On the Hardness of Point-Set Embeddability⋆

A point-set embedding of a plane graph G with n vertices on a set S of n points is a straight-line drawing of G, where the vertices of G are mapped to distinct points of S. The problem of deciding whether a plane graph admits a point-set embedding on a given set of points is NPcomplete for 2-connected planar graphs, but polynomial-time solvable for outerplanar graphs and plane 3-trees. In this paper we prove that the problem remains NP-complete for 3-connected planar graphs, which settles an open question posed by Cabello (Journal of Graph Algorithms and Applications, 10(2), 2000). We then show that the constraint of convexity makes the problem easier for klee graphs, which is a subclass of 3-connected planar graphs. We give a polynomial-time algorithm to decide whether a klee graph with exactly three outer vertices admits a convex point-set embedding on a given set of points and compute such an embedding if one exists.