Upper Bounds and Algorithms for Parallel Knock-Out Numbers

Abstract

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 show that, for a reducible graph G, the minimum number of required rounds is $O{({\sqrt{\alpha}})}$, where α is the independence number of G. This upper bound is tight and the result implies the square-root conjecture which was first posed in MFCS 2004. We also show that for reducible K _1,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.