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<math><mi is="true">G</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">−</mo><mi is="true">S</mi></math> 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)<math><mi mathvariant="script" is="true">O</mi><mo stretchy="false" is="true">(</mo><msup is="true"><mrow is="true"><mi is="true">s</mi></mrow><mrow is="true"><msup is="true"><mrow is="true"><mi is="true">s</mi></mrow><mrow is="true"><mi mathvariant="script" is="true">O</mi><mo stretchy="false" is="true">(</mo><mi is="true">s</mi><mo stretchy="false" is="true">)</mo></mrow></msup></mrow></msup><msup is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></msup><mo stretchy="false" is="true">)</mo></math> 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)<math><msup is="true"><mrow is="true"><mi is="true">s</mi></mrow><mrow is="true"><mi mathvariant="script" is="true">O</mi><mo stretchy="false" is="true">(</mo><mi is="true">s</mi><mo stretchy="false" is="true">)</mo></mrow></msup><msup is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mi mathvariant="script" is="true">O</mi><mo stretchy="false" is="true">(</mo><mn is="true">1</mn><mo stretchy="false" is="true">)</mo></mrow></msup></math> 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<math><mi is="true">k</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">+</mo><mi is="true">s</mi></math>. 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)<math><msup is="true"><mrow is="true"><mn is="true">2</mn></mrow><mrow is="true"><mi mathvariant="script" is="true">O</mi><mo stretchy="false" is="true">(</mo><msqrt is="true"><mrow is="true"><mi is="true">s</mi></mrow></msqrt><mi mathvariant="normal" is="true">log</mi><mo is="true">⁡</mo><mi is="true">s</mi><mo stretchy="false" is="true">)</mo></mrow></msup><msup is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mi mathvariant="script" is="true">O</mi><mo stretchy="false" is="true">(</mo><mn is="true">1</mn><mo stretchy="false" is="true">)</mo></mrow></msup></math> 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<math><mi is="true">k</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">+</mo><mi is="true">s</mi></math>. We complement this hardness result by designing polynomial-time algorithms for Vertex k-Way Cut on interval graphs, circular-arc graphs and permutation graphs.