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

Johnson, M. and Paulusma, D. and Wood, C. (2010) 'Path factors and parallel knock-out schemes of almost claw-free graphs.', in Proceedings of the International Workshop on Combinatorial Algorithms 2008. , pp. 27-41.


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).

Item Type:Book chapter
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Date accepted:No date available
Date deposited:19 April 2016
Date of first online publication:2010
Date first made open access:No date available

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