A digraph $D=(V, A)$ has a good pair at a vertex $r$ if $D$ has a pair of arc-disjoint in- and out-branchings rooted at $r$. Let $T$ be a digraph with $t$ vertices $u_1,\dots , u_t$ and let $H_1,\dots H_t$ be digraphs such that $H_i$ has vertices $u_{i,j_i},\ 1\le j_i\le n_i.$ Then the composition $Q=T[H_1,\dots , H_t]$ is a digraph with vertex set $\{u_{i,j_i}\mid 1\le i\le t, 1\le j_i\le n_i\}$ and arc set $$A(Q)=\cup^t_{i=1}A(H_i)\cup \{u_{ij_i}u_{pq_p}\mid u_iu_p\in A(T), 1\le j_i\le n_i, 1\le q_p\le n_p\}.$$ When $T$ is arbitrary, we obtain the following result: every strong digraph composition $Q$ in which $n_i\ge 2$ for every $1\leq i\leq t$, has a good pair at every vertex of $Q.$ The condition of $n_i\ge 2$ in this result cannot be relaxed. When $T$ is semicomplete, we characterize semicomplete compositions with a good pair, which generalizes the corresponding characterization by Bang-Jensen and Huang (J. Graph Theory, 1995) for quasi-transitive digraphs. As a result, we can decide in polynomial time whether a given semicomplete composition has a good pair rooted at a given vertex.

中文翻译：

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在有向图的组合中根植于同一顶点的弧不相交的内支和外支

一个有向图 $D=(V, A)$ 在顶点 $r$ 处有一个好的对，如果 $D$ 有一对以 $r$ 为根的弧不相交的内支和外支。令 $T$ 是具有 $t$ 顶点 $u_1,\dots , u_t$ 的有向图，并令 $H_1,\dots H_t$ 是有向图，使得 $H_i$ 具有顶点 $u_{i,j_i},\ 1\le j_i\le n_i.$ 那么组合 $Q=T[H_1,\dots , H_t]$ 是一个顶点集为 $\{u_{i,j_i}\mid 1\le i\le t, 1\le 的有向图j_i\le n_i\}$ 和弧集 $$A(Q)=\cup^t_{i=1}A(H_i)\cup \{u_{ij_i}u_{pq_p}\mid u_iu_p\in A(T ), 1\le j_i\le n_i, 1\le q_p\le n_p\}.$$ 当 $T$ 是任意的时，我们得到以下结果：每个强有向图组合 $Q$ 其中 $n_i\ge 2$对于每个$1\leq i\leq t$，在$Q 的每个顶点都有一个好的对。$n_i\ge 2$ 在这个结果中的条件不能放宽。当 $T$ 是半完全时，我们用一个好的对来表征半完全组合，它概括了 Bang-Jensen 和 Huang (J. Graph Theory, 1995) 对准传递有向图的相应表征。因此，我们可以在多项式时间内决定给定的半完全组合是否有一个以给定顶点为根的好对。