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 two 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.