This paper considers * -graphs in which all vertices have degree 4 or 6, and studies the question of calculating the genus of orientable 2-surfaces into which such graphs may be embedded. A * -graph is a graph endowed with a formal adjacency structure on the half-edges * -graphs in which all vertices have degree 4 or 6. This gives an algorithm in quadratic time to determine whether a * -graph admits an embedding into the plane.

This paper considers * -graphs in which all vertices have degree 4 or 6, and studies the question of calculating the genus of nonorientable surfaces into which such graphs may be embedded. In a previous paper [Embeddings of * -graphs into 2-surfaces, preprint (2012), arXiv:1212.5646] by the authors, the problem of calculating whether a given * -graph in which all vertices have degree 4 or 6 admits a Z 2 -homologically trivial embedding into a given orientable surface was shown to be equivalent to a problem on matrices. Here we extend those results to nonorientable surfaces. The embeddability condition that we obtain yields quadratic-time algorithms to determine whether a * -graph with all vertices of degree 4 or 6 admits a Z 2 -homologically trivial embedding into the projective plane or into the Klein bottle.