On a Straight-Line Embedding Problem of Forests

We consider nite planar graphs without loops or multiple edges. Let G be a planar graph with vertex set V (G) and edge set E(G). We denote by jGj the order of G, that is, jGj = jV (G)j. Given a planar graph G, let P be a set of jGj points in the plane (2-dimentional Euclidean space) in general position (i.e., no three of them are collinear). Then G is said to be line embedded onto P or stright-line embedded onto P if G can be embedded in the plane so that every vertex of G corresponds to a point of P , every edge corresponds to a straight-line segment, and no two straight-line segments intersect except their common endpoint. Namely, G is line embedded onto P if there exists a bijection : V (G) ! P such that two points (x) and (y) are joined by a straight-line segment if and only if x and y are joined by an edge of G and all two distinct open straight-line segments have no point in common. We call such a bijection a line embedding or a straight-line embedding of G onto P . In this..