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. [4], 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. [4] that Boxicity remains NP-complete even on graphs of constant treewidth.
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Bruhn, H., Chopin, M., Joos, F., Schaudt, O. (2014). Structural Parameterizations for Boxicity. In: Kratsch, D., Todinca, I. (eds) Graph-Theoretic Concepts in Computer Science. WG 2014. Lecture Notes in Computer Science, vol 8747. Springer, Cham. https://doi.org/10.1007/978-3-319-12340-0_10
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DOI: https://doi.org/10.1007/978-3-319-12340-0_10
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