Partitioning graphs into connected parts

Hof, P. van 't; Paulusma, D.; Woeginger, G. J.

Abstract

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 Pℓ on ℓ vertices. We show that the computational complexity of the Longest Path Contractibility problem restricted to Pℓ-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 for any n-vertex Pℓ-free graph. For ℓ=6, its running time is . We modify this algorithm to solve the Longest Path Contractibility problem for P6-free graphs in time.

Citation

Hof, P. V. '., Paulusma, D., & Woeginger, G. J. (2009). Partitioning graphs into connected parts. Theoretical Computer Science, 410(47-49), 4834-4843. https://doi.org/10.1016/j.tcs.2009.06.028

• Journal Article Type: Article
• Acceptance Date: Jun 13, 2009
• Online Publication Date: Jun 23, 2009
• Publication Date: Nov 6, 2009
• Deposit Date: Oct 14, 2009
• Journal: Theoretical Computer Science
• Print ISSN: 0304-3975
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 410
• Issue: 47-49
• Pages: 4834-4843
• DOI: https://doi.org/10.1016/j.tcs.2009.06.028
• Keywords: Graph partition, Edge contraction, Path, Exact algorithm.
• Public URL: https://durham-repository.worktribe.com/output/1523871