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