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Contracting Graphs to Paths and Trees

  • Conference paper
Parameterized and Exact Computation (IPEC 2011)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7112))

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Abstract

Vertex deletion and edge deletion problems play a central role in Parameterized Complexity. Examples include classical problems like Feedback Vertex Set, Odd Cycle Transversal, and Chordal Deletion. The study of analogous edge contraction problems has so far been left largely unexplored from a parameterized perspective. We consider two basic problems of this type: Tree Contraction and Path Contraction. These two problems take as input an undirected graph G on n vertices and an integer k, and the task is to determine whether we can obtain an acyclic graph or a path, respectively, by a sequence of at most k edge contractions in G. We present an algorithm with running time 4.98k n O(1) for Tree Contraction, based on a variant of the color coding technique of Alon, Yuster and Zwick, and an algorithm with running time 2k + o(k) + n O(1) for Path Contraction. Furthermore, we show that Path Contraction has a kernel with at most 5k + 3 vertices, while Tree Contraction does not have a polynomial kernel unless NP ⊆ coNP/poly. We find the latter result surprising, because of the connection between Tree Contraction and Feedback Vertex Set, which is known to have a kernel with 4k 2 vertices.

Supported by the Research Council of Norway (project SCOPE, 197548/V30) and the French ANR (project AGAPE, ANR-09-BLAN-0159).

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Authors and Affiliations

  1. Department of Informatics, University of Bergen, N-5020, Bergen, Norway

    Pinar Heggernes & Pim van ’t Hof

  2. CNRS, LIRMM, Université Montpellier 2, Montpellier, France

    Benjamin Lévêque & Christophe Paul

  3. Dept. Computer Science and Engineering, University of California, San Diego, USA

    Daniel Lokshtanov

Authors
  1. Pinar Heggernes
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  2. Pim van ’t Hof
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  3. Benjamin Lévêque
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  4. Daniel Lokshtanov
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  5. Christophe Paul
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Editors and Affiliations

  1. MTA SZTAKI, Lágymányosi u. 11, 1111, Budapest, Hungary

    Dániel Marx

  2. Lehrgebiet Theoretische Informatik, RWTH Aachen, Ahornstraße 55, 52056, Aachen, Germany

    Peter Rossmanith

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Heggernes, P., van ’t Hof, P., Lévêque, B., Lokshtanov, D., Paul, C. (2012). Contracting Graphs to Paths and Trees. In: Marx, D., Rossmanith, P. (eds) Parameterized and Exact Computation. IPEC 2011. Lecture Notes in Computer Science, vol 7112. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-28050-4_5

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  • DOI: https://doi.org/10.1007/978-3-642-28050-4_5

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-28049-8

  • Online ISBN: 978-3-642-28050-4

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