Disk intersection representations realize graphs by mapping vertices bijectively to disks in the plane such that two disks intersect each other if and only if the corresponding vertices are adjacent in the graph. If intersections are restricted to touching points of the boundaries, we call them disk contact representations. Deciding whether a vertex-weighted planar graph can be realized such that the disks\u27 radii coincide with the vertex weights is known to be NP-hard for both contact and intersection representations. In this work, we reduce the gap between hardness and tractability by analyzing the problem for special graph classes. We show that in the contact scenario 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 a constructive linear-time recognition algorithm for embedded stars with arbitrary weights.
We also consider a version of the problem in which the disks of a representation are supposed to cover preassigned points, called seeds. We show that both for contact and intersection representations this problem is NP-hard for unit weights even if the given graph is a path. If the disks\u27 radii are not prescribed, the problem remains NP-hard for trees in the contact scenario
