Embedding Plane 3-Trees in ℝ2 and ℝ3

A point-set embedding of a planar graph G with n vertices on a set P of n points in Rd , d ≥ 1, is a straight-line drawing of G, where the vertices of G are mapped to distinct points of P . The problem of computing a point-set embedding of G on P is NP-complete in R2 , even when G is 2-outerplanar and the points are in general position. On the other hand, if the points of P are in general position in R3 , then any bijective mapping of the vertices of G to the points of P determines a point-set embedding of G on P . In this paper, we give an O(n4/3+ )-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 R2 and compute such an embedding if it exists, for any fixed >0. We extend our algorithm to embed a subclass of 4-trees on a point set in R3 in the form of nested tetrahedra. We also prove that given a plane 3-tree G with n vertices, a set P of n points in R3 that are not necessarily in general position and a mapping of the three outer vertices of G to three different points of P , it is NP-complete to decide if G admits a point-set embedding on P respecting the given mapping