  It is known that an ideal triangulation of a compact $3$-manifold with
nonempty boundary is minimal if and only if it contains the minimum number of
edges among all ideal triangulations of the manifold. Therefore, any ideal
one-edge triangulation (i.e., an ideal singular triangulation with exactly one
edge) is minimal. Vesnin, Turaev, and the first author showed that an ideal
two-edge triangulation is minimal if no $3$-$2$ Pachner move can be applied. In
this paper we show that any of the so-called poor ideal three-edge
triangulations is minimal. We exploit this property to construct minimal ideal
triangulations for an infinite family of hyperbolic $3$-manifolds with totally
geodesic boundary.
