Let $G$ be a simple graph. A subset $S\subseteq V$ is a dominating set of $G$, if for any vertex $v\in V-S$ there exists a vertex $u\in S$ such that $uv\in E\left(G\right)$. The domination number, denoted by $\gamma \left(G\right)$, is the minimum cardinality of a dominating set. In this paper we prove that if $G$ is a 4-regular graph with order $n$, then $\gamma \left(G\right)\le \frac{4}{11}n$.

Let $f$ be an integer-valued function defined on the vertex set $V\left(G\right)$ of a graph $G$. A subset $D$ of $V\left(G\right)$ is an $f$-dominating set if each vertex $x$ outside $D$ is adjacent to at least $f\left(x\right)$ vertices in $D$. The minimum number of vertices in an $f$-dominating set is defined to be the $f$-domination number, denoted by ${\gamma }_f\left(G\right)$. In a similar way one can define the connected and total $f$-domination numbers ${\gamma }_{c,f}\left(G\right)$ and ${\gamma }_{t,f}\left(G\right)$. If $f\left(x\right)=1$ for all vertices $x$, then these are the ordinary domination number, connected domination number and total...

Let $G$ be a graph with vertex set $V\left(G\right)$, and let $k\ge 1$ be an integer. A subset $D\subseteq V\left(G\right)$ is called a if every vertex $v\in V\left(G\right)-D$ has at least $k$ neighbors in $D$. The $k$-domination number ${\gamma }_k\left(G\right)$ of $G$ is the minimum cardinality of a $k$-dominating set in $G$. If $G$ is a graph with minimum degree $\delta \left(G\right)\ge k+1$, then we prove that $${\gamma }_{k+1}\left(G\right)\le \frac{\left|V\left(G\right)\right|+{\gamma }_k\left(G\right)}{2}.$$ In addition, we present a characterization of a special class of graphs attaining equality in this inequality.