Vizing's conjecture (open since 1968) relates the product of the domination numbers of two graphs to the domination number of their Cartesian product graph. In this paper, we formulate Vizing's conjecture as a Positivstellensatz existence question. In particular, we select classes of graphs according to their number of vertices and their domination number and encode the conjecture as an ideal/polynomial pair such that the polynomial is non-negative on the variety associated with the ideal if and only if the conjecture is true for this graph class. Using semidefinite programming we obtain numeric sum-of-squares certificates, which we then manage to transform into symbolic certificates confirming non-negativity of our polynomials. Specifically, we obtain exact low-degree sparse sum-of-squares certificates for particular classes of graphs.

The obtained certificates allow generalizations for larger graph classes. Besides computational verification of these more general certificates, we also present theoretical proofs as well as conjectures and questions for further investigations.

Vizing的猜想（自1968年开放）将两个图的支配数的乘积与其直角乘积图的支配数相关联。在本文中，我们将维辛的猜想表达为一个Positivstellensatz存在问题。特别是，我们根据图的顶点数和支配数选择图的类别，并将猜想编码为理想/多项式对，这样，当且仅当猜想为，多项式在与理想相关的变数上为非负数对于此图类为true。使用半定编程，我们获得了数字平方和证书，然后将其转换为符号证书，以确认多项式的非负性。具体来说，我们为特定类别的图获得了精确的低度稀疏平方和证书。

获得的证书允许对较大的图类进行概括。除了对这些更通用的证书进行计算验证外，我们还提供理论证明以及猜想和问题，以供进一步研究。