Generating minimal redundant and maximal irredundant sets in incidence graphs

TL;DR

This paper investigates the computational complexity of enumerating minimal redundant and maximal irredundant sets in various graph classes, revealing intractability in some cases and tractability in others, thus advancing understanding of these problems.

Contribution

It characterizes the complexity of enumerating these sets in specific graph classes, solving open questions and identifying new tractable classes.

Findings

Maximal irredundant sets enumeration is tractable in split and strongly orderable graphs.

Minimal redundant sets enumeration is intractable in split and co-bipartite graphs.

Tractable enumeration of minimal redundant sets is possible in $(C_3,C_5,C_6,C_8)$-free graphs.

Abstract

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.