Margulies, S.Hicks, I.V.2013-07-122013-07-122012Margulies, S. and Hicks, I.V.. "An Algebraic Exploration of Dominating Sets and Vizing's Conjecture." <i>The Electronic Journal of Combinatorics,</i> 19, no. 2 (2012) The Electronic Journal of Combinatorics: <a href="https://hdl.handle.net/1911/71548">https://hdl.handle.net/1911/71548</a>.https://hdl.handle.net/1911/71548Systems of polynomial equations are commonly used to model combinatorial problems such as independent set, graph coloring, Hamiltonian path, and others. We formulate the dominating set problem as a system of polynomial equations in two di erent ways: rst, as a single, high-degree polynomial, and second as a collection of polynomials based on the complements of domination-critical graphs. We then provide a su cient criterion for demonstrating that a particular ideal representation is already the universal Gr obner bases of an ideal, and show that the second representation of the dominating set ideal in terms of domination-critical graphs is the universal Gr obner basis for that ideal. We also present the rst algebraic formulation of Vizing's conjecture, and discuss the theoretical and computational rami cations to this conjecture when using either of the two dominating set representations described above.engArticle is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.Journal articledominating setsVizing's conjectureuniversal Grobner bases