Greedoids from flames

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 $ D $ with $ r\in V(D) $ is an $ r $-flame if for every $ {v\in V(D)-r} $, the in-degree of $ v $ is equal to the local edge-connectivity $ \lambda_D(r,v) $. We show that for every digraph $ D $ and $ r\in V(D) $, the edge sets of the $ r $-flame subgraphs of $ D $ form a greedoid. Our method yields a new proof of Lov\'asz' theorem stating: for every digraph $ D $ and $ r\in V(D) $, there is an $ r $-flame subdigraph $ F $ of $ D $ such that $ \lambda_F(r,v) =\lambda_D(r,v) $ for $ v\in V(D)-r $. We also give a strongly polynomial algorithm to find such an $ F $ working with a fractional generalization of Lov\'asz' theorem.