paper - Algorithmica manuscript No. (will be inserted by...

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Algorithmica manuscript No. (will be inserted by the editor) Diameter and Treewidth in Minor-Closed Graph Families, Revisited Erik D. Demaine, MohammadTaghi Hajiaghayi MIT Computer Science and Artifcial Intelligence Laboratory, 32 Vassar St., Cambridge, MA 02139, USA. { edemaine,hajiagha } @mit.edu The date oF receipt and acceptance will be inserted by the editor Abstract Eppstein [5] characterized the minor-closed graph families for which the treewidth is bounded by a function of the diameter, which includes, e.g., planar graphs. This characterization has been used as the basis for several (approxima- tion) algorithms on such graphs (e.g., [2,5–8]). The proof of Eppstein is compli- cated. In this short paper we obtain the same characterization with a simple proof. In addition, the relation between treewidth and diameter is slightly better and ex- plicit. Key words apex graphs, graph minors, bounded local treewidth, graph algo- rithms, approximation algorithms 1 Introduction Eppstein [5] introduced the diameter-treewidth property for a class of graphs, which requires that the treewidth of a graph in the class is upper bounded by a function of its diameter. This notion has been used extensively in a slightly mod- iFed form called the bounded-local-treewidth property , which requires that the treewidth of any connected subgraph of a graph in the class is upper bounded by a function of its diameter. ±or minor-closed graph families, which is the focus of most work in this context, these properties are identical. The reason for introducing graphs of bounded local treewidth is that they have many similar properties to both planar graphs and graphs of bounded treewidth, two classes of graphs on which many problems are substantially easier. In partic- ular, Baker’s approach for polynomial-time approximation schemes (PTASs) on planar graphs [1] applies to this setting. As a result, PTASs are known for heredi- tary maximization problems such as maximum independent set, maximum triangle matching, maximum H -matching, maximum tile salvage, minimum vertex cover,
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2 Erik D. Demaine, MohammadTaghi Hajiaghayi minimum dominating set, minimum edge-dominating set, and subgraph isomor- phism for a Fxed pattern [2,5,8]. Graphs of bounded local treewidth also admit several efFcient Fxed-parameter algorithms. In particular, ±rick and Grohe [6] give a general framework for deciding any property expressible in Frst-order logic in graphs of bounded local treewidth. The foundation of these results is the following characterization by Eppstein [5] of minor-closed families with the diameter-treewidth property. An apex graph is a graph in which the removal of some vertex leaves a planar graph.
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This note was uploaded on 01/20/2012 for the course CS 6.889 taught by Professor Erikdemaine during the Fall '11 term at MIT.

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paper - Algorithmica manuscript No. (will be inserted by...

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