
TL;DR
This paper introduces a new class of subgraphs called $r$-flames in digraphs, proves they form a greedoid, and provides a polynomial algorithm to find optimal $r$-flame subgraphs, offering a new proof of Lovász's theorem.
Contribution
It establishes that $r$-flame subgraphs form a greedoid and presents a strongly polynomial algorithm for their construction, extending Lovász's theorem.
Findings
$r$-flame subgraphs form a greedoid.
A new proof of Lovász's theorem is provided.
A strongly polynomial algorithm is developed.
Abstract
A digraph with is an -flame if for every , the in-degree of is equal to the local edge-connectivity . We show that for every digraph and , the edge sets of the -flame subgraphs of form a greedoid. Our method yields a new proof of Lov\'asz' theorem stating: for every digraph and , there is an -flame subdigraph of such that for . We also give a strongly polynomial algorithm to find such an working with a fractional generalization of Lov\'asz' theorem.
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