An algebraic exploration of dominating sets and Vizing's conjecture

Abstract

Systems 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 different ways: first, 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 suficient criterion for demonstrating that a particular ideal representation is already the universal Gröbner bases of an ideal, and show that the second representation of the dominating set ideal in terms of domination-critical graphs is the universal Gröbner basis for that ideal. We also present the first algebraic formulation of Vizing's conjecture, and discuss the theoretical and computational ramifications to this conjecture when using either of the two dominating set representations described above.