Partitioning graphs into connected parts

Abstract

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*(1.5790 n) time. © 2009 Springer.