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Bidimensional Parameters and Local Treewidth

  • Erik D. Demaine
  • Fedor V. Fomin
  • Mohammad Taghi Hajiaghayi
  • Dimitrios M. Thilikos
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2976)

Abstract

For several graph theoretic parameters such as vertex cover and dominating set, it is known that if their values are bounded by k then the treewidth of the graph is bounded by some function of k. This fact is used as the main tool for the design of several fixed-parameter algorithms on minor-closed graph classes such as planar graphs, single-crossing-minor-free graphs, and graphs of bounded genus. In this paper we examine the question whether similar bounds can be obtained for larger minor-closed graph classes, and for general families of parameters including all the parameters where such a behavior has been reported so far.

Given a graph parameter P, we say that a graph family \(\mathcal{F}\) has the parameter-treewidth property for P if there is a function f(p) such that every graph \(G \in \mathcal{F}\) with parameter at most p has treewidth at most f(p). We prove as our main result that, for a large family of parameters called contraction-bidimensional parameters, a minor-closed graph family \(\mathcal{F}\) has the parameter-treewidth property if \(\mathcal{F}\) has bounded local treewidth. We also show “if and only if” for some parameters, and thus this result is in some sense tight. In addition we show that, for a slightly smaller family of parameters called minor-bidimensional parameters, all minor-closed graph families \(\mathcal{F}\) excluding some fixed graphs have the parameter-treewidth property. The bidimensional parameters include many domination and covering parameters such as vertex cover, feedback vertex set, dominating set, edge-dominating set, q-dominating set (for fixed q). We use these theorems to develop new fixed-parameter algorithms in these contexts.

Keywords

Planar Graph Vertex Cover Tree Decomposition Domination Number Apex Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Erik D. Demaine
    • 1
  • Fedor V. Fomin
    • 2
  • Mohammad Taghi Hajiaghayi
    • 1
  • Dimitrios M. Thilikos
    • 3
  1. 1.MIT Laboratory for Computer ScienceCambridgeUSA
  2. 2.Department of InformaticsUniversity of BergenBergenNorway
  3. 3.Departament de Llenguatges i Sistemes InformàticsUniversitat Politècnica de CatalunyaBarcelonaSpain

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