Packing spanning arborescences with extra large one

The celebrated Nash-Williams and Tutte’s theorem states that a graph $G=(V,E)$ contains $k$ edge disjoint spanning trees if and only if ${\nu }_f(G)\ge k$, where $${\nu }_f(G):=\underset{\mathrm{\mid }\mathcal{P}\mathrm{\mid }>1,\mathcal{P}\text{ is a partition of }V(G)}{\min }\frac{\mathrm{\mid }E(\mathcal{P})\mathrm{\mid }}{\mathrm{\mid }\mathcal{P}\mathrm{\mid }-1}\mathrm{.}$$ Inspired by the NDT theorem as structural explanations for the fractional part of Nash-Williams’ forest decomposition theorem, Fang and Yang extended Nash-Williams and Tutte’s theorem and proved that if ${\nu }_f(G)>k+\frac{d-1}{d}$, then $G$ contains $k$ edge disjoint spanning trees and another forest $F$ with $\mathrm{\mid }E(F)\mathrm{\mid }>\frac{d-1}{d}(\mathrm{\mid }V(G)\mathrm{\mid }-1)\mathrm{\mid }$, and if $F$ is not a spanning tree, then $F$ has a component with at least $d$ edges. In this paper, we give a digraphic version of their result; however, the mixed graphic version remains open.