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Structural Parameterizations for Boxicity

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Abstract

The boxicity of a graph G is the least integer d such that G has an intersection model of axis-aligned d-dimensional boxes. Boxicity, the problem of deciding whether a given graph G has boxicity at most d, is NP-complete for every fixed \(d \ge 2\). We show that Boxicity is fixed-parameter tractable when parameterized by the cluster vertex deletion number of the input graph. This generalizes the result of Adiga et al. (2010), that Boxicity is fixed-parameter tractable in the vertex cover number. Moreover, we show that Boxicity admits an additive 1-approximation when parameterized by the pathwidth of the input graph. Finally, we provide evidence in favor of a conjecture of Adiga et al. (2010) that Boxicity remains NP-complete even on graphs of constant treewidth.

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Notes

  1. It was observed by one of the referees that indeed \({{\mathrm{box}}}(H) \le k + 1\) holds. Since this does not affect the running time of our algorithm but needs a little bit more discussion, we stick to the weaker bound.

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Authors and Affiliations

  1. Institut für Optimierung und Operations Research, Universität Ulm, Helmholtzstraße 18, 89081, Ulm, Germany

    Henning Bruhn, Morgan Chopin & Felix Joos

  2. Institut für Informatik, Universität zu Köln, Weyertal 80, 50931, Köln, Germany

    Oliver Schaudt

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  1. Henning Bruhn
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  2. Morgan Chopin
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  3. Felix Joos
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  4. Oliver Schaudt
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Correspondence to Morgan Chopin.

Additional information

An extended abstract was accepted for publication in the Proceedings of the 40th International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2014).

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Bruhn, H., Chopin, M., Joos, F. et al. Structural Parameterizations for Boxicity. Algorithmica 74, 1453–1472 (2016). https://doi.org/10.1007/s00453-015-0011-0

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