Reconstruction of the path graph

Let P be a set of n≥5 points in convex position in the plane. The path graph G(P) of P is an abstract graph whose vertices are non-crossing spanning paths of P, such that two paths are adjacent if one can be obtained from the other by deleting an edge and adding another edge. We prove that the automorphism group of G(P) is isomorphic to D_n, the dihedral group of order 2n. The heart of the proof is an algorithm that first identifies the vertices of G(P) that correspond to boundary paths of P, where the identification is unique up to an automorphism of K(P) as a geometric graph, and then identifies (uniquely) all edges of each path represented by a vertex of G(P). The complexity of the algorithm is O(Nlog⁡N) where N is the number of vertices of G(P).