Parameterized complexity in multiple-interval graphs: Domination, partition, separation, irredundancy

  • Authors:

  • Minghui Jiang;Yong Zhang

  • Affiliations:

  • Department of Computer Science, Utah State University, Logan, UT 84322, USA;Department of Computer Science, Kutztown University of PA, Kutztown, PA 19530, USA

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2012

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Abstract

We show that the problem k-dominating set and its several variants including k-connected dominating set, k-independent dominating set, and k-dominating clique, when parameterized by the solution size k, are W[1]-hard in either multiple-interval graphs or their complements or both. On the other hand, we show that these problems belong to W[1] when restricted to multiple-interval graphs and their complements. This answers an open question of Fellows et al. In sharp contrast, we show that d-distance k-dominating set for d=2 is W[2]-complete in multiple-interval graphs and their complements. We also show that k-perfect code and d-distance k-perfect code for d=2 are W[1]-complete even in unit 2-track interval graphs. In addition, we present various new results on the parameterized complexities of k-vertex clique partition and k-separating vertices in multiple-interval graphs and their complements, and present a very simple alternative proof of the W[1]-hardness of k-irredundant set in general graphs.