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M. Nöllenburg, B. Klemz, R. Prutkin:
"Recognizing Weighted Disk Contact Graphs";
Vortrag: Graph Drawing and Network Visualization (GD´15), Los Angeles, USA; 24.09.2015 - 26.09.2015; in: "Graph Drawing and Network Visualization (GD'15)", E. Di Giacomo, A. Lubiw (Hrg.); Springer, LNCS 9411 (2015), ISBN: 978-3-319-27260-3; S. 433 - 446.

Disk contact representations realize graphs by mapping vertices bijectively to interior-disjoint disks in the plane such that two disks touch each other if and only if the corresponding vertices are adjacent in the graph. Deciding whether a vertex-weighted planar graph can be realized such that the disks´ radii coincide with the vertex weights is known to be NP-hard. In this work, we reduce the gap between hardness and tractability by analyzing the problem for special graph classes. We show that it remains NP-hard for outerplanar graphs with unit weights and for stars with arbitrary weights, strengthening the previous hardness results. On the positive side, we present constructive linear-time recognition algorithms for caterpillars with unit weights and for embedded stars with arbitrary weights.

http://dx.doi.org/10.1007/978-3-319-27261-0