Dominating sequences in grid-like and toroidal graphs

Bresar, Bostjan and Bujtás, Csilla and Gologranc, Tanja and Klavzar, Sandi and Kosmrlj, Gasper and Patkós, Balázs and Tuza, Zsolt and Vizer, Máté (2016) Dominating sequences in grid-like and toroidal graphs. ELECTRONIC JOURNAL OF COMBINATORICS. ISSN 1077-8926 (Submitted)

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Abstract

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.

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