  Refining the notion of an ideal triangulation of a compact three-manifold, we
provide in this paper a combinatorial presentation of the set of pairs (M,a),
where M is a three-manifold and a is a collection of properly embedded arcs. We
also show that certain well-understood combinatorial moves are sufficient to
relate to each other any two refined triangulations representing the same
(M,a). Our proof does not assume the Matveev-Pergallini calculus for ideal
triangulations, and actually easily implies this calculus.
