EEMCS

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Official URL: http://dx.doi.org/10.1016/j.tcs.2008.03.024

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 resolve the square-root conjecture, first posed at MFCS 2004, by showing that for a reducible graph , the minimum number of required rounds is ; in fact, our result is stronger than the conjecture as we show that the minimum number of required rounds is , where is the independence number of . This upper bound is tight. We also show that for reducible -free graphs at most 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.

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