Equitable k-Fair Domination Under Some Binary Operations

Abstract

A subset $S$ of the vertex set $V(G)$ of a graph $G$ is called an equitable fair dominating set of $G$ if $S$ is an equitable dominating set of $G$ and for any $v,$ $w \in V(G) \backslash S$, $|N_G(v) \cap S| = |N_G(w) \cap S| \geq 1$. The equitable fair domination number of $G$ denoted by $\gamma_{efd}(G)$ is the minimum cardinality of an EFD-set of $G$. $S$ is  called an equitable $k$-fair dominating set (abbreviated E$k$FD-set) of $G$ if $|N_G(v) \cap S| = k$ for any $v \in V(G) \backslash S$ where $k$ is a positive integer. The equitable $k$-fair domination number of $G$ denoted by $\gamma_{_{kfd}}^e (G)$ is the minimum cardinality of an E$k$FD-set. An equitable $k$-fair dominating set of cardinality $\gamma_{_{kfd}}^e (G)$ is called a $\gamma_{_{kfd}}^e$-set of $G$. In this paper, we characterize the notions of equitable k-fair domination in graphs, study the EkFD-sets under some binary operations of graphs, and determine exact values or bounds for this domination variant.