Dominating sequences in grid-like and toroidal graphs

TL;DR

This paper investigates the Grundy domination number in various graph products, providing bounds and exact values for paths and cycles, with conjectures for the strong product.

Contribution

It introduces bounds and exact values for the Grundy domination number in four standard graph products, advancing understanding of domination sequences in complex graphs.

Findings

Lower bounds for Grundy domination number in graph products

Exact values for paths and cycles in most cases

Conjecture on exactness for the strong product

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.