Diagonal transformations of triangulation on surfaces

THEOREM 1. (K. Wagner) Any two maximal planar graphs with the same number of vertices are equivalent under diagnal transformations. A maximal planar graph $G$ is a simple graph embedded in the plane such that one can add no new edge to it in the plane, that is, such a one that each region or face is three-edged. The diagonal transformation is to switch the diagonal edge $ac$ in the union of two adjacent triangular faces $abc$ and $acd$ , as shown in Figure 1. We however have to preserve the simpleness of graphs, that is, the diagonal transformation cannot be applied if vertices $b$ and $d$ are adjacent in $G$ .