On k-domination and j-independence in graphs

Let $G$ be a graph and let $k$ and $j$ be positive integers. A subset $D$ of the vertex set of $G$ is a $k$-dominating set if every vertex not in $D$ has at least $k$ neighbors in $D$. The $k$-domination number ${\gamma }_k\left(G\right)$ is the cardinality of a smallest $k$-dominating set of $G$. A subset $I?V\left(G\right)$ is a $j$-independent set of $G$ if every vertex in $I$ has at most $j?1$ neighbors in $I$. The $j$-independence number ${\alpha }_j\left(G\right)$ is the cardinality of a largest $j$-independent set of $G$. In this work, we study the interaction between ${\gamma }_k\left(G\right)$ and ${\alpha }_j\left(G\right)$ in a graph $G$. Hereby, we generalize some known inequalities concerning these parameters and put into relation different known and new bounds on $k$-domination and $j$-independence. Finally, we will discuss several consequences that follow from the given relations, while always highlighting the symmetry that exists between these two graph invariants.