In this paper, we exhibit connections between the following subjects:

• Tree packing in graphs and digraphs (both behave completely different),
• the rigidity matroid of a graph,
• Henneberg moves on trees,
• the conjectures of Thomassen and Matthews and Sumner, and
• (s,t)‐orderings of digraphs.

We do this by studying graphs which admit acyclic orientations that contain an out‐branching and in‐branching which are arc‐disjoint (such an orientation is called good). A 2T‐graph is a graph whose edge set can be decomposed into two edge‐disjoint spanning trees. It is a well‐known result due to Tutte and Nash‐Williams, respectively, that every 4‐edge‐connected graph contains a spanning 2T‐graph. Vertex‐minimal 2T‐graphs with at least two vertices which are known as generic circuits play an important role in rigidity theory for graphs. We prove that every generic circuit has a good orientation. Using this result we prove that if $G$ is 2T‐graph whose vertex set has a partition $V_1,V_2,\text{…},V_k$ so that each $V_i$ induces a generic circuit $G_i$ of $G$ and the set of edges between different $G_i$'s form a matching in $G$, then $G$ has a good orientation. We also obtain a characterization for the case when the set of edges between different $G_i$'s form a double tree, that is, if we contract each $G_i$ to one vertex, and delete parallel edges we obtain a tree. All our proofs are constructive and imply polynomial algorithms for finding the desired good orderings and the pairs of arc‐disjoint branchings which certify that the orderings are good. We identify a structure which can be used to certify that a given 2T‐graph does not have a good orientation.

中文翻译：

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边缘不相交的生成树并集的良好方向

在本文中，我们展示了以下主题之间的联系：

• 图形和有向图中的树包装（两者的行为完全不同），
• 图的刚性拟阵
• 亨内伯格在树上移动，
• 托马森，马修斯和萨姆纳的猜想，以及
• （s，t）有向图的顺序。

为此，我们研究了允许非循环取向的图，这些非循环取向包含向外分支和向内分支的弧线不相交（这种方向称为good）。甲2T-图形是其边缘集可被分解成两个边缘不相交的生成树的曲线图。分别由于Tutte和Nash-Williams，这是一个众所周知的结果，每个4边连接的图都包含一个跨度2T图。至少具有两个顶点的顶点最小2T图（称为通用电路）在图的刚度理论中起着重要作用。我们证明每个通用电路都具有良好的方向性。使用此结果，我们证明$G$ 是2T图，其顶点集具有分区 $V_{\mathrm{1个}}，V_2，\text{…}，V_{ķ}$ 这样每个 $V_{\mathrm{一世}}$ 诱导通用电路 $G_{\mathrm{一世}}$ 的 $G$ 和不同边之间的边集 $G_{\mathrm{一世}}$的形式匹配 $G$， 然后 $G$具有良好的定位。我们还获得了当一组边之间的不同时的特征$G_{\mathrm{一世}}$形成一棵双树，也就是说，如果我们将每个$G_{\mathrm{一世}}$到一个顶点，并删除平行边，我们得到一棵树。我们所有的证明都是构造性的，并且是多项式算法，用于找到所需的良好排序和成对的弧不相交分支，以证明这些排序是良好的。我们确定了可以用来证明给定的2T图没有良好方向的结构。