Résumé : Graph cuts optimization is now well established for their efficiency but remains limited to the minimization of some Markov Random Fields (MRF) over a small number of variables due to the large memory requirement for storing the graphs. An existing strategy to reduce the graph size consists in testing every node and to create the node satisfying a given local condition. The remaining nodes are typically located in a thin band around the object to segment. However, there does not exists any theoretical guarantee that this strategy permits to construct a global minimizer of the MRF. In this paper, we propose a local test similar to already existing test for reducing these graphs. A large part of this paper consists in proving that any node satisfying this new test can be safely removed from the non-reduced graph without modifying its max-flow value. The constructed solution is therefore guanranteed to be a global minimizer of the MRF. Afterwards, we present numerical experiments for segmenting grayscale and color images which confirm this property while globally having memory gains similar to ones obtained with the previous existing local test.