It has been proved by Boros and Makino that there is no output-polynomial-time algorithm enumerating the minimal redundant sets or the maximal irredundant sets of a hypergraph, unless P=NP. The same question was left open for graphs, with only a few tractable cases known to date. In this paper, we focus on graph classes that capture incidence relations such as bipartite, co-bipartite, and split graphs. Concerning maximal irredundant sets, we show that the problem on co-bipartite graphs is as hard as in general graphs and tractable in split and strongly orderable graphs, the latter being a generalization of chordal bipartite graphs. As for minimal redundant sets enumeration, we first show that the problem is intractable in split and co-bipartite graphs, answering the aforementioned open question, and that it is tractable on $(C_3,C_5,C_6,C_8)$-free graphs, a class of graphs incomparable to strongly orderable graphs, and which also generalizes chordal bipartite graphs.

翻译：Boros 和 Makino 已证明，除非 P=NP，否则不存在输出多项式时间算法来枚举超图的最小冗余集或最大非冗余集。对于图而言，该问题至今悬而未决，仅知少数可处理的特例。本文聚焦于刻画关联关系的图类，如二分图、余二分图和分裂图。关于最大非冗余集，我们证明了在余二分图上的问题与一般图同样困难，而在分裂图和强可序图上可处理，后者是弦二分图的推广。至于最小冗余集的枚举，我们首先证明了该问题在分裂图和余二分图上是难解的，从而回答了前述开放性问题；同时证明了该问题在 $(C_3,C_5,C_6,C_8)$-自由图上是可处理的，此类图与强可序图不可比较，且同样推广了弦二分图。