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.
"Hyperbolic Graphs of Small Complexity." Experiment. Math. 19 (2) 211 - 236, 2010.