Exact and parameterized algorithms for the independent cutset problem

The Independent Cutset problem asks whether there is a set of vertices in a given graph that is both independent and a cutset. This problem is

-complete even when the input graph is planar and has maximum degree five. We first present a $O^{⁎}\left({1.4423}^n\right)$-time algorithm to compute a minimum independent cutset (if any). Since the property of having an independent cutset is MSO₁-expressible, our main results are concerned with structural parameterizations for the problem considering parameters incomparable with clique-width. We present

-time algorithms under 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 generalizes key ideas of this article.