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 O(ssO(s)n2) time algorithm, where n denotes the number of vertices in G. The best-known algorithm for this problem runs in time sO(s)nO(1) given by Lokshtanov et al. [ACM Tran. of Algo. 2021]. On the other hand, Vertexk-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 2O(slog⁡s)nO(1) on chordal graphs.•It is “known” that Vertexk-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 Vertexk-Way Cut on interval graphs, circular-arc graphs and permutation graphs.