Background Inferring gene networks from high-throughput data constitutes an important step in the discovery of relevant regulatory relationships in organism cells. Despite the large number of available Gene Regulatory Network inference methods, the problem remains challenging: the underdetermination in the space of possible solutions requires additional constraints that incorporate a priori information on gene interactions. Methods Weighting all possible pairwise gene relationships by a probability of edge presence, we formulate the regulatory network inference as a discrete variational problem on graphs. We enforce biologically plausible coupling between groups and types of genes by minimizing an edge labeling functional coding for a priori structures. The optimization is carried out with Graph cuts, an approach popular in image processing and computer vision. We compare the inferred regulatory networks to results achieved by the mutual-information-based Context Likelihood of Relatedness (CLR) method and by the state-of-the-art GENIE3, winner of the DREAM4 multifactorial challenge. Results Our BRANE Cut approach infers more accurately the five DREAM4 in silico networks (with improvements from 6 % to 11 %). On a real Escherichia coli compendium, an improvement of 11.8 % compared to CLR and 3 % compared to GENIE3 is obtained in terms of Area Under Precision-Recall curve. Up to 48 additional verified interactions are obtained over GENIE3 for a given precision. On this dataset involving 4345 genes, our method achieves a performance similar to that of GENIE3, while being more than seven times faster. The BRANE Cut code is available at: http://www-syscom.univ-mlv.fr/~pirayre/Codes-GRN-BRANE-cut.html. Conclusions BRANE Cut is a weighted graph thresholding method. Using biologically sound penalties and data-driven parameters, it improves three state-of-the art GRN inference methods. It is applicable as a generic network inference post-processing, due to its computational efficiency.