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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: dyk90@kaist.ac.kr)
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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 

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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.

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