  It is well known that determining if a digraph has a kernel is an NP-complete
problem. However, Topp proved that when subdividing every arc of a digraph we
obtain a digraph with a kernel. In this paper we define the kernel subdivision
number $\kappa(D)$ of a digraph $D$ as the minimum number of arcs, such that,
when subdividing them, we obtain a digraph with a kernel. We give a general
bound for $\kappa(D)$ in terms of the number of directed cycles of odd length
and compute $\kappa(D)$ for a few families of digraphs.
  If the digraph is $H$-colored, we can analogously define the $H$-kernel
subdivision number. In this paper we also improve a result for $H$-kernels
given by Galeana et al. to subdividing every arc of a spanning subgraph with
certain properties. Finally we prove that when the directed cycles of a digraph
overlap little enough, we can obtain a good bound for the $H$-kernel
subdivision number.
