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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2502.10052 |
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| _version_ | 1866909493231616000 |
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| author | Szeszlér, Dávid |
| author_facet | Szeszlér, Dávid |
| contents | A directed graph $F$ with a root node $r$ is called a flame if for every vertex $v$ other than $r$ the local edge-connectivity value $λ(r,v)$ from $r$ to $v$ is equal to $\varrho_F(v)$, the in-degree of $v$. It is a classic, simple and beautiful result of Lovász that every digraph $D$ with a root node $r$ has a spanning subgraph $F$ that is a flame and the $λ(r,v)$ values are the same in $F$ as in $D$ for every vertex $v$ other than $r$. However, the complexity of finding the minimum weight of such a subgraph is open. In this paper we prove that this problem is solvable in strongly polynomial time for acyclic digraphs. Besides that, we prove a decomposition result of flames into a chain of smaller flames via edge-disjoint branchings and use this to prove a common generalization of Lovász's above mentioned theorem and Edmonds' classic disjoint arborescences theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_10052 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Some Algorithmic and Structural Results on Flames Szeszlér, Dávid Combinatorics A directed graph $F$ with a root node $r$ is called a flame if for every vertex $v$ other than $r$ the local edge-connectivity value $λ(r,v)$ from $r$ to $v$ is equal to $\varrho_F(v)$, the in-degree of $v$. It is a classic, simple and beautiful result of Lovász that every digraph $D$ with a root node $r$ has a spanning subgraph $F$ that is a flame and the $λ(r,v)$ values are the same in $F$ as in $D$ for every vertex $v$ other than $r$. However, the complexity of finding the minimum weight of such a subgraph is open. In this paper we prove that this problem is solvable in strongly polynomial time for acyclic digraphs. Besides that, we prove a decomposition result of flames into a chain of smaller flames via edge-disjoint branchings and use this to prove a common generalization of Lovász's above mentioned theorem and Edmonds' classic disjoint arborescences theorem. |
| title | On Some Algorithmic and Structural Results on Flames |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2502.10052 |