Disjoint essential cycles (Q1125961)

Graphs that have two disjoint noncontractible cycles in every possible embedding in surfaces are characterized. Similar characterization is given for the class of graphs whose orientable embeddings (embeddings in surfaces different from the projective plane, respectively) always have two disjoint noncontractible cycles. For graphs which admit embeddings in closed surfaces without having two disjoint noncontractible cycles, such embeddings are characterized in terms of their geometric structure. This result shows close relationship with Dirac's characterization of graphs which do not have two disjoint cycles.