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

An H 1 , { H 2 } -factor of a graph G is a spanning subgraph of G with exactly one component isomorphic to the graph H 1 and all other components (if there are any) isomorphic to the graph H 2 . We completely characterise the class of connected almost claw-free graphs that have a P 7 , { P 2 } -factor, where P 7 and P 2 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 claw-free 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 ( n 5.376 ) time algorithm for claw-free graphs on n vertices).