Hyperbolic graphs of small complexity

Matveev's definition of the complexity c(M) of a (closed, irreducible)
3-manifold M gives a very natural measure of how complicated M is.  In
addition, c has very nice properties, including additivity under connected
sum.  In these talks I will describe a variation of the definition of c
which applies to trivalent graphs embedded in 3-manifolds, whence in
particular to supports of 3-orbifolds.  I will describe two ways in which
a graph can be given a hyperbolic structure (both implied by the fact that
an orbifold supported by the graph is hyperbolic).  And I will explain
why the theory of complexity works particularly well for hyperbolic
graphs.  Starting from these theoretical results I will then describe a
census recently carried out jointly with D. Heard, C. Hodgson and B.
Martelli of hyperbolic graphs up to complexity 5 (including knots and
links up to complexity 4).