Minimal triangulation size of Seifert fibered spaces with boundary

One measure of the complexity of a 3-manifold is its triangulation complexity: the minimal number of tetrahedra in a triangulation of it. A natural question is whether we can relate this quantity to its topology. We determine the triangulation complexity of Seifert fibered spaces with non-empty boundary in terms of their Seifert data, up to a multiplicative constant. We also show that all singular fibres of such a Seifert fibered space (aside from those of multiplicity two) can be made simplicial in the 79th barycentric subdivision of any triangulation of it.