Reconstruction of the Path Graph

TL;DR

This paper characterizes the automorphism group of the path graph of convex point sets, proving it is dihedral, and provides an efficient algorithm to identify boundary paths and edges within the graph.

Contribution

It establishes the automorphism group of the path graph for convex point sets as dihedral and introduces an $O(N \, log \, N)$ algorithm for identifying boundary paths and edges.

Findings

Automorphism group of $G(P)$ is isomorphic to the dihedral group $D_{n}$.

Developed an $O(N \, log \, N)$ algorithm for path and edge identification.

Proved the uniqueness of boundary path identification up to automorphism.

Abstract

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 $G(P)$.