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Enumerating Minimal Dominating Sets and Variants in Chordal Bipartite Graphs

Authors Emanuel Castelo , Oscar Defrain , Guilherme C. M. Gomes

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ACM Subject Classification
  • Mathematics of computing → Enumeration
  • Mathematics of computing → Graph enumeration
Keywords
  • algorithmic enumeration
  • minimal dominating sets
  • connected dominating sets
  • total dominating sets
  • chordal bipartite graphs
  • sequential method
  • polynomial delay

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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, Kanté, Kratsch, Sæther, 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.

Cite As Get BibTex

Emanuel Castelo, Oscar Defrain, and Guilherme C. M. Gomes. Enumerating Minimal Dominating Sets and Variants in Chordal Bipartite Graphs. In 19th International Symposium on Algorithms and Data Structures (WADS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 349, pp. 15:1-15:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.WADS.2025.15

Author Details

Emanuel Castelo
  • Aix-Marseille Université, CNRS, LIS, Marseille, France
Oscar Defrain
  • Aix-Marseille Université, CNRS, LIS, Marseille, France
Guilherme C. M. Gomes
  • LIRMM, Université de Montpellier, CNRS, Montpellier, France
  • Universidade Federal de Minas Gerais, Belo Horizonte, Brazil

Funding

  • Defrain, Oscar: Supported by the ANR project PARADUAL (ANR-24-CE48-0610-01).
  • C. M. Gomes, Guilherme: Funded by the European Union, project PACKENUM, grant number 101109317. Views and opinions expressed are however those of the author only and do not necessarily reflect those of the European Union. Neither the European Union nor the granting authority can be held responsible for them.

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