Extending Grundy domination to $k$-Grundy domination

TL;DR

This paper extends the concept of Grundy domination to $k$-Grundy domination, analyzing its properties, bounds, and relationships with other graph parameters across various graph families.

Contribution

It introduces the $k$-Grundy domination number, computes it for specific graph families, and explores bounds and relationships with other graph invariants.

Findings

Determined $k$-Grundy domination number for certain graph families.

Established degree-based bounds for the $k$-$L$-Grundy domination number.

Linked $k$-$Z$-Grundy domination number with the $k$-forcing number.

Abstract

The Grundy domination number of a graph $G = (V,E)$ is the length of the longest sequence of unique vertices $S = (v_1, \ldots, v_k)$ satisfying $N[v_i] \setminus \cup_{j=1}^{i-1}N[v_j] \neq \emptyset$ for each $i \in [k]$. Recently, a generalization of this concept called $k$-Grundy domination was introduced. In $k$-Grundy domination, a vertex $v$ can be included in $S$ if it has a neighbor $u$ such that $u$ appears in the closed neighborhood of fewer than $k$ vertices of $S$. In this paper, we determine the $k$-Grundy domination number for some families of graphs, find degree-based bounds for the $k$-$L$-Grundy domination number, and define a relationship between the $k$-$Z$-Grundy domination number and the $k$-forcing number of a graph.