Manifolds spines and hyperbolicity equations

We give a combinatorial representation of compact connected orientable 3-manifolds with boundary and their special spines by a class of graphs with extrastructure which are strictly related to o-graphs. Then we describe a simple algorithm for constructing the boundary of these manifolds by using a list of 6-tuples of non-negative integers. Finally we discuss some combinatorial methods for determining the hyperbolicity equations. Examples of hyperbolic 3-manifolds of low complexity illustrate in particular cases the constructions and algorithms presented in the paper.