Selected papers from the second Krakow conference on Graph theory
Journal of Combinatorial Theory Series B
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
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 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(@a), where @a is the independence number of G. This upper bound is tight. 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. We also pinpoint a relationship with (locally bijective) graph homomorphisms.