FPT algorithms for packing k-safe spanning rooted sub(di)graphs

We study three problems introduced by Bang-Jensen and Yeo (2015) and by Bang-Jensen et al. (2016) about finding disjoint "balanced"spanning rooted substructures in graphs and digraphs, which generalize classic packing problems such as detecting the existence of multiple arc-disjoint spanning arborescences. Namely, given a positive integer k, a digraph D = (V, A), and a root r is an element of V, we first consider the problem of finding two arc-disjoint k-safe spanning r-arborescences, meaning arborescences rooted at a vertex r such that deleting any arc rv and every vertex in the sub-arborescence rooted at v leaves at least k vertices. Then, we consider the problem of finding two arc-disjoint (r, k)-flow branchings meaning arc sets admitting a flow that distributes one unit from r to every other vertex while respecting a capacity limit of n - k on every arc. 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 et al. (2016). Further, given a positive integer k, a graph G = (V, E), and r is an element of V, we consider the problem of finding two edge-disjoint (r, k)-safe spanning trees meaning spanning trees such that the component containing r has size at least k when deleting any vertex different from r. 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.