A Proof of the Grundy domination strong product conjecture

TL;DR

This paper proves a conjecture regarding the Grundy domination number of the strong product of two graphs and explores its implications for the zero forcing number, advancing understanding in graph theory.

Contribution

It provides a proof of a recent conjecture about the Grundy domination number in the strong product of graphs, linking it to zero forcing numbers.

Findings

Proof of the Grundy domination conjecture for strong product

Relationship established between Grundy domination and zero forcing numbers

Advancement in theoretical understanding of graph products

Abstract

The Grundy domination number of a simple graph $G = (V,E)$ is the length of the longest sequence of unique vertices $S = (v_1, \ldots, v_k)$, $v_i \in V$, that satisfies the property $N[v_i] \setminus \cup_{j=1}^{i-1}N[v_j] \neq \emptyset$ for each $i \in [k]$. Here, $N(v) = \{u : uv \in E\}$ and $N[v] = N(v) \cup \{v\}$. In this note, we prove a recent conjecture about the Grundy domination number of the strong product of two graphs. We then discuss how this result relates to the zero forcing number of the strong product of graphs.