A (δ ≥ k1, δ ≥ k2)-partition of a graph G is a vertex-partition (V1, V2) of G satisfying that δ(G[Vi]) ≥ ki for i = 1, 2. We determine, for all positive integers k1, k2, the complexity of deciding whether a given graph has a (δ ≥ k1, δ ≥ k2)-partition. We also address the problem of finding a function g(k1, k2) such that the (δ ≥ k1, δ ≥ k2)-partition problem is N P-complete for the class of graphs of minimum degree less than g(k1, k2) and polynomial for all graphs with minimum degree at least g(k1, k2). We prove that g(1, k) = k for k ≥ 3, that g(2, 2) = 3 and that g(2, 3), if it exists, has value 4 or 5.