Abstract

The PDS problem in graphs mathematically models the difficulty of monitoring electrical networks, which is inspired by the deployment of Phasor Measuring Units (PMUs) in power systems. This paper introduces a novel algebraic framework for studying power domination based on graph-derived monomial algebras. By encoding propagation rules as algebraic saturation operations, the power domination number is described in terms of minimal generating sets of monomial ideals. Examples of standard graph families are presented. The relationships with algebraic invariants, such as regularity and projective dimension, are investigated. Algebraic methods for computing and finding bounds for the power domination number are proposed. This paper connects commutative algebra and practical graph theory, with implications for electrical network research.

Keywords

Monomial Algebras, Electrical Network Monitoring Problem, Ideals.

References

[1] Winfried Bruns, Aldo Conca, and Matteo Varbaro, “Castelnuovo–Mumford Regularity and Powers,” Commutative Algebra, pp. 147-158, 2021.
[CrossRef] [Google Scholar] [Publisher Link]

[2] Sara Saeedi Madani, “Binomial Edge Ideals: A Survey,” Multigraded Algebra and Applications, pp. 83-94, 2020.
[CrossRef] [Google Scholar] [Publisher Link]

[3] Delio Jaramillo-Velez, and Lisa Seccia, “Connected Domination in Graphs and v-Numbers of Binomial Edge Ideals,” Collectanea Mathematica, vol. 75, pp. 771-793, 2023.
[CrossRef] [Google Scholar] [Publisher Link]

[4] A.V. Jayanthan, and Rajiv Kumar, “Regularity of Symbolic Powers of Edge Ideals,” Journal of Pure and Applied Algebra, vol. 224, no. 7, pp. 1-12, 2020.
[CrossRef] [Publisher Link]

[5] Leila Sharifan, “t-Closed Neighborhood Ideal of a Graph,” Contributions to Algebra and Geometry, vol. 66, pp. 927-940, 2024.
[CrossRef] [Google Scholar] [Publisher Link]

[6] Josep Àlvarez Montaner, Oscar Fernández-Ramos, and Philippe Gimenez, “Pruned Cellular Free Resolutions of Monomial Ideals,” Journal of Algebra, vol. 541, pp. 126-145, 2020.
[CrossRef] [Google Scholar] [Publisher Link]

[7] Timothy B.P. Clark, and Alexandre Tchernev, “Regular CW-Complexes and Poset Resolutions of Monomial Ideals,” Communications in Algebra, vol. 44, no. 6, pp. 2707-2718, 2016.
[CrossRef] [Google Scholar] [Publisher Link]

[8] Rachelle R. Bouchat, Huy Tài Hà, and Augustine O’Keefe, “Path Ideals of Rooted Trees and Their Graded Betti Numbers,” Journal of Combinatorial Theory, Series A, vol. 118, no. 8, pp. 2411-2425, 2011.
[CrossRef] [Google Scholar] [Publisher Link]

[9] Claude Berge, “Optimization and Hypergraph Theory,” European Journal of Operational Research, vol. 46, no. 3, pp. 297-303, 1990.
[CrossRef] [Google Scholar] [Publisher Link]

[10] Teresa W. Haynes et al., “Domination in Graphs Applied to Electric Power,” SIAM Journal on Discrete Mathematics, vol. 15, no. 4, 2002.
[CrossRef] [Google Scholar] [Publisher Link]