Graphs with equal Grundy domination and independence number

TL;DR

This paper explores graphs where the Grundy domination number equals the upper domination number and the independence number, characterizing these classes and identifying hypercubes as members.

Contribution

It characterizes graphs with equal Grundy domination and independence numbers, and introduces new classes with specific properties, including hypercubes.

Findings

Graphs with equal Grundy and independence numbers include hypercubes.

Characterizations of twin-free connected graphs with these properties.

Necessary conditions for graphs where Grundy and upper domination numbers are equal.

Abstract

The Grundy domination number, ${\gamma_{\rm gr}}(G)$, of a graph $G$ is the maximum length of a sequence $(v_1,v_2,\ldots, v_k)$ of vertices in $G$ such that for every $i\in \{2,\ldots, k\}$, the closed neighborhood $N[v_i]$ contains a vertex that does not belong to any closed neighborhood $N[v_j]$, where $j<i$. It is well known that the Grundy domination number of any graph $G$ is greater than or equal to the upper domination number $\Gamma(G)$, which is in turn greater than or equal to the independence number $\alpha(G)$. In this paper, we initiate the study of the class of graphs $G$ with $\Gamma(G)={\gamma_{\rm gr}}(G)$ and its subclass consisting of graphs $G$ with $\alpha(G)={\gamma_{\rm gr}}(G)$. We characterize the latter class of graphs among all twin-free connected graphs, provide a number of properties of these graphs, and prove that the hypercubes are members of this class. In addition, we give several necessary conditions for graphs $G$ with $\Gamma(G)={\gamma_{\rm gr}}(G)$ and present large families of such graphs.