A graph is well-dominated if all of its minimal dominating sets have the same cardinality. It is proved that there are exactly eleven connected, well-dominated, triangle-free graphs whose domination number is at most 3. We prove that at least one of the factors is well-dominated if the Cartesian product of two graphs is well-dominated. In addition, we show that the Cartesian product of two connected, triangle-free graphs is well-dominated if and only if both graphs are complete graphs of order 2. Under the assumption that at least one of the connected graphs G or H has no isolatable vertices, we prove that the direct product of G and H is well-dominated if and only if either \(G=H=K_3\) or \(G=K_2\) and H is either the 4-cycle or the corona of a connected graph. Furthermore, we show that the disjunctive product of two connected graphs is well-dominated if and only if one of the factors is a complete graph and the other factor has domination number at most 2.

如果所有图的最小支配集都具有相同的基数，则该图是支配的。事实证明，恰好有11个连通的，支配的，无三角形的图，其支配数最多为3。如果两个图的笛卡尔积是支配的，则证明至少一个因素是支配的。 。此外，我们表明，当且仅当两个图都是阶数为2的完整图时，两个连通的无三角形图的笛卡尔积才是主导的。在假定连通图G或H中至少有一个不存在的情况下可分离的顶点，我们证明G和H的直接乘积在且仅当\（G = H = K_3 \）或\（G = K_2 \），H是连接图的4周期或电晕。此外，我们表明，当且仅当一个因素是一个完整的图，而另一个因素具有至多2的支配数时，两个连通图的析取乘积才是主导的。