Abstract
Branchings play an important role in digraph theory and algorithms. In particular, a chapter in the monograph of Bang-Jensen and Gutin, Digraphs: Theory, Algorithms and Application, Ed. 2, 2009 is wholly devoted to branchings. The well-known Edmonds Branching Theorem provides a characterization for the existence of k arc-disjoint out-branchings rooted at the same vertex. A short proof of the theorem by Lovász (1976) leads to a polynomial-time algorithm for finding such out-branchings. A natural related problem is to characterize digraphs having an out-branching and an in-branching which are arc-disjoint. Such a pair of branchings is called a good pair.
Bang-Jensen, Bessy, Havet and Yeo (2020) pointed out that it is NP-complete to decide if a given digraph has a good pair. They also showed that every digraph of independence number at most 2 and arc-connectivity at least 2 has a good pair, which settled a conjecture of Thomassen for digraphs of independence number 2. Then they asked for the smallest number \(n_{ngp}\) of vertices in a 2-arc-strong digraph which has no good pair. They proved that \(7\le n_{ngp}\le 10.\) In this paper, we prove that \(n_{ngp}=10\), which solves the open problem.
Gu was supported by Natural Science Foundation of Jiangsu Province (No. BK20170860), National Natural Science Foundation of China (No. 11701143), and Fundamental Research Funds for the Central Universities. Li was supported by National Natural Science Foundation of China (No. 11301480), Zhejiang Provincial Natural Science Foundation of China (No. LY18A010002), and the Natural Science Foundation of Ningbo, China. Shi and Taoqiu are supported by the National Natural Science Foundation of China (No. 11922112), the Natural Science Foundation of Tianjin (Nos. 20JCJQJC00090 and 20JCZDJC00840) and the Fundamental Research Funds for the Central Universities, Nankai University.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Bang-Jensen, J.: Edge-disjoint in- and out-branchings in tournaments and related path problems. J. Combin. Theory Ser. B 51(1), 1–23 (1991)
Bang-Jensen J., Bessy S., Havet F., Yeo, A.: Arc-disjoint in- and out-branchings in digraphs of independence number at most 2. arXiv:2003.02107 (2020)
Bang-Jensen J., Bessy, S., Yeo, A.: Non-separating spanning trees and out-branchings in digraphs of independence number 2. arXiv:2007.02834v1 (2020)
Bang-Jensen, J., Gutin, G. (eds.): Classes of Directed Graphs. SMM, Springer, Cham (2018). https://doi.org/10.1007/978-3-319-71840-8
Bang-Jensen, J., Gutin, G.: Digraphs: Theory, Algorithms and Applications, 2nd edn. Springer Verlag, London (2009). https://doi.org/10.1007/978-1-84800-998-1
Bang-Jensen, J., Gutin, G., Yeo, A.: Arc-disjoint strong spanning subdigraphs of semicomplete compositions. J. Graph Theory 95(2), 267–289 (2020)
Bang-Jensen, J., Huang, J.: Decomposing locally semicomplete digraphs into strong spanning subdigraphs. J. Combin. Theory Ser. B 102(3), 701–714 (2010)
Bang-Jensen, J., Saurabh, S., Simonsen, S.: Parameterized algorithms for non-separating trees and branchings in digraphs. Algorithmica 76(1), 279–296 (2016). https://doi.org/10.1007/s00453-015-0037-3
Gu R., Gutin G., Li S., Shi Y., Taoqiu Z.: The smallest number of vertices in a 2-arc-strong digraph which has no good pair. arXiv:2012.03742 (2020)
Gutin, G., Reidl, F., Wahlström, M.: \(k\)-distinct in- and out-branchings in digraphs. J. Comput. Syst. Sci. 95, 86–97 (2018)
Lovász, L.: On two min-max theorems in graph theory. J. Comb. Theory Ser. B 21, 96–103 (1976)
Thomassen, C.: Configurations in graphs of large minimum degree, connectivity, or chromatic number. Ann. New York Acad. Sci. 555, 402–412 (1989)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this paper
Cite this paper
Gu, R., Gutin, G., Li, S., Shi, Y., Taoqiu, Z. (2021). The Smallest Number of Vertices in a 2-Arc-Strong Digraph Without Pair of Arc-Disjoint In- and Out-Branchings. In: Du, DZ., Du, D., Wu, C., Xu, D. (eds) Combinatorial Optimization and Applications. COCOA 2021. Lecture Notes in Computer Science(), vol 13135. Springer, Cham. https://doi.org/10.1007/978-3-030-92681-6_26
Download citation
DOI: https://doi.org/10.1007/978-3-030-92681-6_26
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-92680-9
Online ISBN: 978-3-030-92681-6
eBook Packages: Computer ScienceComputer Science (R0)