  In this paper we enumerate and classify the ``simplest'' pairs (M,G) where M
is a closed orientable 3-manifold and G is a trivalent graph embedded in M.
  To enumerate the pairs we use a variation of Matveev's definition of
complexity for 3-manifolds, and we consider only (0,1,2)-irreducible pairs,
namely pairs (M,G) such that any 2-sphere in M intersecting G transversely in
at most 2 points bounds a ball in M either disjoint from G or intersecting G in
an unknotted arc. To classify the pairs our main tools are geometric invariants
defined using hyperbolic geometry. In most cases, the graph complement admits a
unique hyperbolic structure with parabolic meridians; this structure was
computed and studied using Heard's program Orb and Goodman's program Snap.
  We determine all (0,1,2)-irreducible pairs up to complexity 5, allowing
disconnected graphs but forbidding components without vertices in complexity 5.
The result is a list of 129 pairs, of which 123 are hyperbolic with parabolic
meridians. For these pairs we give detailed information on hyperbolic
invariants including volumes, symmetry groups and arithmetic invariants.
Pictures of all hyperbolic graphs up to complexity 4 are provided. We also
include a partial analysis of knots and links.
  The theoretical framework underlying the paper is twofold, being based on
Matveev's theory of spines and on Thurston's idea (later developed by several
authors) of constructing hyperbolic structures via triangulations. Many of our
results were obtained (or suggested) by computer investigations.
