Partitioning graphs into connected partsPim van 't Hof, Daniël Paulusma and Gerhard J. Woeginger
A preliminary version of this paper appeared in the proceedings of CSR 2009, the 4th International Computer Science Symposium in Russia (held on August 18-23, 2009 in Novosibirsk, Russia), Lecture Notes in Computer Science, vol. 5675, pp. 143-154, 2009.
The 2-Disjoint Connected Subgraphs problem asks if a given graph has two disjoint connected subgraphs containing predefined sets of vertices. We show that this problem is NP-complete, even if one of the sets has cardinality 2. The Longest Path Contractibility problem asks for the largest integer k for which an input graph can be contracted to the path Pk on k vertices. We show that the computational complexity of the Longest Path Contractibility problem restricted to Pk-free graphs jumps from being polynomially solvable to being NP-complete at k=6, while this jump occurs at k=5 for the 2-Disjoint Connected Subgraphs problem. We also present an exact algorithm that solves the 2-Disjoint Connected Subgraphs problem in O*(1.5790n) time for P6-free graphs, and show how this algorithm can be modified to solve the Longest Path Contractibility problem for P6-free graphs in in O*(1.5790n) time.