Contracting graphs to paths and trees

  • Authors:

  • Pinar Heggernes;Pim van 't Hof;Benjamin Lévêque;Daniel Lokshtanov;Christophe Paul

  • Affiliations:

  • Department of Informatics, University of Bergen, Bergen, Norway;Department of Informatics, University of Bergen, Bergen, Norway;CNRS, LIRMM, Université Montpellier 2, Montpellier, France;Dept. Computer Science and Engineering, University of California, San Diego;CNRS, LIRMM, Université Montpellier 2, Montpellier, France

  • Venue:
  • IPEC'11 Proceedings of the 6th international conference on Parameterized and Exact Computation
  • Year:
  • 2011

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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 4.98knO(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)+nO(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 4k2 vertices.