Generating minimal redundant and maximal irredundant subhypergraphs

Given a hypergraph H⊆2V on a finite base set V, a vertex v∈V is called the private vertex of a hyperedge H∈H if H is the only hyperedge of H containing it. A hypergraph is called irredundant if every edge of it has a private vertex, and it is called redundant otherwise. Motivated by some graph domination problems, Uno (2015) posed the problems of generating all minimal redundant and maximal irredundant subhypergraphs of a given hypergraph. Here we prove that these are NP-hard generation problems, and present positive results for certain special cases.