2 research outputs found

    Sparse highly connected spanning subgraphs in dense directed graphs

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    Mader proved that every strongly kk-connected nn-vertex digraph contains a strongly kk-connected spanning subgraph with at most 2kn−2k22kn - 2k^2 edges, where the equality holds for the complete bipartite digraph DKk,n−k{DK}_{k,n-k}. For dense strongly kk-connected digraphs, this upper bound can be significantly improved. More precisely, we prove that every strongly kk-connected nn-vertex digraph DD contains a strongly kk-connected spanning subgraph with at most kn+800k(k+Δ‾(D))kn + 800k(k+\overline{\Delta}(D)) edges, where Δ‾(D)\overline{\Delta}(D) denotes the maximum degree of the complement of the underlying undirected graph of a digraph DD. Here, the additional term 800k(k+Δ‾(D))800k(k+\overline{\Delta}(D)) is tight up to multiplicative and additive constants. As a corollary, this implies that every strongly kk-connected nn-vertex semicomplete digraph contains a strongly kk-connected spanning subgraph with at most kn+800k2kn + 800k^2 edges, which is essentially optimal since 800k2800k^2 cannot be reduced to the number less than k(k−1)/2k(k-1)/2. We also prove an analogous result for strongly kk-arc-connected directed multigraphs. Both proofs yield polynomial-time algorithms.Comment: 31 page

    Minimally k-edge-connected directed graphs of maximal size

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    Minimally k-edge-connected directed graphs of maximal siz
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