Exact and Parameterized Algorithms for the Independent Cutset Problem
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The Independent Cutset problem asks whether there is a set of vertices in a given graph that is both independent and a cutset. Such a problem is -complete even when the input graph is planar and has maximum degree five. In this paper, we first present a 𝒪^*(1.4423^n)-time algorithm for the problem. We also show how to compute a minimum independent cutset (if any) in the same running time. Since the property of having an independent cutset is MSO_1-expressible, our main results are concerned with structural parameterizations for the problem considering parameters that are not bounded by a function of the clique-width of the input. We present -time algorithms for the problem considering the following parameters: the dual of the maximum degree, the dual of the solution size, the size of a dominating set (where a dominating set is given as an additional input), the size of an odd cycle transversal, the distance to chordal graphs, and the distance to P_5-free graphs. We close by introducing the notion of α-domination, which allows us to identify more fixed-parameter tractable and polynomial-time solvable cases.
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