Bibliographic Details
| Title: |
FPT algorithms for packing [formula omitted]-safe spanning rooted sub(di)graphs. |
| Authors: |
Bessy, Stéphane1 (AUTHOR) stephane.bessy@lirmm.fr, Hörsch, Florian1,2 (AUTHOR) florian.hoersch@cispa.de, Maia, Ana Karolinna3 (AUTHOR) karolmaia@ufc.br, Rautenbach, Dieter4 (AUTHOR) dieter.rautenbach@uni-ulm.de, Sau, Ignasi1 (AUTHOR) ignasi.sau@lirmm.fr |
| Superior Title: |
Discrete Applied Mathematics. Mar2024, Vol. 346, p80-94. 15p. |
| Subject Terms: |
*SPANNING trees, *ALGORITHMS, *INTEGERS |
| Abstract: |
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 ∈ 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 r v 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 ∈ 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. [ABSTRACT FROM AUTHOR] |
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| Database: |
Academic Search Premier |
