Where can I find a data set of graphs with known domination numbers?
One place to look is House of Graphs, which is a database of "interesting" graphs. For instance, there are graphs that have been used as counterexamples to various conjectures, have extremal properties with respect to some parameter, and so on. For more, see [1].

Another possibility is using Mathematica, which also has a database of graphs accessed through its GraphData-function.



[1] Brinkmann, G., Coolsaet, K., Goedgebeur, J. and Mélot, H., 2013. House of Graphs: A database of interesting graphs. Discrete Applied Mathematics, 161(1), pp.311-314. --- Free version at arXiv