Enumerating minimal dominating sets and variants in chordal bipartite graphs

TL;DR

This paper presents a polynomial delay and space algorithm for enumerating minimal dominating sets in chordal bipartite graphs, solving an open problem and extending to related variants, with implications for complexity in bipartite graphs.

Contribution

It introduces a polynomial delay enumeration algorithm for minimal dominating sets in chordal bipartite graphs and addresses open questions for total and connected variants.

Findings

Polynomial delay algorithm for minimal dominating sets in chordal bipartite graphs.

Polynomial and incremental-polynomial delay algorithms for total and connected variants.

Hardness results indicating limitations of techniques for bipartite graphs.

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.