MAX (k, n−k)-CUT (resp., MIN (k, n−k)-CUT) is a constrained version of MAX-CUT (resp.,MIN-CUT) where one has to find a bipartition of the vertex set into two subsets with respectively k and n − k vertices (n being the total number of vertices of the input graph) which maximizes (resp., minimizes) the number of edges going from one subset to the other. In this paper, we investigate the parameterized complexity of these two graph problems by considering several parameters, such as the value p of the solution, k, the size á of a minimum vertex cover and the treewidth tw of the input graph. We also give approximation schemata in FPT time for parameterizations which turn out to be W[1]-hard.