Path factors and parallel knock-out schemes of almost claw-free graphs

Johnson, M.; Paulusma, D.; Wood, C.

Abstract

An H1, {H2}-factor of a graph G is a spanning subgraph of G with exactly one component isomorphic to the graph H1 and all other components (if there are any) isomorphic to the graph H2.We completely characterise the class of connected almost claw-free graphs that have a P7, {P2}-factor, where P7 and P2 denote the paths on seven and two vertices, respectively. We apply this result to parallel knock-out schemes for almost claw-free graphs. These schemes proceed in rounds in each of which each surviving vertex eliminates one of its surviving neighbours. A graph is reducible if such a scheme eliminates every vertex in the graph. Using our characterisation we are able to classify all reducible almost claw-free graphs, and we can show that every reducible almost clawfree graph is reducible in at most two rounds. This leads to a quadratic time algorithm for determining if an almost claw-free graph is reducible (which is a generalisation and improvement upon the previous strongest result that showed that there was a O(n5.376) time algorithm for claw-free graphs on n vertices).

Citation

Johnson, M., Paulusma, D., & Wood, C. (2010, December). Path factors and parallel knock-out schemes of almost claw-free graphs. Presented at 19th International Workshop on Combinatorial Algorithms, Nagoya

• Presentation Conference Type: Conference Paper (published)
• Conference Name: 19th International Workshop on Combinatorial Algorithms
• Publication Date: Jan 1, 2010
• Deposit Date: Oct 6, 2010
• Publicly Available Date: Apr 19, 2016
• Pages: 27-41
• Book Title: Proceedings of the International Workshop on Combinatorial Algorithms 2008.
• DOI: https://doi.org/10.1016/j.disc.2009.04.022
• Public URL: https://durham-repository.worktribe.com/output/1158556
• Publisher URL: http://www.iwoca.org/iwoca2008/default.htm