# Upper bounds and algorithms for parallel knock-out numbers

Haitze J. Broersma, Matthew Johnson, Daniël Paulusma

Research output: Chapter in Book/Report/Conference proceedingConference contributionAcademicpeer-review

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

### Cite this

Broersma, H. J., Johnson, M., & Paulusma, D. (2007). Upper bounds and algorithms for parallel knock-out numbers. In G. Prencipe, & S. Zaks (Eds.), SIROCCO 2007: 14th International Colloquium on Structural Information and Communication Complexity (pp. 328-340). [10.1007/978-3-540-72951-8_26] (Lecture Notes in Computer Science; Vol. 4474, No. 1). Berlin: Springer. https://doi.org/10.1007/978-3-540-72951-8_26
Broersma, Haitze J. ; Johnson, Matthew ; Paulusma, Daniël. / Upper bounds and algorithms for parallel knock-out numbers. SIROCCO 2007: 14th International Colloquium on Structural Information and Communication Complexity. editor / Giuseppe Prencipe ; Shmuel Zaks. Berlin : Springer, 2007. pp. 328-340 (Lecture Notes in Computer Science; 1).
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title = "Upper bounds and algorithms for parallel knock-out numbers",
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.",
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Broersma, HJ, Johnson, M & Paulusma, D 2007, Upper bounds and algorithms for parallel knock-out numbers. in G Prencipe & S Zaks (eds), SIROCCO 2007: 14th International Colloquium on Structural Information and Communication Complexity., 10.1007/978-3-540-72951-8_26, Lecture Notes in Computer Science, no. 1, vol. 4474, Springer, Berlin, pp. 328-340. https://doi.org/10.1007/978-3-540-72951-8_26

Upper bounds and algorithms for parallel knock-out numbers. / Broersma, Haitze J.; Johnson, Matthew; Paulusma, Daniël.

SIROCCO 2007: 14th International Colloquium on Structural Information and Communication Complexity. ed. / Giuseppe Prencipe; Shmuel Zaks. Berlin : Springer, 2007. p. 328-340 10.1007/978-3-540-72951-8_26 (Lecture Notes in Computer Science; Vol. 4474, No. 1).

Research output: Chapter in Book/Report/Conference proceedingConference contributionAcademicpeer-review

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Broersma HJ, Johnson M, Paulusma D. Upper bounds and algorithms for parallel knock-out numbers. In Prencipe G, Zaks S, editors, SIROCCO 2007: 14th International Colloquium on Structural Information and Communication Complexity. Berlin: Springer. 2007. p. 328-340. 10.1007/978-3-540-72951-8_26. (Lecture Notes in Computer Science; 1). https://doi.org/10.1007/978-3-540-72951-8_26