Reconstruction of the Path Graph

Let $P$ be a set of $n \geq 5$ 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 $O(N \log N)$ where $N$ is the number of vertices of $G(P)$.