Trivial Cycles in Graphs Embedded in 3-Manifolds

A cycle C of a graph embedded in a 3-manifold M is said tobe trivial in if it bounds a disk with interior disjoint from. Let e be an edge of with ends on C. We shall study the relationbetween triviality of cycles in and that of – e and/e. Let C₁ be one of the two cycles in C e containing e. Themain theorem says that if C is trivial in – e and C₁/eis trivial in /e, then either C or C₁ is trivial in . Some applicationsto cycle trivial graphs will be given in Section 2.