On the orientable genus of graphs embedded in the klein bottle
Neil R. Robertson, Robin B. Thomas · Journal of Graph Theory · 1991
Abstract Let G be a graph embedded in the Klein bottle with “representativity” at least four. We give a formula for the orientable genus of G, which also implies a polynomially bounded algorithm. The formula is in terms of the number of times certain closed curves on the Klein bottle intersect the graph. In particular, it shows that a cut‐and‐paste technique for re‐embedding graphs is the best possible.