Obedient Plane Drawings for Disk Intersection Graphs

Banyassady, Bahareh; Hoffmann, Michael M.; Klemz, Boris; Löffler, Maarten; Miltzow, Tillmann

doi:10.1007/978-3-319-62127-2_7

Cited by 2 publications

(3 citation statements)

References 11 publications

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“…Given a planar clustered graph C = (G, V), a disk arrangement D C that is not planar, i.e., D C satisfies condition (C1) and (C2) but not (P1) and (P2), and a D C -framed embedding ψ of G, is there a D Cframed straight-line drawing Γ that is homeomorphic to ψ and D C ? Note that if the disks D C are allowed to overlap (condition (P1)) and G is the intersection graph of D C , the problem is known to be N P-hard [6]. Thus, in the following we require that the disks do not overlap, but there can be disk-pipe intersections, i.e, D C satisfies conditions (C1), (C1) and (P1) but not (P2).…”

Section: Drawing On General Disk Arrangementsmentioning

confidence: 99%

“…This requires that in the transition from ψ to Γ at any point in time an edge uv does not intersect a geometric object other than its own clusters. Note that if the disks D C are allowed to overlap and C is the intersection graph of D C , the problem is known to be N P-hard [6]. Thus, in the following we require that the disk may not overlap, but there can be disk-pipe intersection.…”

Section: A Drawing On Non-planar Disk Arrangementsmentioning

confidence: 99%

“…Banyassady et al [6] study whether the intersection graph of unit disks has a straight-line drawing such that each vertex lies in its disk. They proved that this problem is N P-hard regardless of whether the embedding of the intersection graph is prescribed or not.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Drawing Clustered Graphs on Disk Arrangements

Mchedlidze

Radermacher

Rutter

et al. 2018

WALCOM: Algorithms and Computation

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Let G = (V, E) be a planar graph and let V be a partition of V . We refer to the graphs induced by the vertex sets in V as clusters. Let DC be an arrangement of disks with a bijection between the disks and the clusters. Akitaya et al. [2] give an algorithm to test whether (G, V) can be embedded onto DC 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 N P-hard to decide whether a clustered graph has such a straight-line drawing, if we permit pipe-disk intersections.

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Section: Drawing On General Disk Arrangementsmentioning

confidence: 99%

Section: A Drawing On Non-planar Disk Arrangementsmentioning

confidence: 99%

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Drawing Clustered Graphs on Disk Arrangements

Mchedlidze

Radermacher

Rutter

et al. 2018

WALCOM: Algorithms and Computation

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Drawing Clustered Planar Graphs on Disk Arrangements

Mchedlidze¹,

Radermacher²,

Rutter³

et al. 2020

JGAA

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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].

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Obedient Plane Drawings for Disk Intersection Graphs

Cited by 2 publications

References 11 publications

Drawing Clustered Graphs on Disk Arrangements

Drawing Clustered Graphs on Disk Arrangements

Drawing Clustered Planar Graphs on Disk Arrangements

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