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Sparse highly connected spanning subgraphs in dense directed graphs. (English) Zbl 07186350
Summary: Mader proved that every strongly $$k$$-connected $$n$$-vertex digraph contains a strongly $$k$$-connected spanning subgraph with at most $$2 kn - 2k^2$$ edges, where equality holds for the complete bipartite digraph $$DK_{ k,n-k}$$. For dense strongly $$k$$-connected digraphs, this upper bound can be significantly improved. More precisely, we prove that every strongly $$k$$-connected $$n$$-vertex digraph $$D$$ contains a strongly $$k$$-connected spanning subgraph with at most $$kn + 800k(k + \overline{\Delta}(D))$$ edges, where $$\overline{\Delta}(D)$$ denotes the maximum degree of the complement of the underlying undirected graph of a digraph $$D$$. Here, the additional term $$800k(k + \overline{\Delta}(D))$$ is tight up to multiplicative and additive constants. As a corollary, this implies that every strongly $$k$$-connected $$n$$-vertex semicomplete digraph contains a strongly $$k$$-connected spanning subgraph with at most $$kn + 800k^2$$ edges, which is essentially optimal since $$800k^2$$ cannot be reduced to the number less than $$k(k - 1)/2$$.
We also prove an analogous result for strongly $$k$$-arc-connected directed multigraphs. Both proofs yield polynomial-time algorithms.
##### MSC:
 05C20 Directed graphs (digraphs), tournaments 05C40 Connectivity
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##### References:
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