Enumerating minimal dominating sets and variants in chordal bipartite graphs

2025-02-20

AI Summary

This paper resolves the long-standing open problem of efficiently enumerating minimal dominating sets—and their variants (total dominating sets and connected dominating sets)—in chordal bipartite graphs. Methodologically, it leverages the double-convex vertex ordering and C₄-free structure of chordal bipartite graphs to design a recursive enumeration framework integrating local domination-based pruning with a structured search tree. The contributions are threefold: (i) the first polynomial-delay (O(n + m)) and linear-space algorithm for enumerating all minimal dominating sets; (ii) a concise new proof of polynomial-delay enumerability for total dominating sets; and (iii) the first incremental polynomial-delay algorithm for connected dominating sets. Crucially, the paper also establishes that such polynomial-delay enumeration is impossible for general bipartite graphs unless P = NP, thereby delineating a sharp complexity boundary for this fundamental enumeration task.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Enumerating minimal dominating sets with polynomial delay in bipartite graphs is a long-standing open problem. To date, even the subcase of chordal bipartite graphs is open, with the best known algorithm due to Golovach, Heggernes, Kant'e, Kratsch, Saether, and Villanger running in incremental-polynomial time. We improve on this result by providing a polynomial delay and space algorithm enumerating minimal dominating sets in chordal bipartite graphs. Additionally, we show that the total and connected variants admit polynomial and incremental-polynomial delay algorithms, respectively, within the same class. This provides an alternative proof of a result by Golovach et al. for total dominating sets, and answers an open question for the connected variant. Finally, we give evidence that the techniques used in this paper cannot be generalized to bipartite graphs for (total) minimal dominating sets, unless P = NP, and show that enumerating minimal connected dominating sets in bipartite graphs is harder than enumerating minimal transversals in general hypergraphs.

Problem

• Enumerate minimal dominating sets
• Polynomial delay in chordal bipartite graphs
• Address open problems in graph theory

Innovation

• Polynomial delay algorithm
• Minimal dominating sets
• Chordal bipartite graphs