### An algebraic approach to sets defining minimal dominating sets of regular graphs

#### Abstract

Suppose that *V* = {1, …, *n*} is a non-empty set of *n* elements, *S* = {*S*_{1}, …, *S*_{m}} a non-empty set of *m* non-empty subsets of *V*. In this paper, by using some algebraic notions in commutative algebra, we investigate the question arises whether there exists an undirected finite simple graph *G* with *V*(*G*)=*V*, where *S* is the set whose elements are the minimal dominating sets of *G*.

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PDFDOI: http://dx.doi.org/10.5614/ejgta.2023.11.2.5

#### References

J. A. Bondy and U. S. R. Murty, Graph theory. Graduate Texts in Mathematics, 244. Springer, New York, 2008.

J. Herzog and T. Hibi, Monomial Ideals. Graduate Texts in Mathematics 260 Springer-Verlag, 2011.

J. Honeycutt and S. K. Sather-Wagstaff, Closed neighborhood ideals of finite simple graphs, La Matematica 1 (2022), 387–394.

J. Mart' i-Farr' e, Sets defining minimal vertex covers, Discrete Math. 197/198 (1999), 555–559.

D. A. Mojdeh M. Alishahi, Outer independent global dominating set of trees and unicyclic graphs, Electron. J. Graph Theory Appl. 7 (1) (2019), 121–145.

D. A. Mojdeh, S. R. Musawi, and E. Nazari Kiashi, On the distance domination number of bipartite graphs, Electron. J. Graph Theory Appl. 8 (2) (2020), 353–364.

M. Nasernejad, S. Bandari, and L. G. Roberts, Normality and associated primes of closed neighborhood ideals and dominating ideals, J. Algebra Appl., 2023, doi:10.1142/S0219498825500094.

M. Nasernejad and K. Khashyarmanesh, Sets defining minimal vertex covers of uniform hypergraphs, Ars Combin. 147 (2019), 135–142.

M. Nasernejad, A. A. Qureshi, S. Bandari, and A. Musapaşaoğlu, Dominating ideals and closed neighborhood ideals of graphs, Mediterranean Journal of Mathematics, 19, Article number: 152 (2022).

L. Sharifan and S. Moradi, Closed neighborhood ideal of a graph, Rocky Mountain J. Math. 50 (3) (2020), 1097–1107.

R. H. Villarreal, Cohen–Macaulay graphs, Manuscripta Math. 66 (1990), 277–293.

R. H. Villarreal, Monomial Algebras. 2nd. Edition, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2015.

D. B. West, Introduction to graph theory. Prentice Hall, Inc., Upper Saddle River, NJ, 1996. xvi+512 pp. ISBN: 0-13-227828-6.

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