Graph minors. VIII. A kuratowski theorem for general surfaces

Kuratowski’s famous theorem of 1930 [6] asserts that a graph is planar if and only if it does not topologically contain K5 or K3,3. (Graphs are finite, and may have loops or multiple edges. A graph G topologically contains H if we may obtain a graph isomorphic to H from some subgraph of G by suppressing some divalent vertices.) There arose in the 1930’s the proposal to lind parallels to this theorem which apply to other surfaces. If C is a surface, let T(L) denote the class of all graphs which cannot be drawn in Z: and which are minimal with this property under topological containment. If 22 is the plane or sphere then the members of T(C) are precisely the graphs isomorphic to K5 or K,,,-this is another way to state Kuratowski’s theorem. In general, a graph can be drawn in Z: if and only if it topologically contains no members of T(Z). It remains then to determine T(C) explicitly. This appears to be very difficult, and there was little progress on the