Drawing Clustered Planar Graphs on Disk Arrangements

Tamara Mchedlidze ; Marcel Radermacher ; Ignaz Rutter ; Nina Zimbel

Journal of Graph Algorithms and Applications, Special Issue on Selected Papers from the 13th International Conference and Workshops on Algorithms and Computation, WALCOM 2019 , Tome 24 (2020) no. 2, pp. 105-131.

Résumé

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].

Détail

DOI : 10.7155/jgaa.00521

Keywords: Clustered Planar Graphs, Disk Arrangements, straight-line drawings, NP-Hardness