Exact and Parameterized Algorithms for the Independent Cutset Problem

TL;DR

This paper introduces exact and fixed-parameter tractable algorithms for the NP-complete Independent Cutset problem, including structural parameterizations beyond clique-width, and explores new tractability notions like alpha-domination.

Contribution

It provides the first subexponential algorithm for the problem and develops FPT algorithms for various structural parameters, expanding the understanding of its computational complexity.

Findings

Presented a $ ilde{O}(1.4423^n)$-time exact algorithm.

Developed FPT algorithms for parameters like dual degree, solution size, and graph distances.

Introduced the concept of $oldsymbol{oldsymbol{oldsymbol{ ext{alpha}}}}$-domination for broader tractability.

Abstract

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 $\textsf{NP}$-complete even when the input graph is planar and has maximum degree five. In this paper, we first present a $\mathcal{O}^*(1.4423^{n})$-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 $\textsf{FPT}$-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 $\alpha$-domination, which allows us to identify more fixed-parameter tractable and polynomial-time solvable cases.