Connectivity of the flip graph of plane spanning paths

Prove that for every finite point set S in the plane in general position (no three points collinear), the flip graph G(S)—whose vertices are all plane spanning paths on S (non-crossing straight-line Hamiltonian paths) and whose edges connect two paths that differ by exactly two segments (removing one segment and adding one segment)—is connected.

Background

The paper studies the flip graph G(S) where vertices are non-crossing straight-line spanning paths on a point set S and edges represent flips that replace one segment with another so that the two paths differ by exactly two segments. The central question is whether G(S) is connected for every S in general position.

Prior work has established connectivity for special classes: points in convex position, one interior point, and generalized double circle configurations. This paper extends these results by proving connectivity (with diameter bounds) when exactly one point lies outside the convex hull of the remaining points and shows every connected component has at least three vertices. The general connectivity question, however, remains open and is formalized by the stated conjecture.