On a straight-line embedding problem of graphs

Let G be a planar graph with n vertices, v be a specified vertex of G, and P be a set of n points in the Euclidian plane in general position. A straight-line embedding of G onto P is an embedding of G onto whose images of vertices are distinct points in P and whose images of edges are (straight) line segments. In this paper, we classify into five classes those pairs of G and v such that for any P and any p P, G has a straight-line embedding onto P which maps v to p. We then show that all graphs in three of the classes indeed have such an embedding. This result gives a solution to a generalized version of the rooted-tree embedding problem raised by M. Perles.