Klemz, B., Nöllenburg, M., & Prutkin, R. (2015). Recognizing Weighted Disk Contact Graphs. In E. Di Giacomo & A. Lubiw (Eds.), Graph Drawing and Network Visualization (pp. 433–446). Springer. https://doi.org/10.1007/978-3-319-27261-0_36
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.
