  For $c\in \mathbb{R}^{+}\cup \{\infty \}$ and a graph $G$, a function
$f:V(G)\rightarrow \{0,1,c\}$ is called a $c$-self dominating function of $G$
if for every vertex $u\in V(G)$, $f(u)\geq c$ or $\max\{f(v):v\in
N_{G}(u)\}\geq 1$ where $N_{G}(u)$ is the neighborhood of $u$ in $G$. The
minimum weight $w(f)=\sum _{u\in V(G)}f(u)$ of a $c$-self dominating function
$f$ of $G$ is called the $c$-self domination number of $G$. The $c$-self
domination concept is a common generalization of three domination-type
invariants; (original) domination, total domination and Roman domination. In
this paper, we study a behavior of the $c$-self domination number in random
graphs for small $c$.
