Abstract
Let \(G=(V, E)\) be a planar graph and let \(\mathcal V\) be a partition of V. We refer to the graphs induced by the vertex sets in 
as clusters. Let \(\mathcal {D}_{\mathcal C}\) be an arrangement of disks with a bijection between the disks and the clusters. Akitaya et al. [2] give an algorithm to test whether 
can be embedded onto \(\mathcal {D}_{\mathcal C}\) with the additional constraint that edges are routed through a set of pipes between the disks. Based on such an embedding, we prove that every clustered graph and every disk arrangement without pipe-disk intersections has a planar straight-line drawing where every vertex is embedded in the disk corresponding to its cluster. This result can be seen as an extension of the result by Alam et al. [3] who solely consider biconnected clusters. Moreover, we prove that it is \(\mathcal {NP} \)-hard to decide whether a clustered graph has such a straight-line drawing, if we permit pipe-disk intersections.
Work was partially supported by grant WA 654/21-1 of the German Research Foundation (DFG).
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Mchedlidze, T., Radermacher, M., Rutter, I., Zimbel, N. (2019). Drawing Clustered Graphs on Disk Arrangements. In: Das, G., Mandal, P., Mukhopadhyaya, K., Nakano, Si. (eds) WALCOM: Algorithms and Computation. WALCOM 2019. Lecture Notes in Computer Science(), vol 11355. Springer, Cham. https://doi.org/10.1007/978-3-030-10564-8_13
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