Critical Paired Dominating Sets and Irreducible Decompositions of Powers of Edge Ideals

Abstract

Let G be a finite simple graph. A set S of vertices is a critical paired dominating set of G, if every vertex is adjacent to a vertex in S and the removal of any vertex does not change the matching number of G. In this paper, we give a characterization of graphs G which has a critical paired dominating set in terms of the irreducible decomposition of powers of the edge ideal associated to G.

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Acknowledgements

The authors would like to thank Professor Marcel Morales for his interesting discussions on this subject. The authors would like to thank the referee for the valuable comments to improve this article.

Funding

This research was funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.04 − 2017.14.

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Correspondence to Hoang Le Truong.

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Dedicated to Professor Le Tuan Hoa on the Occasion of His 60th Birthday

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Dung, N.T., Tam, N.T.T., Truong, H.L. et al. Critical Paired Dominating Sets and Irreducible Decompositions of Powers of Edge Ideals. Acta Math Vietnam 44, 587–601 (2019). https://doi.org/10.1007/s40306-018-0273-0

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Keywords

  • Edge ideals
  • Irreducible ideals
  • Irreducible decomposition
  • Matching
  • Factor-critical graphs
  • Critical paired dominating set

Mathematics Subject Classification (2010)

  • 13A30
  • 13D45
  • 13E05
  • 13H10