Equitable and Outdegree Equitable Domination in Product Graphs

An equitable dominating set of a graph $G = (V , E)$, is a subset of vertices $D \subseteq V$ such that each vertex $v\in V-D$, is adjacent to at least one vertex $u\in D$ and the degree of $u$ and $v$ differ by at most 1. The \textbf{equitable domination number} $\gamma^e(G)$ is the size of the smallest equitable dominating set of $G$. For a vertex $u$ in the dominating set $D$ of a graph $G$, the number of edges from $u$ to $V-D$ is called the outdegree of $u$ with respect to $D$, $d_{D, G}^o(u)$. A dominating set $D$ is called the outdegree equitable dominating set if the absolute value of the differences of outdegrees of any two vertices in $D$ is at most one. The \textbf{outdegree equitable domination number} of $G$ $\gamma_{oe}(G)$ is the size of the smallest outdegree equitable dominating set of $G$. In this paper equitability related domination parameters such as equitable domination number and outdegree equitable domination number are analyzed within the realm of some graph products specifically the cartesian and the strong product of graphs.