A sequence of vertices in a graph is called a \emph{(total) legal dominating sequence} if every vertex in the sequence (total) dominates at least one vertex not dominated by those ones that precede it, and at the end all vertices of the graph are (totally) dominated. The \emph{Grundy (total) domination number} of a graph is the size of the largest (total) legal dominating sequence. In this work, we address the problems of determining these two parameters by introducing a generalized version of them. We explicitly calculate the corresponding (general) parameter for paths and web graphs. We propose integer programming formulations for the new problem and we study the polytope associated to one of them. We find families of valid inequalities and derive conditions under which they are facet-defining. Finally, we perform computational experiments to compare the formulations as well as to test valid inequalities as cuts in a B\&C framework.