DISJOINT ESSENTIAL CYCLES

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 embed dings in closed surfaces without having two disjoint noncontractible c ycles, such embeddings are structurally characterized. (C) 1996 Academ ic Press, Inc.