  For nonnegative integers $k, d_1, \ldots, d_k$, a graph is $(d_1, \ldots,
d_k)$-colorable if its vertex set can be partitioned into $k$ parts so that the
$i$th part induces a graph with maximum degree at most $d_i$ for all $i\in\{1,
\ldots, k\}$. A class $\mathcal C$ of graphs is {\it balanced
$k$-partitionable} and {\it unbalanced $k$-partitionable} if there exists a
nonnegative integer $D$ such that all graphs in $\mathcal C$ are $(D, \ldots,
D)$-colorable and $(0, \ldots, 0, D)$-colorable, respectively, where the tuple
has length $k$.
  A set $X$ of cycles is a {\it cycle obstruction set} of a class $\mathcal C$
of planar graphs if every planar graph containing none of the cycles in $X$ as
a subgraph belongs to $\mathcal C$. This paper characterizes all cycle
obstruction sets of planar graphs to be balanced $k$-partitionable and
unbalanced $k$-partitionable for all $k$; namely, we identify all
inclusion-wise minimal cycle obstruction sets for all $k$.
