On the parameterized complexity of lineal topologies (depth-first spanning trees) with many or few leaves

This paper considers four problems with possible applications in network design: Given a graph G with $\left|G\right|=n$ and an integer $k\ge 0$, does G have a DFS tree with (i) ≤k leaves, (ii) ≥k leaves, (iii) $\le n-k$ leaves, and (iv) $\ge n-k$ leaves? We show that all four problems are NP-hard. When parameterized by k, we prove that while (i) is para-NP-hard and (ii) is W[1]-hard, both (iii) and (iv) admit polynomial kernels with $O\left(k^3\right)$ vertices, implying FPT algorithms running in $k^{O\left(k\right)}\cdot n^{O\left(1\right)}$ time. Our polynomial kernels are based on a $O\left(k\right)$-sized vertex cover structure associated with the solution of these problems. As a byproduct, we obtain polynomial kernels for these problems parameterized by the vertex cover number of the input graph.