Learning the structure of Bayesian networks from data is a NP-Hard problem that involves optimization over a super-exponential sized space. In this work, we show that in most real life datasets, a number of the arcs contained in the final structure can be pre-screened at low computational cost with a limited impact on the global graph score. We formalize the identification of these arcs via the notion of quasi-determinism, and propose an associated algorithm that narrows the structure learning task down to a subset of the original variables. We show, on diverse benchmark datasets, that this algorithm exhibits a significant decrease in computational time and complexity for only a little decrease in performance score.