Graphs with equal Grundy domination and independence number

Abstract

The Grundy domination number, ${\gamma }_{\mathrm{gr}}\left(G\right)$, of a graph $G$ is the maximum length of a sequence $(v_1,v_2,\dots ,v_k)$ of vertices in $G$ such that for every $i\in \{2,\dots ,k\}$, the closed neighborhood $N\left[v_i\right]$ contains a vertex that does not belong to any closed neighborhood $N\left[v_j\right]$, 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 $\Gamma \left(G\right)$, which is in turn greater than or equal to the independence number $\alpha \left(G\right)$. In this paper, we initiate the study of the class of graphs $G$ with $\Gamma \left(G\right)={\gamma }_{\mathrm{gr}}\left(G\right)$ and its subclass consisting of graphs $G$ with $\alpha \left(G\right)={\gamma }_{\mathrm{gr}}\left(G\right)$. 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 $\Gamma \left(G\right)={\gamma }_{\mathrm{gr}}\left(G\right)$ and present large families of such graphs.