Wagner's theorem for Torus graphs

Abstmct. Wagner’s theorem (any two maximal plane graphs having p vertices are equivalent under diagonal transformations) is extended to maximal torus graphs, graphs embedded in the torus with a maximal set of edges present. Thus any maximal torus graph havingp vertices may be diagonally transformed into any other maximal torus p;aph having p vertices. As with Wag mr’s theorem, a normal form representing an intermediate stage in the above transformation is displayed. This result. along with Wagner’s theorem, may make possible constructive characterizations of planar and toroidal graphs, through a wholly combinatorial definition of diagonal transformation.