  Given a digraph $D=(V,A)$ and a positive integer $k$, a subset $B\subseteq A$
is called a \textbf{$k$-union-arborescence}, if it is the disjoint union of $k$
spanning arborescences. When also arc-costs $c:A\to \mathbb{R}$ are given,
minimizing the cost of a $k$-union-arborescence is well-known to be tractable.
In this paper we take on the following problem: what is the minimum cardinality
of a set of arcs the removal of which destroys every minimum $c$-cost
$k$-union-arborescence. Actually, the more general weighted problem is also
considered, that is, arc weights $w:A\to \mathbb{R}_+$ (unrelated to $c$) are
also given, and the goal is to find a minimum weight set of arcs the removal of
which destroys every minimum $c$-cost $k$-union-arborescence. An equivalent
version of this problem is where the roots of the arborescences are fixed in
advance. In an earlier paper [A. Bern\'ath and Gy. Pap, \emph{Blocking optimal
arborescences}, Integer Programming and Combinatorial Optimization, Springer,
2013] we solved this problem for $k=1$. This work reports on other partial
results on the problem. We solve the case when both $c$ and $w$ are uniform --
that is, find a minimum size set of arcs that covers all
$k$-union-arbosercences. Our algorithm runs in polynomial time for this
problem. The solution uses a result of [M. B\'ar\'asz, J. Becker, and A. Frank,
\emph{An algorithm for source location in directed graphs}, Oper. Res. Lett.
\textbf{33} (2005)] saying that the family of so-called insolid sets (sets with
the property that every proper subset has a larger in-degree) satisfies the
Helly-property, and thus can be (efficiently) represented as a subtree
hypergraph. We also give an algorithm for the case when only $c$ is uniform but
$w$ is not. This algorithm is only polynomial if $k$ is not part of the input.
