
Minimum 2vertex strongly biconnected spanning directed subgraph problem
A directed graph G=(V,E) is strongly biconnected if G is strongly connec...
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Counting orientations of graphs with no strongly connected tournaments
Let S_k(n) be the maximum number of orientations of an nvertex graph G ...
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Twinless articulation points and some related problems
Let G=(V,E) be a twinless strongly connected graph. a vertex v∈ V is a t...
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Parameterized Complexity of MinPower Asymmetric Connectivity
We investigate parameterized algorithms for the NPhard problem MinPowe...
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An Optimal Rounding for HalfIntegral Weighted Minimum Strongly Connected Spanning Subgraph
In the weighted minimum strongly connected spanning subgraph (WMSCSS) pr...
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Random kout subgraph leaves only O(n/k) intercomponent edges
Each vertex of an arbitrary simple graph on n vertices chooses k random ...
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Strongly Connected Components in Stream Graphs: Computation and Experimentations
Stream graphs model highly dynamic networks in which nodes and/or links ...
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Sparse highly connected spanning subgraphs in dense directed graphs
Mader (1985) proved that every strongly kconnected nvertex digraph contains a strongly kconnected spanning subgraph with at most 2kn  2k^2 edges, and this is sharp. For dense strongly kconnected digraphs, the bound can be significantly improved for dense strongly kconnected digraphs. Let Δ(D) be the maximum degree of the complement of the underlying undirected graph of a digraph D. We prove that every strongly kconnected nvertex digraph D contains a strongly kconnected spanning subgraph with at most kn + 800k(k+Δ(D)) edges. The additional term 800k(k+Δ(D)) is sharp up to a multiplicative constant. In particular, it follows that every strongly kconnected nvertex semicomplete digraph contains a strongly kconnected spanning subgraph with at most kn + 800k^2 edges, improving the recent result of Kang, Kim, Kim, and Suh (2017) for tournaments and establishing the tight bound, as 800k^2 cannot be improved to the number less than k(k1)/2. We also prove an analogous result for strongly karcconnected directed multigraphs, which generalises the earlier result of BangJensen, Huang, and Yeo (2004) for strongly connected digraphs. The proofs yield polynomialtime algorithms.
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