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

Abstract

For a set S of n points in convex position in the plane, let P ( S ) denote the set of all plane spanning paths of S. The geometric path graph of S, denoted by G n , is the graph with P ( S ) as its vertex set and two vertices P , Q ∈ P ( S ) are adjacent if and only if P and Q can be transformed to each other by means of a single edge replacement. Recently, Akl et al. [S.G. Akl, K. Islam, H. Meijer, On planar path transformation, Inform. Process. Lett. 104 (2007) 59–64] showed that the diameter of G n is at most 2 n − 5 . In this note, we derive the exact diameter of G n for n ⩾ 3 .