  A $(\delta\geq k_1,\delta\geq k_2)$-partition of a graph $G$ is a
vertex-partition $(V_1,V_2)$ of $G$ satisfying that $\delta(G[V_i])\geq k_i$
for $i=1,2$. We determine, for all positive integers $k_1,k_2$, the complexity
of deciding whether a given graph has a $(\delta\geq k_1,\delta\geq
k_2)$-partition.
  We also address the problem of finding a function $g(k_1,k_2)$ such that the
$(\delta\geq k_1,\delta\geq k_2)$-partition problem is ${\cal
  NP}$-complete for the class of graphs of minimum degree less than
$g(k_1,k_2)$ and polynomial for all graphs with minimum degree at least
$g(k_1,k_2)$. We prove that $g(1,k)=k$ for $k\ge 3$, that $g(2,2)=3$ and that
$g(2,3)$, if it exists, has value 4 or 5.
