Blocking optimal $k$-arborescences

Given a digraph $D=(V,A)$ and a positive integer $k$, an arc set $F\subseteq Ais called a \textbf{k-arborescence} if it is the disjoint union ofk$spanningarborescences\mathrm{.}Theproblemoffindingaminimumcost$k$-arborescence is known to be polynomial-time solvable using matroid intersection. In this paper we study the following problem: find a minimum cardinality subset of arcs that contains at least one arc from every minimum cost $k$-arborescence. For $k=1$, the problem was solved in [A. Bern'ath, G. Pap , Blocking optimal arborescences, IPCO 2013]. In this paper we give an algorithm for general $k$ that has polynomial running time if $k$ is fixed.