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 (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 \(\frac{c}{3}\ln{n}\) 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.

中文翻译：

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弦和近弦图中支配集问题的参数化复杂性和不可逼近性。

在本文中，我们研究了弦图和近弦图中支配集问题的参数化复杂度。我们证明该问题是 W[2]-困难问题，并且在弦图和s -弦图 ( s >3) 中无法在时间n ^{o ( k )}内解决，除非 W[1]=FPT。此外，我们还获得了计算弦图和近弦图中的最小支配集的不可逼近性结果。我们的结果证明，除非 NP=P，否则弦图或s -弦图 ( s >3) 图中的最小支配集不能在\(\frac{c}{3}\ln{n}\)的比率内近似在多项式时间内，其中n是图中的顶点数，0 < c < 1 是一般图中最小支配集的不可逼近性的常数。换句话说，我们的结果表明，限制为弦图或s弦图可以将近似率提高不超过 3 倍。然后，我们扩展我们的技术，以找到独立支配集问题和连通支配集问题的类似结果弦图或近弦图中的问题。