Let $G$ be a graph and $T_1,T_2$ be two spanning trees of $G$. We say that $T_1$ can be transformed into $T_2$ via an edge flip if there exist two edges $e \in T_1$ and $f$ in $T_2$ such that $T_2= (T_1 \setminus e) \cup f$. Since spanning trees form a matroid, one can indeed transform a spanning tree into any other via a sequence of edge flips, as observed by Ito et al. We investigate the problem of determining, given two spanning trees $T_1,T_2$ with an additional property $\Pi$, if there exists an edge flip transformation from $T_1$ to $T_2$ keeping property $\Pi$ all along. First we show that determining if there exists a transformation from $T_1$ to $T_2$ such that all the trees of the sequence have at most $k$ (for any fixed $k \ge 3$) leaves is PSPACE-complete. We then prove that determining if there exists a transformation from $T_1$ to $T_2$ such that all the trees of the sequence have at least $k$ leaves (where $k$ is part of the input) is PSPACE-complete even restricted to split, bipartite or planar graphs. We complete this result by showing that the problem becomes polynomial for cographs, interval graphs and when $k=n-2$.

中文翻译：

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多叶或少叶生成树的重构

令 $G$ 是一个图，$T_1,T_2$ 是 $G$ 的两个生成树。如果在 $T_2$ 中存在两条边 $e \in T_1$ 和 $f$ 使得 $T_2= (T_1 \setminus e) \cup f ，我们说 $T_1$ 可以通过边翻转转换为 $T_2$ $. 由于生成树形成一个拟阵，正如 Ito 等人所观察到的那样，确实可以通过一系列边缘翻转将生成树转换为任何其他生成树。我们研究了确定的问题，给定两个具有附加属性 $\Pi$ 的生成树 $T_1,T_2$，是否存在从 $T_1$ 到 $T_2$ 的边缘翻转变换始终保持属性 $\Pi$。首先，我们证明确定是否存在从 $T_1$ 到 $T_2$ 的转换，使得序列的所有树最多具有 $k$（对于任何固定的 $k \ge 3$）叶子是 PSPACE 完全的。然后我们证明，确定是否存在从 $T_1$ 到 $T_2$ 的转换，使得序列的所有树至少有 $k$ 的叶子（其中 $k$ 是输入的一部分）是 PSPACE 完全的，甚至是受限的分裂，二分或平面图。我们通过展示问题在 cographs、区间图和 $k=n-2$ 时变成多项式来完成这个结果。