Embedding Plane 3-Trees in R2 and R3

A point-set embedding of a planar graph C with n vertices on a set P of n points in R-d, d >= 1, is a straight-line drawing of C, where the vertices of C are mapped to distinct points of P. The problem of computing a point-set embedding of C on P is NP-complete in R-2, even when C is 2-outerplanar and the points are in general position. On the other hand, if the points of P are in general position in R-3, then any bijective mapping of the vertices of C to the points of P determines a point-set embedding of C; on P. In this paper, we give an O(n(4/3+epsilon))-expected time algorithm to decide whether a plane 3-tree with n vertices admits a point-set embedding on a. given set of n points in general position in R-2 and compute such an embedding if it exists, for any fixed epsilon>0. We extend our algorithm to embed a subclass of 4-trees on a point set in R-3 in the form of nested tetrahedra. We also prove that given a plane 3-tree C with n vertices, a set P of n points in R-3 that are not necessarily in general position and a mapping of the three outer vertices of C to three different points of P, it is NP-complete to decide if C; admits a point-set embedding on P respecting the given mapping.