Upper Bounds and Algorithms for Parallel Knock-Out Numbers

We study parallel knock-out schemes for graphs. These schemes proceed in rounds in each of which each surviving vertex simultaneously eliminates one of its surviving neighbours; a graph is reducible if such a scheme can eliminate every vertex in the graph. We resolve the square-root conjecture, first posed at MFCS 2004, by showing that for a reducible graph G, the minimum number of required rounds is O( √ n); in fact, our result is stronger than the conjecture as we show that the minimum number of required rounds is O( √ α), where α is the independence number of G. This upper bound is tight. We also show that for reducible K1,p-free graphs at most p − 1 rounds are required. It is already known that the problem of whether a given graph is reducible is NP-complete. For claw-free graphs, however, we show that this problem can be solved in polynomial time. We also pinpoint a relationship with (locally bijective) graph homomorphisms. © 2008 Elsevier B.V. All rights reserved.