On the diameter of geometric path graphs of points in convex position

For a set S of n points in convex position in the plane, let P(S) denote the set of all plane spanning paths on S. The geometric path graph Gn of S is the graph with P(S) as its vertex set and two vertices P,Q ∈ P(S) are adjacent if one of the corresponding paths can be obtained from the other by means of a single edge replacement. Recently, Akl et al. [On planar path transformation, Inform. Process. Lett. 104 (2007) 59–64] showed that the diameter of Gn is at most 2n−5. In this note, we derive the exact diameter of Gn for n > 4.