Selected papers from the second Krakow conference on Graph theory
Graph Theory With Applications
Graph Theory With Applications
Eliminating graphs by means of parallel knock-out schemes
Discrete Applied Mathematics
The computational complexity of the parallel knock-out problem
LATIN'06 Proceedings of the 7th Latin American conference on Theoretical Informatics
The computational complexity of the parallel knock-out problem
Theoretical Computer Science
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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(√α), 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 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.