Type 3 diminimal maps on the torus (Q1196321)

A map on a surface is an embedding of a graph such that each face is homeomorphic to a planar disk. It is polyhedral if it may be realized as the edge-complex of a polyhedron. It is called diminimal if its polyhedrality is lost by shrinking any edge (collapsing the edge into one vertex and identifying possible multiple edges) or removal of any edge (the dual operation of deleting the edge followed by replacement of any 2-valent vertex and its two adjacent edges into one edge). The type of a map on the torus is the maximal number of disjoint circuits which are homotopically nontrivial (nonplanar). It is shown that any diminimal map on the torus is either of type 2 or 3, and that there are exactly two of type 3: the triangular picture frame (three triangular cylinders joined without twist into a torus) and the (unique) twisted version. The characterization of type 2 diminimal toroidal maps remains open.