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Contracting Graphs to Paths and Trees

  • Pinar Heggernes
  • Pim van ’t Hof
  • Benjamin Lévêque
  • Daniel Lokshtanov
  • Christophe Paul
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7112)

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 4.98 k 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 2 k + 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.

Keywords

Outer Loop Input Graph Polynomial Kernel Graph Class Edge Contraction 
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 2012

Authors and Affiliations

  • Pinar Heggernes
    • 1
  • Pim van ’t Hof
    • 1
  • Benjamin Lévêque
    • 2
  • Daniel Lokshtanov
    • 3
  • Christophe Paul
    • 2
  1. 1.Department of InformaticsUniversity of BergenBergenNorway
  2. 2.CNRS, LIRMM, Université Montpellier 2MontpellierFrance
  3. 3.Dept. Computer Science and EngineeringUniversity of CaliforniaSan DiegoUSA

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