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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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