For a given graph $G$, a depth-first search (DFS) tree $T$ of $G$ is an $r$-rooted spanning tree such that every edge of $G$ is either an edge of $T$ or is between a \textit{descendant} and an \textit{ancestor} in $T$. A graph $G$ together with a DFS tree is called a \textit{lineal topology} $\mathcal{T} = (G, r, T)$. Sam et al. (2023) initiated study of the parameterized complexity of the \textsc{Min-LLT} and \textsc{Max-LLT} problems which ask, given a graph $G$ and an integer $k\geq 0$, whether $G$ has a DFS tree with at most $k$ and at least $k$ leaves, respectively. Particularly, they showed that for the dual parameterization, where the tasks are to find DFS trees with at least $n-k$ and at most $n-k$ leaves, respectively, these problems are fixed-parameter tractable when parameterized by $k$. However, the proofs were based on Courcelle's theorem, thereby making the running times a tower of exponentials. We prove that both problems admit polynomial kernels with $\Oh(k^3)$ vertices. In particular, this implies FPT algorithms running in $k^{\Oh(k)}\cdot n^{O(1)}$ time. We achieve these results by making use of a $\Oh(k)$-sized vertex cover structure associated with each problem. This also allows us to demonstrate polynomial kernels for \textsc{Min-LLT} and \textsc{Max-LLT} for the structural parameterization by the vertex cover number.

翻译：对于给定图 $G$，$G$ 的深度优先搜索（DFS）树 $T$ 是一棵以 $r$ 为根的生成树，使得 $G$ 的每条边要么是 $T$ 的边，要么是 $T$ 中某对 \textit{后代} 与 \textit{祖先} 之间的边。将图 $G$ 与其 DFS 树一同称为 \textit{线性拓扑} $\mathcal{T} = (G, r, T)$。Sam 等人（2023）首次研究了 \textsc{Min-LLT} 和 \textsc{Max-LLT} 问题的参数化复杂度，这两个问题要求给定图 $G$ 和整数 $k\geq 0$，判断 $G$ 是否存在一棵叶子数分别至多为 $k$ 和至少为 $k$ 的 DFS 树。特别地，他们证明了在对偶参数化（即分别寻找叶子数至少为 $n-k$ 和至多为 $n-k$ 的 DFS 树）下，这些问题在以 $k$ 为参数时是固定参数可解的。然而，其证明基于 Courcelle 定理，导致运行时间为指数塔复杂度。我们证明这两个问题均存在包含 $\Oh(k^3)$ 个顶点的多项式核。特别地，这蕴涵了运行时间为 $k^{\Oh(k)}\cdot n^{O(1)}$ 的 FPT 算法。我们通过利用每个问题关联的一个大小为 $\Oh(k)$ 的顶点覆盖结构实现了这些结果，这也使我们能够展示 \textsc{Min-LLT} 和 \textsc{Max-LLT} 关于顶点覆盖数这一结构参数化的多项式核。