Transitions of Hexangulations on the Sphere.

A hexangulation G is a 2-connected simple plane graph such that each face of G is bounded by a 6-cycle. It was recently proved that any two hexangulations with the same number of vertices can be transformed into each other by three specifically defined transformations $${\mathbb{A}, \mathbb{B}}$$ and $${\mathbb{C}}$$ . We prove that any two hexangulations G and G′ with bipartitions { B, W} and { B′, W′}, respectively, such that | B| = | B′| and | W| = | W′|, can be transformed into each other by successive applications of operations in $${\{\mathbb{A}, \mathbb{B}^2, \mathbb{C}\}}$$ . Moreover, we completely describe the role of the four operations $${\mathbb{A}, \mathbb{B}, \mathbb{B}^2}$$ and $${\mathbb{C}}$$ in the transition diagram of hexangulations. [ABSTRACT FROM AUTHOR]