Parameterized complexity and inapproximability of dominating set problem in chordal and near chordal graphs

  • Authors:

  • Chunmei Liu;Yinglei Song

  • Affiliations:

  • Dept. of Systems and Computer Science, Howard University, Washington, USA 20059;Dept. of Mathematics and Computer Science, University of Maryland Eastern Shore, Princess Anne, USA 21853

  • Venue:
  • Journal of Combinatorial Optimization
  • Year:
  • 2011

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Abstract

In this paper, we study the parameterized complexity of Dominating Set problem in chordal graphs and near chordal graphs. We show the problem is W[2]-hard and cannot be solved in time n o(k) in chordal and s-chordal (s3) graphs unless W[1]=FPT. In addition, we obtain inapproximability results for computing a minimum dominating set in chordal and near chordal graphs. Our results prove that unless NP=P, the minimum dominating set in a chordal or s-chordal (s3) graph cannot be approximated within a ratio of $\frac{c}{3}\ln{n}$ in polynomial time, where n is the number of vertices in the graph and 0cs-chordal graphs can improve the approximation ratio by no more than a factor of 3. We then extend our techniques to find similar results for the Independent Dominating Set problem and the Connected Dominating Set problem in chordal or near chordal graphs.