  We define the induced arboricity of a graph $G$, denoted by ${\rm ia}(G)$, as
the smallest $k$ such that the edges of $G$ can be covered with $k$ induced
forests in $G$. This notion generalizes the classical notions of the arboricity
and strong chromatic index.
  For a class $\mathcal{F}$ of graphs and a graph parameter $p$, let
$p(\mathcal{F}) = \sup\{p(G) \mid G\in \mathcal{F}\}$. We show that ${\rm
ia}(\mathcal{F})$ is bounded from above by an absolute constant depending only
on $\mathcal{F}$, that is ${\rm ia}(\mathcal{F})\neq\infty$ if and only if
$\chi(\mathcal{F} \nabla \frac{1}{2}) \neq\infty$, where $\mathcal{F} \nabla
\frac{1}{2}$ is the class of $\frac{1}{2}$-shallow minors of graphs from
$\mathcal{F}$ and $\chi$ is the chromatic number.
  Further, we give bounds on ${\rm ia}(\mathcal{F})$ when $\mathcal{F}$ is the
class of planar graphs, the class of $d$-degenerate graphs, or the class of
graphs having tree-width at most $d$. Specifically, we show that if
$\mathcal{F}$ is the class of planar graphs, then $8 \leq {\rm ia}(\mathcal{F})
\leq 10$.
  In addition, we establish similar results for so-called weak induced
arboricities and star arboricities of classes of graphs.
