Let \(I(G)\) be a graphical invariant defined for any graph \(G\). For several choices of \(I\) representing domination parameters, we characterize sequences of positive integers \(a_1,a_2,\ldots,a_n\) which have an associated sequence of graphs \(G_1,G_2,\ldots,G_n\) such that \(G_i\) has \(i\) vertices, \(G_i\) is an induced subgraph of \(G_{i+1}\), and \(I(G_i) = a_i\).