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.
Keywords
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.
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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
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DOI: https://doi.org/10.1007/978-3-030-92681-6_26
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