On signed diagonal flip sequences

Eliahou (1999) [1] and Kryuchkov (1992) [3] conjectured a proposition that Gravier and Payan (2002) [2] proved to be equivalent to the Four Color Theorem. It states that any triangulation of a polygon can be transformed into another triangulation of the same polygon by a sequence of signed diagonal flips. It is well known that any pair of polygonal triangulations are connected by a sequence of (non-signed) diagonal flips. In this paper we give a sufficient and necessary condition for a diagonal flip sequence to be a signed diagonal flip sequence.