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SIAM Journal on Algebraic and Discrete Methods
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Simplicial decompositions of graphs: a survey of applications
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STOC '90 Proceedings of the twenty-second annual ACM symposium on Theory of computing
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WADS '93 Proceedings of the Third Workshop on Algorithms and Data Structures
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LATIN '02 Proceedings of the 5th Latin American Symposium on Theoretical Informatics
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UAI '01 Proceedings of the 17th Conference in Uncertainty in Artificial Intelligence
WG '01 Proceedings of the 27th International Workshop on Graph-Theoretic Concepts in Computer Science
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ISAAC '02 Proceedings of the 13th International Symposium on Algorithms and Computation
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STOC '78 Proceedings of the tenth annual ACM symposium on Theory of computing
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SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
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SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
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FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
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SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
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ACM Transactions on Algorithms (TALG)
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WG'10 Proceedings of the 36th international conference on Graph-theoretic concepts in computer science
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Note: max-cut and containment relations in graphs
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Computer Science Review
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ICALP'07 Proceedings of the 34th international conference on Automata, Languages and Programming
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Many problems that are intractable for general graphs allow polynomial-time solutions for structured classes of graphs, such as planar graphs and graphs of bounded treewidth. In this paper, we demonstrate structural properties of larger classes of graphs and show how to exploit the properties to obtain algorithms. The classes considered are those formed by excluding as a minor a graph that can be embedded in the plane with at most one crossing. We show that graphs in these classes can be decomposed into planar graphs and graphs of small treewidth; we use the decomposition to show that all such graphs have locally bounded treewidth (all subgraphs of a certain form are graphs of bounded treewidth). Finally, we make use of the structural properties to derive polynomial-time algorithms for approximating treewidth within a factor of 1.5 and branchwidth within a factor of 2.25 as well as polynomial-time approximation schemes for both minimization and maximization problems and fixed-parameter algorithms for problems such as vertex cover, edge-dominating set, feedback vertex set, and others.