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} 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 kβ₯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 nβk and
at most nβk 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.Comment: 16 pages, accepted for presentation at FCT 202