WALCOM 2019: WALCOM: Algorithms and Computation pp 160-171

# Drawing Clustered Graphs on Disk Arrangements

• Tamara Mchedlidze
• Ignaz Rutter
• Nina Zimbel
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11355)

## 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.

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© Springer Nature Switzerland AG 2019

## Authors and Affiliations

• Tamara Mchedlidze
• 1