Partitioning Subclasses of Chordal Graphs with Few Deletions

Abstract

In the (Vertex) k -Way Cut problem, input is an undirected graph G, an integer s, and the goal is to find a subset S of edges (vertices) of size at most s, such that $G-S$ has at least k connected components. Downey et al. [Electr. Notes Theor. Comput. Sci. 2003] showed that k -Way Cut is W[1]-hard parameterized by k. However, Kawarabayashi and Thorup [FOCS 2011] showed that the problem is fixed-parameter tractable (FPT) in general graphs with respect to the parameter s and provided a ${\mathcal {O}} (s^{s^{{\mathcal {O}} (s)}} n^2)$ time algorithm, where n denotes the number of vertices in G. The best-known algorithm for this problem runs in time $s^{{\mathcal {O}} (s)} n^{{\mathcal {O}} (1)}$ given by Lokshtanov et al. [ACM Tran. of Algo. 2021]. On the other hand, Vertex k -Way Cut is W[1]-hard with respect to either of the parameters, k or s or $k+s$. These algorithmic results motivate us to look at the problems on special classes of graphs.

In this paper, we consider the (Vertex) k -Way Cut problem on subclasses of chordal graphs and obtain the following results.

• We first give a sub-exponential FPT algorithm for k -Way Cut running in time $2^{{\mathcal {O}} (\sqrt{s} \log s)} n^{{\mathcal {O}} (1)}$ on chordal graphs.

• It is “known" that Vertex k -Way Cut is W[1]-hard on chordal graphs, in fact on split graphs, parameterized by $k+s$. We complement this hardness result by designing polynomial-time algorithms for Vertex k -Way Cut on interval graphs, circular-arc graphs and permutation graphs.