Quickly excluding a planar graph
01 Nov 1994-Journal of Combinatorial Theory, Series B (Academic Press, Inc.)-Vol. 62, Iss: 2, pp 323-348
TL;DR: A much better bound is proved on the tree-width of planar graphs with no minor isomorphic to a g × g grid and this is the best known bound.
About: This article is published in Journal of Combinatorial Theory, Series B.The article was published on 1994-11-01 and is currently open access. It has received 488 citations till now. The article focuses on the topics: Graph minor & Planar straight-line graph.
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27 Jul 2015
TL;DR: This comprehensive textbook presents a clean and coherent account of most fundamental tools and techniques in Parameterized Algorithms and is a self-contained guide to the area, providing a toolbox of algorithmic techniques.
Abstract: This comprehensive textbook presents a clean and coherent account of most fundamental tools and techniques in Parameterized Algorithms and is a self-contained guide to the area. The book covers many of the recent developments of the field, including application of important separators, branching based on linear programming, Cut & Count to obtain faster algorithms on tree decompositions, algorithms based on representative families of matroids, and use of the Strong Exponential Time Hypothesis. A number of older results are revisited and explained in a modern and didactic way. The book provides a toolbox of algorithmic techniques. Part I is an overview of basic techniques, each chapter discussing a certain algorithmic paradigm. The material covered in this part can be used for an introductory course on fixed-parameter tractability. Part II discusses more advanced and specialized algorithmic ideas, bringing the reader to the cutting edge of current research. Part III presents complexity results and lower bounds, giving negative evidence by way of W[1]-hardness, the Exponential Time Hypothesis, and kernelization lower bounds. All the results and concepts are introduced at a level accessible to graduate students and advanced undergraduate students. Every chapter is accompanied by exercises, many with hints, while the bibliographic notes point to original publications and related work.
1,544 citations
Cites background from "Quickly excluding a planar graph"
...23 (Planar excluded grid theorem, [239, 403])....
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...The best known lower bound for function g(t) in the Excluded Grid Minor Theorem is Ω(t2 log t), given by Robertson, Seymour, and Thomas [403]; this refines by a log t factor the lower bound given in Exercise 7....
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...23) was first proved by Robertson, Seymour and Thomas [403]; their proof gave constant 6 in the relation between treewidth and the largest grid minor in a graph....
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...There were several improvements on the exponential dependence on treewidth of the size of the grid minor: by Robertson, Seymour and Thomas [403], by Kawarabayashi and Kobayashi [286], and by Leaf and Seymour [317]....
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TL;DR: This survey paper wants to give an overview of many classes of graphs that can be seen to have a uniform upper bound on the treewidth of graphs in the class.
1,197 citations
Cites background from "Quickly excluding a planar graph"
...In [91], it is shown that one can take in Theorem 13 cH = 20 4jVH j+8jEH j 5 ....
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Journal Article•
TL;DR: A short overview of recent results in algorithmic graph theory that deal with the notions treewidth and pathwidth can be found in this paper, where the authors discuss algorithms that find tree-decomposition, algorithms that use treedecompositions to solve hard problems efficiently, graph minor theory, and some applications.
Abstract: A short overview is given of many recent results in algorithmic graph theory that deal with the notions treewidth, and pathwidth. We discuss algorithms that find tree-decompositions, algorithms that use tree-decompositions to solve hard problems efficiently, graph minor theory, and some applications. The paper contains an extensive bibliography.
755 citations
TL;DR: It is proved that, under some complexity theoretic assumption from parameterized complexity theory, HOM(C,−) is in polynomial time if and only if C has bounded tree width modulo homomorphic equivalence.
Abstract: We give a complexity theoretic classification of homomorphism problems for graphs and, more generally, relational structures obtained by restricting the left hand side structure in a homomorphism. For every class C of structures, let HOM(C,−) be the problem of deciding whether a given structure A ∈C has a homomorphism to a given (arbitrary) structure s. We prove that, under some complexity theoretic assumption from parameterized complexity theory, HOM(C,−) is in polynomial time if and only if C has bounded tree width modulo homomorphic equivalence.Translated into the language of constraint satisfaction problems, our result yields a characterization of the tractable structural restrictions of constraint satisfaction problems. Translated into the language of database theory, it implies a characterization of the tractable instances of the evaluation problem for conjunctive queries over relational databases.
501 citations
TL;DR: It is found that every graph with no minor isomorphic to L may be constructed by piecing together in a tree-structure graphs each of which "almost" embeds in some surface in which L cannot be embedded.
400 citations