  Arboricity is a graph parameter akin to chromatic number, in that it seeks to
partition the vertices into the smallest number of sparse subgraphs. Where for
the chromatic number we are partitioning the vertices into independent sets,
for the arboricity we want to partition the vertices into cycle-free subsets
(i.e., forests). Arboricity is NP-hard in general, and our focus is on the
arboricity of cographs. For arboricity two, we obtain the complete list of
minimal cograph obstructions. These minimal obstructions do generalize to
higher arboricities; however, we no longer have a complete list, and in fact,
the number of minimal cograph obstructions grows exponentially with arboricity.
We obtain bounds on their size and the height of their cotrees.
  More generally, we consider the following common generalization of colouring
and partition into forests: given non-negative integers $p$ and $q$, we ask if
a given cograph $G$ admits a vertex partition into $p$ forests and $q$
independent sets. We give a polynomial-time dynamic programming algorithm for
this problem. In fact, the algorithm solves a more general problem which also
includes several other problems such as finding a maximum $q$-colourable
subgraph, maximum subgraph of arboricity-$p$, minimum vertex feedback set and
minimum $q$ of a $q$-colourable vertex feedback set.
