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 for which an input graph can be contracted to the path P on vertices. We show that the computational complexity of the Longest Path Contractibility problem restricted to P -free graphs jumps from being polynomially solvable to being NP-hard at = 6, while this jump occurs at = 5 for the 2Disjoint 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 -free graph. For = 6, its running time is O∗

• Type: Conference
• Year: 2009
• Where: CSR
• Authors: Pim van 't Hof, Daniël Paulusma, Gerhard J. Woeginger
• Keywords: 2-disjoint Connected Subgraphs, Connected Subgraphs Problem, CSR 2009, Path Contractibility Problem, Theoretical Computer Science