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

An H1,{H2}H1,{H2}-factor of a graph GG is a spanning subgraph of GG with exactly one component isomorphic to the graph H1H1 and all other components (if there are any) isomorphic to the graph H2H2. We completely characterise the class of connected almost claw-free graphs that have a P7,{P2}P7,{P2}-factor, where P7P7 and P2P2 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(n5.376)O(n5.376) time algorithm for claw-free graphs on nn vertices).