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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 (2022) 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 n g p of vertices in a 2-arc-strong digraph which has no good pair. They proved that 7 ≤ n n g p ≤ 10. In this paper, we prove that n n g p = 10 , which solves the open problem. • The study of good pair has relations with Tutte's Packing Theorem and Edmonds Branching Theorem. • We answer a problem by Bang-Jensen, Bessy, Havet and Yeo about the minimum order of 2-arc-strong digraphs without good pairs. • We study the relationship between Hamilton dipath and good pair. • We generalize some results by Bang-Jensen, Bessy, Havet and Yeo on good pair. [ABSTRACT FROM AUTHOR]