Grundy domination and zero forcing in regular graphs

TL;DR

This paper investigates the Grundy domination and zero forcing numbers in regular graphs, establishing bounds and characterizations for connected cubic graphs and other regular graphs, enhancing understanding of domination and propagation processes.

Contribution

It provides new bounds for the Grundy domination number in regular graphs and characterizes extremal cases, linking these to zero forcing numbers in cubic graphs.

Findings

Bound: $ ext{γ}_{ m gr}(G) ext{ ≥ } rac{n + ext{⌈}k/2 ext{⌉} - 2}{k-1}$ for connected k-regular graphs.

Characterization of cubic graphs with $ ext{γ}_{ m gr}(G) = n/2$.

Identification of cubic graphs where the zero forcing number equals $n/2$.

Abstract

Given a finite graph $G$, the maximum length of a sequence $(v_1,\ldots,v_k)$ of vertices in $G$ such that each $v_i$ dominates a vertex that is not dominated by any vertex in $\{v_1,\ldots,v_{i-1}\}$ is called the Grundy domination number, $\gamma_{\rm gr}(G)$, of $G$. A small modification of the definition yields the Z-Grundy domination number, which is the dual invariant of the well-known zero forcing number. In this paper, we prove that $\gamma_{\rm gr}(G) \geq \frac{n + \lceil \frac{k}{2} \rceil - 2}{k-1}$ holds for every connected $k$-regular graph of order $n$ different from $K_{k+1}$ and $\bar{2C_4}$. The bound in the case $k=3$ reduces to $\gamma_{\rm gr}(G) \geq \frac{n}{2}$, and we characterize the connected cubic graphs with $\gamma_{\rm gr}(G)=\frac{n}{2}$. If $G$ is different from $K_4$ and $K_{3,3}$, then $\frac{n}{2}$ is also an upper bound for the zero forcing number of a connected cubic graph, and we characterize the connected cubic graphs attaining this bound.