Given a graph $G$ and sets $\{\alpha_v~|~v \in V(G)\}$ and $\{\beta_v~|~v \in V(G)\}$ of non-negative integers, it is known that the decision problem whether $G$ contains a spanning tree $T$ such that $\alpha_v \le d_T (v) \le \beta_v $ for all $v \in V(G)$ is $NP$-complete. In this article, we relax the problem by demanding that the degree restrictions apply to vertices $v\in U$ only, where $U$ is a stable set of $G$. In this case, the problem becomes tractable. A. Frank presented a result characterizing the positive instances of that relaxed problem. Using matroid intersection developed by J. Edmonds, we give a new and short proof of Frank's result and show that if $U$ is stable and the edges of $G$ are weighted by arbitrary real numbers, then even a minimum-cost tree $T$ with $\alpha_v \le d_T (v) \le \beta_v $ for all $v \in U$ can be found in polynomial time if such a tree exists.

翻译：给定图$G$以及非负整数集合$\{\alpha_v~|~v \in V(G)\}$和$\{\beta_v~|~v \in V(G)\}$，已知判定$G$是否包含生成树$T$使得对所有$v \in V(G)$满足$\alpha_v \le d_T (v) \le \beta_v $的问题是$NP$完全问题。本文通过要求度约束仅适用于顶点$v\in U$来松弛该问题，其中$U$是$G$的稳定集。在此条件下，该问题变得可处理。A. Frank提出了刻画该松弛问题正实例的结果。利用J. Edmonds发展的拟阵交理论，我们给出了Frank结果的新颖简洁证明，并证明若$U$是稳定集且$G$的边被任意实数加权，则即使需要寻找满足所有$v \in U$具有$\alpha_v \le d_T (v) \le \beta_v $约束的最小代价树$T$，只要这样的树存在，就可在多项式时间内找到。