Reconstruction of the Geometric Structure of a Set of Points in the Plane from Its Geometric Tree Graph

Let P be a finite set of points in general position in the plane. The structure of the complete graph K(P) as a geometric graph includes, for any pair [a, b], [c, d] of vertex-disjoint edges, the information whether they cross or not. The simple (i.e., non-crossing) spanning trees (SSTs) of K(P) are the vertices of the so-called Geometric Tree Graph of P, G(P). Two such vertices are adjacent in G(P) if they differ in exactly two edges, i.e., if one can be obtained from the other by deleting an edge and adding another edge. In this paper we show how to reconstruct from G(P) (regarded as an abstract graph) the structure of K(P) as a geometric graph. We first identify within G(P) the vertices that correspond to spanning stars. Then we regard each star S(z) with center z as the representative in G(P) of the vertex z of K(P). (This correspondence is determined only up to an automorphism of K(P) as a geometric graph.) Finally we determine for any four distinct stars S(a), S(b), S(c),  and S(d), by looking at their relative positions in G(P), whether the corresponding segments cross.