  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 $G(P)$.
