Concetti Chiave
The article presents a solution to the problem of matroid-reachability-based decomposition of arborescences, which is more complicated than the corresponding packing problem.
Sintesi
The article focuses on packing and decomposition problems related to arborescences in directed graphs. It provides the following key insights:
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Matroid-Reachability-Based Packing of Arborescences:
- The authors solve the problem of matroid-reachability-based (ℓ, ℓ′)-limited packing of arborescences, where a lower bound ℓ and an upper bound ℓ′ are given on the total number of arborescences in the packing.
- This result generalizes previous work on packing spanning arborescences, reachability arborescences, and matroid-based packing of arborescences.
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Matroid-Reachability-Based Decomposition of Arborescences:
- The authors solve the problem of matroid-reachability-based decomposition of arborescences, which turns out to be more complicated than the corresponding packing problem.
- This result is obtained from the solution of the more general matroid-reachability-based (ℓ, ℓ′)-limited packing problem.
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Extensions to Hypergraphs:
- The authors mention that the results have been extended to directed hypergraphs, including packing and decomposition problems for hyperarborescences.
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Implications and Connections:
- The authors show how their results imply and connect to previous work in the field, such as Edmonds' theorem on packing spanning arborescences, Kamiyama-Katoh-Takizawa's theorem on packing reachability arborescences, and Durand de Gevigney-Nguyen-Szigeti's theorem on matroid-based packing of arborescences.