Résumé
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.