Maximal Irredundant Set Enumeration in Bounded-Degeneracy and Bounded-Degree Hypergraphs

Abstract

An irredundant set of a hypergraph \(G = (V,\mathcal {H})\) 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 \(O(2^d \varDelta ^{d+1} n^2)\), where d is the degeneracy of the hypergraph and \(\varDelta \) the maximum node degree. As \(d\le \varDelta \), we immediately obtain an algorithm with delay \(O(2^\varDelta \varDelta ^{\varDelta +1} n^2)\) 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.