Geodesic
The structure of minimum-weight directed forests: related forests and convexity inequalities
Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part X, Tome 475 (2018), pp. 5-21

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A toolkit has been developed that allows to build directed forests from another directed forests. this toolkit helps to prove inequalities that connect the weights of minimal directed forests with different numbers of trees in them. Theorem on related forests hoe to transform a the minimal directed forest into another minimal directed forest with the number of roots less or greater by one. 
@article{ZNSL_2018_475_a0,
     author = {V. A. Buslov},
     title = {The structure of minimum-weight directed forests: related forests and convexity inequalities},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {5--21},
     publisher = {mathdoc},
     volume = {475},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2018_475_a0/}
}
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V. A. Buslov. The structure of minimum-weight directed forests: related forests and convexity inequalities. Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part X, Tome 475 (2018), pp. 5-21. http://geodesic.mathdoc.fr/item/ZNSL_2018_475_a0/