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 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*(2(n)) for any n-vertex P(l)-free graph. For l = 6, its running time is O*(1.5790(n)). We modify this algorithm to solve the LONGEST PATH CONTRACTIBILITY problem for PG-free graphs in O*(1.5790(n)) time. (C) 2009 Elsevier B.V. All rights reserved.
path exact algorithm graph partition edge contraction