A longest sequence $S$ of distinct vertices of a graph $G$ such that each vertex of $S$ dominates some vertex that is not dominated by its preceding vertices, is called a Grundy dominating sequence; the length of $S$ is the Grundy domination number of $G$. In this paper we study the Grundy domination number in the four standard graph products: the Cartesian, the lexicographic, the direct, and the strong product. For each of the products we present a lower bound for the Grundy domination number which turns out to be exact for the lexicographic product and is conjectured to be exact for the strong product. In most of the cases exact Grundy domination numbers are determined for products of paths and/or cycles.
COBISS.SI-ID: 17829465
We continue the study of the Grundy domination number of a graph. A linear algorithm to determine the Grundy domination number of an interval graph is presented. The exact value of the Grundy domination number of an arbitrary Sierpiński graph is proven, and efficient algorithms to construct the corresponding sequences are presented. These results are obtained by using sharp bounds for the Grundy domination number of a vertex- and edge-removed graph, proven in this paper.
COBISS.SI-ID: 17807705