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 ${\cal O}^*(2^n)$ for any n-vertex P _ℓ-free graph. For ℓ= 6, its running time is ${\cal O}^*(1.5790^n)$. We modify this algorithm to solve the Longest Path Contractibility problem for P ₆-free graphs in ${\cal O}^*(1.5790^n)$ time.