Generating minimal redundant and maximal irredundant sets in incidence graphs

2026-02-20

AI Summary

This study investigates the enumeration complexity of minimal redundant sets and maximal irredundant sets in graph structures, with a focus on associated graph classes such as bipartite graphs, co-bipartite graphs, and split graphs. Through structural graph analysis and the design of enumeration algorithms, the work establishes for the first time that enumerating minimal redundant sets is computationally intractable on split graphs and co-bipartite graphs. Conversely, it proves that the problem is solvable in polynomial time on (C₃, C₅, C₆, C₈)-free graphs and strongly orderable graphs. These results delineate precise tractability boundaries across multiple graph classes, resolve a long-standing open question, and significantly extend the landscape of graph classes admitting efficient solutions.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

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.

Problem

• minimal redundant sets
• maximal irredundant sets
• incidence graphs
• output-polynomial enumeration
• computational complexity

Innovation

• minimal redundant sets
• maximal irredundant sets
• incidence graphs
• tractability
• graph classes