关于核与叶：寻找稀疏与繁茂的树
On Kernels and Leaves: Searching for Bare and Lush Trees
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中文总结 AI 辅助
本研究探讨最大（最小）叶生成树问题在后进树上的变体，证明DFS的困难性结果可转移至LDFS，给出指数核并证明多项式核不存在，且多种搜索的最小化问题是para-NP难的。
中文摘要 AI 辅助
我们研究经典的最大（最小）叶生成树问题的一个变体。在许多应用中，深度优先搜索（DFS）被用来计算图的生成树。这种搜索树通过将每个顶点 $v$ 与搜索在访问 $v$ 之前最后访问的顶点相连来构造，我们称之为后进树。通过将最大（最小）叶生成树问题限制在图搜索的后进树上，我们要求一个搜索排序，使得其搜索树中的叶子数量最大（最小）。最近，Bergougnoux 等人 [Journal of Computer and System Sciences 154 (2025)] 研究了这些问题在 DFS 下的参数化复杂性。他们证明了当以叶子数量为参数时，最小化问题是 para-NP 难的，最大化问题是 W[1] 难的。当以内部顶点数量为参数时，两个问题都有多项式核。这里，我们考察这些结果是否也适用于字典序深度优先搜索（LDFS）变体。我们证明了 DFS 的困难性结果可以转移到 LDFS。我们还提出了以内部顶点数量为参数的指数核。我们通过证明除非 NP ⊆ coNP / poly，否则多项式核不存在来补充这一点。我们还考虑了不遵循 DFS 方案的搜索的后进树。与（L）DFS 相反，对于包括广度优先搜索在内的几种搜索，最小化内部顶点数量是 para-NP 难的。
英文摘要
We study a variation of the classical Maximum (Minimum) Leaf Spanning Tree problem. In many applications, Depth-First Search (DFS) is used to compute a spanning tree of a graph. Such a search tree is constructed by connecting each vertex $v$ with the last vertex the search has visited before $v$ and we call this a last-in tree. By restricting the Maximum (Minimum) Leaf Spanning Tree problem to last-in trees of a graph search, we ask for a search ordering that leads to the largest (smallest) number of leaves in its search tree. Recently, Bergougnoux et al. [Journal of Computer and System Sciences 154 (2025)] have studied the parameterized complexity of these problems for DFS. They showed that the minimization problem is para-$\mathsf{NP}$-hard and the maximization problem is $\mathsf{W}[1]$-hard when parameterized by the number of leaves. When parameterized by the number of internal vertices, both problems have polynomial kernels. Here, we examine whether these results also hold for the variant Lexicographic DFS (LDFS). We show that the hardness results of DFS can be transferred to LDFS. We also present exponential kernels for the number of internal vertices as the parameter. We complement this by showing that polynomial kernels do not exist, unless $\mathsf{NP} \subseteq \mathsf{coNP} / \mathsf{poly}$. We also consider last-in trees of searches that do not follow the DFS scheme. In contrast to (L)DFS, minimizing the number of internal vertices is para-$\mathsf{NP}$-hard for several searches including Breadth-First Search.
发表机构
- Brandenburg University of Technology（勃兰登堡工业大学）
- Westsächsische Hochschule Zwickau（茨维考西萨克森应用技术大学）
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