Partitioning graphs into connected parts

The 2-Disjoint Connected Subgraphs problem asks if a given graph has two vertex-disjoint connected subgraphs containing prespecified 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 l for which an input graph can be contracted to the path Pl on l vertices. We show that the computational complexity of the Longest Path Contractibility problem restricted to Pl-free graphs jumps from being polynomially solvable to being NP-hard at l=6, while this jump occurs at l=5 for the 2-Disjoint Connected Subgraphs problem. We also present an exact algorithm that solves the 2-Disjoint Connected Subgraphs problem faster than for any n-vertex Pl-free graph. For l=6, its running time is . We modify this algorithm to solve the Longest Path Contractibility problem for P6-free graphs in time.