Partitioning graphs into connected parts

The 2-DISJOINT CONNECTED SUBGRAPHS problem asks if a given graph has two vertex-disjoint connected subgraphs containing pre-specified 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 P l on l vertices. We show that the computational complexity of the LONGEST PATH CONTRACTIBILITY problem restricted to P l-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 O*(2n) for any n-vertex P l-free graph. For l=¿6, its running time is O*(1.5790n). We modify this algorithm to solve the LONGEST PATH CONTRACTIBILITY problem for P 6-free graphs in O*(1.5790n) time