Dominating sequences in graphs

A sequence of vertices in a graph G is called a legal dominating sequence if every vertex in the sequence dominates at least one vertex not dominated by those vertices that precede it, and at the end all vertices of G are dominated. While the length of a shortest such sequence is the domination number of G, in this paper we investigate legal dominating sequences of maximum length, which we call the Grundy domination number of G. We prove that every graph has a legal dominating sequence of each possible length between its domination number and its Grundy domination number, and characterize the graphs for which the domination number and Grundy domination number are both equal to k, for k ≤ 3. We also prove that the decision version of the Grundy domination number is NP-complete, even when restricted to the class of chordal graphs, and present linear time algorithms for determining this number in trees, cographs and split graphs. Several results are extended to the context of edge covers in hypergraphs.