Drawing Clustered Planar Graphs on Disk Arrangements

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 $\mathcal V$ as clusters. Let $\mathcal C_{\mathcal C}$ be an arrangement of pairwise disjoint disks with a bijection between the disks and the clusters. Akitaya et al.[Akytaia et al. SODA 2018] give an algorithm to test whether $(G, \mathcal V)$ can be embedded onto $\mathcal C_{\mathcal C}$ with the additional constraint that edges are routed through a set of pipes between the disks. If such an embedding exists, 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.[Alam et al. JGAA 2015] 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, even if all disks have unit size. This answers an open question of Angelini et al.[Angelini et al. GD 2014]