  A sequence of vertices in a graph is called a legal dominating sequence if
every vertex in the sequence dominates at least one vertex not dominated by
those that precede it, and at the end all vertices of the graph are dominated.
The Grundy domination number of a graph is the size of a largest legal
dominating sequence. In this work, we introduce a generalized version of the
Grundy domination problem. We explicitly calculate the corresponding parameter
for paths and web graphs. We propose integer programming formulations for the
new problem, find families of valid inequalities and perform extensive
computational experiments to compare the formulations as well as to test these
inequalities as cuts in a branch-and-cut framework. We also design and evaluate
the performance of a heuristic for finding good initial lower and upper bounds
and a tabu search that improves the initial lower bound. The test instances
include randomly generated graphs, structured graphs, classical benchmark
instances and two instances from a real application. Our approach is exact for
graphs with 20-50 vertices and provides good solutions for graphs up to 10000
vertices.
