Diagonal Transformations in Pentangulations on the Sphere

Abstract

A pentangulation is a simple plane graph such that each face is bounded by a cycle of length \(5\). We consider two diagonal transformations in pentangulations, called \(\mathcal{A}\) and \(\mathcal{B}\). In this paper, we shall prove that any two pentangulations with the same number of vertices can be transformed into each other by \(\mathcal{A}\) and \(\mathcal{B}\). In particular, if they are not isomorphic to a special pentangulation, then we do not need \(\mathcal{B}\).