The computational complexity of the parallel knock-out problem

Broersma, H.; Johnson, M.; Paulusma, D.; Stewart, I.A.

Abstract

We consider computational complexity questions related to parallel knock-out schemes for graphs. In such schemes, in each round, each remaining vertex of a given graph eliminates exactly one of its neighbours. We show that the problem of whether, for a given bipartite graph, such a scheme can be found that eliminates every vertex is NP-complete. Moreover, we show that, for all fixed positive integers k≥2, the problem of whether a given bipartite graph admits a scheme in which all vertices are eliminated in at most (exactly) k rounds is NP-complete. For graphs with bounded tree-width, however, both of these problems are shown to be solvable in polynomial time. We also show that r-regular graphs with r≥1, factor-critical graphs and 1-tough graphs admit a scheme in which all vertices are eliminated in one round.

Citation

Broersma, H., Johnson, M., Paulusma, D., & Stewart, I. (2008). The computational complexity of the parallel knock-out problem. Theoretical Computer Science, 393(1-3), 182-195. https://doi.org/10.1016/j.tcs.2007.11.021

• Journal Article Type: Article
• Publication Date: Mar 1, 2008
• Deposit Date: Jun 18, 2008
• Publicly Available Date: Jul 2, 2009
• Journal: Theoretical Computer Science
• Print ISSN: 0304-3975
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 393
• Issue: 1-3
• Pages: 182-195
• DOI: https://doi.org/10.1016/j.tcs.2007.11.021
• Keywords: Parallel knock-out schemes, Computational complexity, NP-completeness, Bounded tree-width, Monadic second-order logic.
• Public URL: https://durham-repository.worktribe.com/output/1591554
• Publisher URL: http://www.dur.ac.uk/i.a.stewart/Papers/knockout.pdf