2 research outputs found
Sparse highly connected spanning subgraphs in dense directed graphs
Mader proved that every strongly -connected -vertex digraph contains a
strongly -connected spanning subgraph with at most edges, where
the equality holds for the complete bipartite digraph . For dense
strongly -connected digraphs, this upper bound can be significantly
improved. More precisely, we prove that every strongly -connected -vertex
digraph contains a strongly -connected spanning subgraph with at most
edges, where denotes
the maximum degree of the complement of the underlying undirected graph of a
digraph . Here, the additional term is tight
up to multiplicative and additive constants. As a corollary, this implies that
every strongly -connected -vertex semicomplete digraph contains a
strongly -connected spanning subgraph with at most edges,
which is essentially optimal since cannot be reduced to the number
less than .
We also prove an analogous result for strongly -arc-connected directed
multigraphs. Both proofs yield polynomial-time algorithms.Comment: 31 page
Minimally k-edge-connected directed graphs of maximal size
Minimally k-edge-connected directed graphs of maximal siz