Kernelization for Finding Lineal Topologies (Depth-First Spanning Trees) with Many or Few Leaves

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 $\mathit{\text{descendant}}$ and an $\mathit{\text{ancestor}}$ in $T$. A graph $G$ together with a DFS tree is called a $\mathit{\text{lineal}}\mathit{\text{topology}}$ $\mathcal{T}=(G,r,T)$. Sam et al. (2023) initiated study of the parameterized complexity of the $\text{textsc}Min-LLT$ and $\text{textsc}Max-LLT$ problems which ask, given a graph $G$ and an integer $k\ge 0$, 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 $\text{Oh}(k^3)$ vertices. In particular, this implies FPT algorithms running in $k^{\text{Oh}(k)}\cdot n^{O(1)}$ time. We achieve these results by making use of a $\text{Oh}(k)$-sized vertex cover structure associated with each problem. This also allows us to demonstrate polynomial kernels for $\text{textsc}Min-LLT$ and $\text{textsc}Max-LLT$ for the structural parameterization by the vertex cover number.