Acta Mathematica Vietnamica

, Volume 44, Issue 3, pp 587–601

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

• Nguyen Thi Dung
• Nguyen Thi Thanh Tam
• Hoang Le Truong
• Hoang Ngoc Yen
Article

## 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.

## Keywords

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

## Mathematics Subject Classification (2010)

13A30 13D45 13E05 13H10

## Notes

### 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 Information

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

## References

1. 1.
Chen, J., Morey, S., Sung, A.: The stable set of associated primes of the ideal of a graph. Rocky Mountain J. Math. 32(1), 71–89 (2002)
2. 2.
Lou, D., Rao, D.: Characterizing factor critical graphs and an algorithm. Australas. J. Combin. 30, 51–56 (2004)
3. 3.
Edmonds, J.: Paths, trees and flowers. Canad. J. Math. 17, 449–467 (1965)
4. 4.
Gallai, T.: Neuer Beweis eines Tutte’schen Satzes. (German) Magyar Tud. Akad. Mat. Kutató. Int. Kö,zl. 8, 135–139 (1963)
5. 5.
Hà, H. T., Morey, S.: Embedded associated primes of powers of square-free monomial ideals. J. Pure Appl. Algebra 214(4), 301–308 (2010)
6. 6.
Herzog, J., Hibi, T.: The depth of powers of an ideal. J. Algebra 291(2), 534–550 (2005)
7. 7.
Herzog, J., Hibi, T.: Bounding the socles of powers of squarefree monomial ideals. Commutative Algebra and Noncommutative Algebraic Geometry. Vol. II. Math. Sci. Res. Inst. Publ., vol. 68, pp 223–229. Cambridge Univ. Press, New York (2015)Google Scholar
8. 8.
Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Fundamentals of domination in graphs. Marcel Dekker, New York (1998)
9. 9.
Hien, H.T.T., Lam, H.M.: Combinatorial characterizations of the saturation and the associated primes of the fourth power of edge ideals. Acta Math. Vietnam. 40(3), 511–526 (2015)
10. 10.
Haynes, T.W., Slater, P.J.: Paired–domination and the paired–domatic number. Congr. Numer. 109, 65–72 (1995)
11. 11.
Haynes, T.W., Slater, P.J.: Paired–domination in graphs. Networks 32(3), 199–206 (1998)
12. 12.
Lovász, L.: A note on a factor-critical graphs. Studia Sci. Math. Hungar. 7, 279–280 (1972)
13. 13.
Lovász, L., Plummer, M.D.: Matching theory. North-Holland Publishing Co., Amsterdam (1986)
14. 14.
Martinez-Bernal, J., Morey, S., Villarreal, R.: Associated primes of powers of edge ideals. Collect. Math. 63(3), 361–374 (2012)
15. 15.
Miller, E., Sturmfels, B.: Combinatorial commutative algebra graduate texts in mathematics, vol. 227. Springer-Verlag, New York (2005)Google Scholar
16. 16.
Noether, E.: Idealtheorie in Ringbereichen. Math. Ann. 83, 24–66 (1921)
17. 17.
Proffitt, K.E., Haynes, T.W., Slater, P.J.: Paired–domination in grid graphs. Congr. Numer. 150, 161–172 (2001)
18. 18.
Shan, E., Kang, L., Henning, M.A.: A characterization of trees with equal total domination and paired–domination numbers. Australas. J. Combin. 30, 31–39 (2004)
19. 19.
Terai, N., Trung, N.V.: On the associated primes and the depth of the second power of a squarefree monomial ideals. J. Pure Appl. Algebra 218(6), 1117–1129 (2014)

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2018

## Authors and Affiliations

• Nguyen Thi Dung
• 1
• Nguyen Thi Thanh Tam
• 2
• Hoang Le Truong
• 3
• 4
Email author
• Hoang Ngoc Yen
• 5
1. 1.Thai Nguyen University of Agriculture and ForestryThai NguyenVietnam
2. 2.Hung Vuong UniversityViet TriVietnam
3. 3.Institute of MathematicsVietnam Academy of Science and TechnologyHanoiVietnam
4. 4.Mathematik und InformatikUniversität des SaarlandesSaarbrückenGermany
5. 5.Thai Nguyen University of EducationThai NguyenVietnam