Realization of Simply Connected Polygonal Linkages and Recognition of Unit Disk Contact Trees

Abstract

We wish to decide whether a simply connected flexible polygonal structure can be realized in Euclidean space. Two models are considered: polygonal linkages (body-and-joint framework) and contact graphs of unit disks in the plane.(1) We show that it is strongly NP-hard to decide whether a given polygonal linkage is realizable in the plane when the bodies are convex polygons and their contact graph is a tree; the problem is weakly NP-hard already for a chain of rectangles, but efficiently decidable for a chain of triangles hinged at distinct vertices. (2) We also show that it is strongly NP-hard to decide whether a given tree is the contact graph of interior-disjoint unit disks in the plane. P and the hinges in H, and edges represent the polygon-hinge incidences.

Recognizing Weighted Disk Contact Graphs

“…Our result holds both for the case that arbitrary embeddings are allowed and the case that a fixed combinatorial embedding is specified. The result for the latter case is also implied by a very recent result by Bowen et al [2] stating that for fixed embeddings the problem is NP-hard even for trees. However, the recognition of trees with a unit disk contact representation remains an interesting open problem if arbitrary embeddings are allowed.…”

“…Note that the NP-hardness of UDC recognition for embedded outerplanar graphs is implied by the recent result of Bowen et al [2] showing the NP-hardness of UDC recognition for embedded trees. It is, however, still open whether Theorem 2 extends to trees without a fixed embedding.…”

Computing Stable Demers Cartograms

“…Cartogram literature emphasizes counting lost adjacencies between regions, not the distance between them. We prefer our measure since 1) there is a big difference if two neighboring countries are set apart by a small or large gap; 2) while the LP can be turned to an integer linear program to count lost adjacencies, it greatly increases computational complexity-optimizing for adjacencies is typically NP-hard, e.g., for disks [4,5] or segments [15]. Our linear program typically admits several optimal solutions, due to translation invariance and since touching squares may slide freely along each other as long as they touch.…”

Triangle-Free Penny Graphs: Degeneracy, Choosability, and Edge Count

“…A minimum-distance representation can be obtained from a contact representation by choosing a point at the center of each circle, and a contact representation can be obtained from a minimum-distance representation by scaling the points so their minimum distance is two and using each point as the center of a unit circle. However, finding either type of representation given only the graph is NP-hard, even for trees [10].…”