A 2k2k-Vertex Kernel for Maximum Internal Spanning Tree

We consider the parameterized version of the maximum internal spanning tree problem, which, given an $n$-vertex graph and a parameter $k$, asks for a spanning tree with at least $k$ internal vertices. Fomin et al. [J. Comput. System Sci., 79:1-6] crafted a very ingenious reduction rule, and showed that a simple application of this rule is sufficient to yield a $3k$-vertex kernel. Here we propose a novel way to use the same reduction rule, resulting in an improved $2k$-vertex kernel. Our algorithm applies first a greedy procedure consisting of a sequence of local exchange operations, which ends with a local-optimal spanning tree, and then uses this special tree to find a reducible structure. As a corollary of our kernel, we obtain a deterministic algorithm for the problem running in time $4^k \cdot n^{O(1)}$

Maximal irredundant set enumeration in bounded-degeneracy and bounded-degree hypergraphs

An irredundant set of a hypergraph G = (V,H) is a subset S of its nodes such that removing any node from S decreases the number of hyperedges it intersects. The concept is deeply related to that of dominating sets, as the minimal dominating sets of a graph correspond exactly to the dominating sets which are also maximal irredundant sets. In this paper we propose an FPT-delay algorithm for listing maximal irredundant sets, whose delay is O(2dΔd+1n2), where d is the degeneracy of the hypergraph and Δ the maximum node degree. As d ≤ Δ,, we immediately obtain an algorithm with delay O(2ΔΔΔ+1n2) that is FPT for bounded-degree hypergraphs. This result opens a gap between known bounds for this problem and those for listing minimal dominating sets in these classes of hypergraphs, as the known running times used to be the same, hinting at the idea that the latter may indeed be harder