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，要么连接$T$中的\textit{后代}与\textit{祖先}。图$G$与其DFS树共同构成一个\textit{线性拓扑} $\mathcal{T} = (G, r, T)$。Sam等人（2023）开创了关于\textsc{Min-LLT}和\textsc{Max-LLT}问题的参数复杂性研究，这两个问题要求给定图$G$和整数$k\geq 0$，分别判断$G$是否具有叶子数不超过$k$和不少于$k$的DFS树。特别地，他们证明了对偶参数化（即任务分别要求找到叶子数至少为$n-k$和至多为$n-k$的DFS树）时，这些问题在以$k$为参数下是固定参数可处理的。然而，其证明基于Courcelle定理，导致运行时间为指数塔形式。我们证明这两个问题均存在具有$\Oh(k^3)$个顶点的多项式核。这尤其意味着运行时间为$k^{\Oh(k)}\cdot n^{O(1)}$的FPT算法。我们通过利用与每个问题相关的$\Oh(k)$规模的顶点覆盖结构来实现这些结果，这还使我们能够展示\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。