Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-25T02:54:20.408Z Has data issue: false hasContentIssue false
  • English
  • Français

Sparse Highly Connected Spanning Subgraphs in Dense Directed Graphs

Part of: Graph theory

Published online by Cambridge University Press:  05 November 2018

DONG YEAP KANG
DONG YEAP KANG*
Affiliation:
Department of Mathematical Sciences, KAIST, 291 Daehak-ro Yuseong-gu Daejeon, 34141South Korea (e-mail: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

Mader proved that every strongly k-connected n-vertex digraph contains a strongly k-connected spanning subgraph with at most 2kn - 2k2 edges, where equality holds for the complete bipartite digraph DKk,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 + 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 800k(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 + 800k2 edges, which is essentially optimal since 800k2 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 classification

Primary: 05C20: Directed graphs (digraphs), tournaments
Secondary: 05C40: Connectivity
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 28 , Issue 3 , May 2019 , pp. 423 - 464
Copyright
Copyright © Cambridge University Press 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

†

Supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (no. NRF-2017R1A2B4005020) and also by a TJ Park Science Fellowship of POSCO TJ Park Foundation.

References

[1] Bang-Jensen, J. (2009) Problems and conjectures concerning connectivity, paths, trees and cycles in tournament-like digraphs. Discrete Math. 309 5655–5667.Google Scholar
[2] Bang-Jensen, J. and Gutin, G. (2009) Digraphs: Theory, Algorithms and Applications, second edition, Springer Monographs in Mathematics, Springer.Google Scholar
[3] Bang-Jensen, J. and Havet, F. (2018) Tournaments and Semicomplete Digraphs, Springer, pp. 35–124.Google Scholar
[4] Bang-Jensen, J., Huang, J. and Yeo, A. (2003) Strongly connected spanning subdigraphs with the minimum number of arcs in quasi-transitive digraphs. SIAM J. Discrete Math. 16 335–343.Google Scholar
[5] Bang-Jensen, J., Huang, J. and Yeo, A. (2004) Spanning k-arc-strong subdigraphs with few arcs in k-arc-strong tournaments. J. Graph Theory 46 265–284.Google Scholar
[6] Bang-Jensen, J. and Yeo, A. (2001) The minimum spanning strong subdigraph problem for extended semicomplete digraphs and semicomplete bipartite digraphs. J. Algorithms 41 1–19.Google Scholar
[7] Berg, A. R. and Jordán, T. (2005) Minimally k-edge-connected directed graphs of maximal size. Graphs Combin. 21 39–50.Google Scholar
[8] Cheriyan, J. and Thurimella, R. (2000) Approximating minimum-size k-connected spanning subgraphs via matching. SIAM J. Comput. 30 528–560.Google Scholar
[9] Dalmazzo, M. (1977) Nombre d'arcs dans les graphes k-arc-fortement connexes minimaux. CR Acad. Sci. Paris Sér. A–B 285 A341–A344.Google Scholar
[10] Edmonds, J. (1973) Edge-disjoint branchings. In Combinatorial Algorithms (Rustin, B., ed.), Academic Press, pp. 91–96.Google Scholar
[11] Garey, M. R. and Johnson, D. S. (1979) Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman.Google Scholar
[12] Hopcroft, J. E. and Karp, R. M. (1973) An n 5/2 algorithm for maximum matchings in bipartite graphs. SIAM J. Comput. 2 225–231.Google Scholar
[13] Kang, D. Y., Kim, J., Kim, Y. and Suh, G. (2017) Sparse spanning k-connected subgraphs in tournaments. SIAM J. Discrete Math. 31 2206–2227.Google Scholar
[14] Kim, J., Kühn, D. and Osthus, D. (2016) Bipartitions of highly connected tournaments. SIAM J. Discrete Math. 30 895–911.Google Scholar
[15] Kühn, D., Lapinskas, J., Osthus, D. and Patel, V. (2014) Proof of a conjecture of Thomassen on Hamilton cycles in highly connected tournaments. Proc. Lond. Math. Soc. (3) 109 733–762.Google Scholar
[16] Kühn, D., Osthus, D. and Townsend, T. (2016) Proof of a tournament partition conjecture and an application to 1-factors with prescribed cycle lengths. Combinatorica 36 451–469.Google Scholar
[17] Mader, W. (1985) Minimal n-fach zusammenhängende Digraphen. J. Combin. Theory Ser. B 38 102–117.Google Scholar
[18] Mader, W. (1996) On vertices of degree n in minimally n-connected graphs and digraphs. In Combinatorics: Paul Erdős is Eighty, (Miklós, D. et al., eds), Vol. 2, János Bolyai Mathematical Society, pp. 423–449.Google Scholar
[19] Pokrovskiy, A. (2015) Highly linked tournaments. J. Combin. Theory Ser. B 115 339–347.Google Scholar
[20] Pokrovskiy, A. (2017) Edge disjoint Hamiltonian cycles in highly connected tournaments. Int. Math. Res. Not. 2017 429–467.Google Scholar
[21] Thomassen, C. (1982) Edge-disjoint Hamiltonian paths and cycles in tournaments. Proc. London Math. Soc. (3) 45 151–168.Google Scholar
[22] Vetta, A. (2001) Approximating the minimum strongly connected subgraph via a matching lower bound. In Proceedings of the Twelfth Annual ACM–SIAM Symposium on Discrete Algorithms, SIAM, pp. 417–426.Google Scholar