For an integer $k$, a $k$-tree is a tree with maximum degree at most $k$. More generally, if $f$ is an integer-valued function on vertices, an $f$-tree is a tree in which each vertex $v$ has degree at most $f(v)$. Let $c(G)$ denote the number of components of a graph $G$. We show that if $G$ is a connected $K_4$-minor-free graph and $$ c(G-S) \;\le\; \sum_{v \in S} (f(v)-1) \quad\hbox{for all $S \subseteq V(G)$ with $S \ne \emptyset$} $$ then $G$ has a spanning $f$-tree. Consequently, if $G$ is a $\frac{1}{k-1}$-tough $K_4$-minor-free graph, then $G$ has a spanning $k$-tree. These results are stronger than results for general graphs due to Win (for $k$-trees) and Ellingham, Nam and Voss (for $f$-trees). The $K_4$-minor-free graphs form a subclass of planar graphs, and are identical to graphs of treewidth at most $2$, and also to graphs whose blocks are series-parallel. We provide examples to show that the inequality above cannot be relaxed by adding $1$ to the right-hand side, and also to show that our result does not hold for general planar graphs. Our proof uses a technique where we incorporate toughness-related information into weights associated with vertices and cutsets.

中文翻译：

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K4-minor-free图中的韧性和生成树

对于整数 $k$，$k$-tree 是最大度数最多为 $k$ 的树。更一般地，如果 $f$ 是顶点上的整数值函数，则 $f$-tree 是其中每个顶点 $v$ 的度数最多为 $f(v)$ 的树。让 $c(G)$ 表示图 $G$ 的分量数。我们证明如果 $G$ 是一个连通的 $K_4$-minor-free 图和 $$ c(GS) \;\le\; \sum_{v \in S} (f(v)-1) \quad\hbox{for all $S \subseteq V(G)$ with $S \ne \emptyset$} $$ 那么 $G$ 有一个跨越$f$-树。因此，如果 $G$ 是 $\frac{1}{k-1}$-tough $K_4$-minor-free 图，则 $G$ 有一个生成的 $k$-tree。由于 Win（对于 $k$-trees）和 Ellingham、Nam 和 Voss（对于 $f$-trees），这些结果强于一般图的结果。$K_4$-minor-free 图形成平面图的一个子类，与树宽至多 $2$ 的图相同，以及块是串并联的图。我们提供了一些例子来证明上面的不等式不能通过在右侧添加 $1$ 来放松，并且还表明我们的结果不适用于一般平面图。我们的证明使用了一种技术，我们将韧性相关信息合并到与顶点和割集相关的权重中。