Sparse Highly Connected Spanning Subgraphs in Dense Directed Graphs

Abstract

Mader proved that every strongly k -connected n -vertex digraph contains a strongly k -connected spanning subgraph with at most 2 kn - 2 k 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 + 800 k ( k + Δ ( D )) edges, where Δ ( D ) denotes the maximum degree of the complement of the underlying undirected graph of a digraph D . Here, the additional term 800 k ( k + Δ ( 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 + 800 k 2 edges, which is essentially optimal since 800 k 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.