Partitioning Subclasses of Chordal Graphs with Few Deletions

Satyabrata Jana, Souvik Saha, Abhishek Sahu, Saket Saurabh, Shaily Verma

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.