# Upper bounds and algorithms for parallel knock-out numbers

Haitze J. Broersma, Matthew Johnson, Daniël Paulusma

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1 Citation (Scopus)

## 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 $\alpha$ 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.
Original language Undefined SIROCCO 2007: 14th International Colloquium on Structural Information and Communication Complexity Giuseppe Prencipe, Shmuel Zaks Berlin Springer 328-340 13 https://doi.org/10.1007/978-3-540-72951-8_26 Published - Jul 2007

### Publication series

Name Lecture Notes in Computer Science Springer Verlag 1 4474 0302-9743 1611-3349

• EWI-11586
• IR-62061
• METIS-245868