Quickly Excluding a Planar Graph

Robertson, Neil; Seymour, Paul; Thomas, Robin

doi:10.1006/jctb.1994.1073

Cited by 370 publications

(321 citation statements)

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“…Another interesting problem is to lower bound the contribution of f in Theorem 5. As mentioned by Robertson, Seymour, and Thomas in [113] there are graphs excluding G k as a minor that have treewidth Ω(k 2 · log k). To see this, one may use the result in [23] (see also [41,122]…”

Section: Theorem 5 ( [105]) There Exists a Recursive Functionmentioning

confidence: 97%

“…The initial estimation of the parameter dependence in Theorem 5 was huge. However, a better one appeared in [113] where it was proven that tw(G) = 20 2·(gm(G)) 5 . An alternative, and relatively simpler, proof of Theorem 5 was given in [33].…”

Section: Theorem 5 ( [105]) There Exists a Recursive Functionmentioning

confidence: 99%

“…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: 99%

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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: Theorem 5 ( [105]) There Exists a Recursive Functionmentioning

confidence: 97%

Section: Theorem 5 ( [105]) There Exists a Recursive Functionmentioning

confidence: 99%

Section: Bidimensionalitymentioning

confidence: 99%

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Graph Minors and Parameterized Algorithm Design

Thilikos

2012

The Multivariate Algorithmic Revolution and Beyond

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“…Indeed it is well-known that every planar graph of branchwidth at least ℓ contains an (⌊ℓ/4⌋ × ⌊ℓ/4⌋)-grid as a minor [RST94]. Therefore, a square grid is inside every planar graph, and any edge-disjoint circuit in a minor of a graph can be easily transformed to an edge-disjoint circuit in the graph itself.…”

Section: Background and Motivationmentioning

confidence: 99%

Edge-Simple Circuits through 10 Ordered Vertices in Square Grids

Coudert

Giroire

2009

Lecture Notes in Computer Science

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A circuit in a simple undirected graph G = (V, E) is a sequence of vertices {v 1 , v 2 , . . . , v k+1 } such that v 1 = v k+1 and {v i , v i+i } ∈ E for i = 1, . . . , k. A circuit C is said to be edge-simple if no edge of G is used twice in C. In this article we study the following problem: which is the largest integer k such that, given any subset of k ordered vertices of an infinite square grid, there exists an edge-simple circuit visiting the k vertices in the prescribed order? We prove that k = 10. To this end, we first provide a counterexample implying that k < 11. To show that k ≥ 10, we introduce a methodology, based on the notion of core graph, to reduce drastically the number of possible vertex configurations, and then we test each one of the resulting configurations with an ILP solver.

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“…The bidimensionality theory is based on algorithmic and combinatorial extensions to various parts of Graph Minors Theory of Robertson and Seymour [30] and provides a simple criteria for checking whether a parameterized problem is solvable in subexponential time on sparse graphs. The theory applies to the graph problems that are bidimensional in the sense that the value of the solution for the problem in question on k × k grid or "grid like graph" is at least Ω(k 2 ) and the value of solution decreases while contracting or sometime deleting the edges.…”

Section: Introductionmentioning

confidence: 99%

Subexponential algorithms for partial cover problems

Fomin

Lokshtanov

Raman

et al. 2011

Information Processing Letters

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Partial Cover problems are optimization versions of fundamental and well studied problems like Vertex Cover and Dominating Set. Here one is interested in covering (or dominating) the maximum number of edges (or vertices) using a given number (k) of vertices, rather than covering all edges (or vertices). In general graphs, these problems are hard for parameterized complexity classes when parameterized by k. It was recently shown by Amini et. al. [FSTTCS 08 ] that Partial Vertex Cover and Partial Dominating Set are fixed parameter tractable on large classes of sparse graphs, namely H-minor free graphs, which include planar graphs and graphs of bounded genus. In particular, it was shown that on planar graphs both problems can be solved in time 2 O(k) n O(1) .During the last decade there has been an extensive study on parameterized subexponential algorithms. In particular, it was shown that the classical Vertex Cover and Dominating Set problems can be solved in subexponential time on H-minor free graphs. The techniques developed to obtain subexponential algorithms for classical problems do not apply to partial cover problems. It was left as an open problem by Amini et al. [FSTTCS 08 ] whether there is a subexponential algorithm for Partial Vertex Cover and Partial Dominating Set. In this paper, we answer the question affirmatively by solving both problems in time 2 O( √ k) n O(1) not only on planar graphs but also on much larger classes of graphs, namely, apex-minor free graphs. Compared to previously known algorithms for these problems our algorithms are significantly faster and simpler.

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Quickly Excluding a Planar Graph

Cited by 370 publications

References 0 publications

Graph Minors and Parameterized Algorithm Design

Graph Minors and Parameterized Algorithm Design

Edge-Simple Circuits through 10 Ordered Vertices in Square Grids

Subexponential algorithms for partial cover problems

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