Let \(\alpha \ge 0\) and \(k \ge 2\) be integers. For a graph G, the total k-excess of G is defined as \(\text{ te }(G;k)=\sum _{v \in V(G)}\max \{d_G(v)-k,0\}\). In this paper, we propose a new closure concept for a spanning tree with bounded total k-excess. We prove that: Let G be a connected graph, and let u and v be two non-adjacent vertices of G. If G satisfies one of the following conditions, then G has a spanning tree T such that \(\text{ te }(T;k) \le \alpha\) if and only if \(G+uv\) has a spanning tree \(T'\) such that \(\text{ te }(T';k) \le \alpha\):

1. (i)

   \(\max \{ \sum _{x \in X} d_G(x): X \text{ is } \text{ a } \text{ subset } \text{ of } S \text{ with } |X|=k \} \ge |G|-1 \text{ for } \text{ every } \text{ independent } \text{ set } S \text{ in } G \text{ of } \text{ order } k+1 \text{ such } \text{ that } \{ u,v \} \subseteq S\); or

2. (ii)

   \(\max \{ \sum _{x \in X} d_G(x): X \text{ is } \text{ a } \text{ subset } \text{ of } S \text{ with } |X|=k \} \ge |G|-\alpha -1 \text{ for } \text{ every } \text{ independent } \text{ set } S \text{ in } G \text{ of } \text{ order } k+\alpha +1 \text{ such } \text{ that } S \cap \{ u,v \} \ne \emptyset .\)

We also show examples to show that these conditions are sharp.

令\（\ alpha \ ge 0 \）和\（k \ ge 2 \）为整数。对于一个图形G ^，总ķ的建筑不动产ģ被定义为\（\ {文本TE}（G; {在V（G）V \} K）= \总和_ \最大\ {D_G（V）-k ，0 \} \）。在本文中，我们为有界总k-过量的生成树提出了一种新的闭合概念。我们证明：令G为连通图，令u和v为G的两个不相邻的顶点。如果G满足以下条件之一，则G具有生成树T，使得\（\ text {te}（T; k）\ le \ alpha \）当且仅当\（G + uv \）具有生成树\（T'\）使得\（\ text {te}（T '; k）\ le \ alpha \）：

1. （一世）

   \（\ max \ {\ sum _ {x \ in X} d_G（x）：X \ text {是} S \ text {与} | X |的\ text {a} \ text {子集} \ text { = k \} \ ge | G | -1 \ text {for} \ text {每个} \ text {独立} \ text {set} S \ text {in} G \ text {of} \ text {顺序} k + 1 \ text {这样} \ text {} \ {u，v \} \ subseteq S \）；或者

2. （ii）

   \（\ max \ {\ sum _ {x \ in X} d_G（x）：X \ text {是} S \ text {与} | X |的\ text {a} \ text {子集} \ text { = k \} \ ge | G |-\ alpha -1 \ text {for} \ text {每个} \ text {独立} \ text {set} S \ text {in} G \ text {of} \ text {顺序} k + \ alpha +1 \ text {这样} \ text {表示} S \ cap \ {u，v \} \ ne \ emptyset。\）

我们还显示了一些示例，以表明这些条件很明显。