  We define a new integer invariant of a finite graph G, the freeness index,
that measures the extent to which G can be embedded in the 3-sphere so that it
and its subgraphs have ``simple" complements, i.e., complements which are
homeomorphic to a connect-sum of handlebodies. We relate the freeness index to
questions of embedding graphs into surfaces, in particular to the orientable
cycle double cover conjecture. We show that a cubic graph satisfying the
orientable double cycle cover conjecture has freeness index at least two.
