Upper Bounds and Algorithms for Parallel Knock-Out Numbers

  • Hajo Broersma
  • Matthew Johnson
  • Daniël Paulusma
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4474)


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.


parallel knock-out schemes claw-free graphs computational complexity 


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Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Hajo Broersma
    • 1
  • Matthew Johnson
    • 1
  • Daniël Paulusma
    • 1
  1. 1.Department of Computer Science, Durham University, Science Laboratories, South Road, Durham DH1 3LEU.K.

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