C v . A graph H is a minor of G if G contains a model of H as a subgraph. We give an algorithm that, given a planar n-vertex graph G and an h-vertex graph H , either finds in time O(2 O(h) · n + n 2 · log n) a model of H in G, or correctly concludes that G does not contain H as a minor. Our algorithm is the first single-exponential algorithm for this problem and improves all previous minor testing algorithms in planar graphs. Our technique is based on a novel approach called partially embedded dynamic programming.An extended abstract of this work appeared in the proceedings of ESA'10 [2].

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Forbidding Kuratowski Graphs as Immersions

2014

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The immersion relation is a partial ordering relation on graphs that is weaker than the topological minor relation in the sense that if a graph G contains a graph H as a topological minor, then it also contains it as an immersion but not vice versa. Kuratowski graphs, namely K 5 and K 3,3 , give a precise characterization of planar graphs when excluded as topological minors. In this note we give a structural characterization of the graphs that exclude Kuratowski graphs as immersions. We prove that they can be constructed by applying consecutive i-edge-sums, for i ≤ 3, starting from graphs that are planar sub-cubic or of branchwidth at most 10.

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

Cited by 10 publications

References 15 publications

Graph Minors and Parameterized Algorithm Design

Graph Minors and Parameterized Algorithm Design

Fast Minor Testing in Planar Graphs

Forbidding Kuratowski Graphs as Immersions

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