The kernel-subdivision number of a digraph

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.