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Parameterized Complexity and Inapproximability of Dominating Set Problem in Chordal and Near Chordal Graphs
1Dept. of Systems and Computer Science, Howard University, Washington, DC 20059, USA chunmei@scs.howard.edu.
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 no(k) in chordal and s-chordal (s > 3) 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 (s > 3) graph cannot be approximated within a ratio of [Formula: see text] ln n in polynomial time, where n is the number of vertices in the graph and 0 < c < 1 is the constant from the inapproximability of the minimum dominating set in general graphs. In other words, our results suggest that restricting to chordal or s-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.
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