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
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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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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DOI: https://doi.org/10.1007/s00453-015-0011-0