Parameterized complexity of vertex deletion into perfect graph classes

  • Authors:

  • Pinar Heggernes;Pim Van 't Hof;Bart M. P. Jansen;Stefan Kratsch;Yngve Villanger

  • Affiliations:

  • University of Bergen, P.O.Box 7803, N-5020 Bergen, Norway;University of Bergen, P.O.Box 7803, N-5020 Bergen, Norway;Utrecht University, P.O. Box 80.089, 3508 TB Utrecht, The Netherlands;Utrecht University, P.O. Box 80.089, 3508 TB Utrecht, The Netherlands;University of Bergen, P.O.Box 7803, N-5020 Bergen, Norway

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2013

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Abstract

Vertex deletion problems are at the heart of parameterized complexity. For a graph class F, the F-Deletion problem takes as input a graph G and an integer k. The question is whether it is possible to delete at most k vertices from G such that the resulting graph belongs to F. Whether Perfect Deletion is fixed-parameter tractable, and whether Chordal Deletion admits a polynomial kernel, when parameterized by k, have been stated as open questions in previous work. We show that Perfect Deletion and Weakly Chordal Deletion are W[2]-hard when parameterized by k. In search of positive results, we study a restricted variant of the F-Deletion problem. In this restricted variant, the deleted vertices must be taken from a specified set X, and we parameterize by |X|. We show that for Perfect Deletion and Weakly Chordal Deletion, although this restriction immediately ensures fixed-parameter tractability, it is not enough to yield polynomial kernels, unless NP @? coNP/poly. On the positive side, for Chordal Deletion, the restriction enables us to obtain a kernel with O(|X|^4) vertices.