Given a connected undirected graph $G$, a spanning tree is a subgraph $T$ of $G$ such that $V(T) = V(G)$ and $T$ is a tree. A collection of $\ell$ spanning trees $T_1,\ldots,T_\ell$ is pairwise $k$-diverse if for every $i \neq j$, $|E(T_i) \triangle E(T_j)| \geq k$. Given a connected undirected graph $G$ and integers $p, q, k, \ell$, Leaf & Internal-Constrained Diverse Spanning Trees asks whether there are $\ell$ distinct spanning trees $T_1,\ldots,T_{\ell}$ of $G$ that are pairwise $k$-diverse such that each tree has at least $p$ leaves and at least $q$ internal vertices. Similarly, Leaf & Non-terminal-Constrained Diverse Spanning Trees takes a connected undirected graph $G$, $V_{NT}\subseteq V(G)$, and three integers $p, k, \ell$, and asks if $G$ has $\ell$ spanning trees that are pairwise $k$-diverse, and each has at least $p$ leaves and conains the vertices of $V_{NT}$ as internal. We consider these two problems from the kernelization perspective and provide polynomial kernels for Leaf & Internal-Constrained Diverse Spanning Trees and Leaf & Non-terminal-Constrained Diverse Spanning Trees, when parameterized by $p + q + k + \ell$ and $p + |V_{\rm NT}| + k + \ell$, respectively.

翻译：给定一个连通无向图 $G$，生成树是 $G$ 的一个子图 $T$，满足 $V(T) = V(G)$ 且 $T$ 是一棵树。一组 $\ell$ 棵生成树 $T_1,\ldots,T_\ell$ 被称为两两 $k$-多样性，如果对于任意 $i \neq j$，有 $|E(T_i) \triangle E(T_j)| \geq k$。给定一个连通无向图 $G$ 及整数 $p, q, k, \ell$，叶子与内部约束多样性生成树问题询问是否存在 $G$ 的 $\ell$ 棵不同的生成树 $T_1,\ldots,T_{\ell}$，它们两两 $k$-多样性，且每棵树至少包含 $p$ 个叶子和至少 $q$ 个内部顶点。类似地，叶子与非终端约束多样性生成树问题输入一个连通无向图 $G$、$V_{NT}\subseteq V(G)$ 及三个整数 $p, k, \ell$，询问 $G$ 是否具有 $\ell$ 棵两两 $k$-多样性的生成树，且每棵树至少包含 $p$ 个叶子并将 $V_{NT}$ 中的顶点作为内部顶点。我们从核化角度考虑这两个问题，并针对叶子与内部约束多样性生成树问题（参数化为 $p + q + k + \ell$）和叶子与非终端约束多样性生成树问题（参数化为 $p + |V_{\rm NT}| + k + \ell$）分别提供了多项式核。