We study three problems introduced by Bang-Jensen and Yeo [Theor. Comput. Sci. 2015] and by Bang-Jensen, Havet, and Yeo [Discret. Appl. Math. 2016] about finding disjoint "balanced" spanning rooted substructures in graphs and digraphs, which generalize classic packing problems. Namely, given a positive integer $k$, a digraph $D=(V,A)$, and a root $r \in V$, we consider the problem of finding two arc-disjoint $k$-safe spanning $r$-arborescences and the problem of finding two arc-disjoint $(r,k)$-flow branchings. We show that both these problems are FPT with parameter $k$, improving on existing XP algorithms. The latter of these results answers a question of Bang-Jensen, Havet, and Yeo [Discret. Appl. Math. 2016]. Further, given an integer $k$, a graph $G=(V,E)$, and $r \in V$, we consider the problem of finding two arc-disjoint $(r,k)$-safe spanning trees. We show that this problem is also FPT with parameter $k$, again improving on a previous XP algorithm. Our main technical contribution is to prove that the existence of such spanning substructures is equivalent to the existence of substructures with size and maximum (out-)degree both bounded by a (linear or quadratic) function of $k$, which may be of independent interest.

翻译：我们研究了由Bang-Jensen和Yeo[Theor. Comput. Sci. 2015]和Bang-Jensen、Havet和Yeo[Discret. Appl. Math. 2016]提出的三个问题,即如何在图表和字典中找到“平衡”的根基子结构脱节的问题,这些问题概括了典型的包装问题。也就是说,考虑到正整数$(V,A)和根值美元(GR),我们考虑了如何找到两个正数美元安全值($-discommt,美元-Jenson,Sci. Sci. Sci. 2015),以及Bang-Jensen 美元(Scient. Appl. Math.