Reconstruction of the path graph

We prove that the automorphism group of G(P) is isomorphic to Dn, 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).