  In this paper, we define a new domination-like invariant of graphs. Let
$\mathbb{R}^{+}$ be the set of non-negative numbers. Let $c\in
\mathbb{R}^{+}-\{0\}$ be a number, and let $G$ be a graph. A function
$f:V(G)\rightarrow \mathbb{R}^{+}$ is a $c$-self-dominating function of $G$ if
for every $u\in V(G)$, $f(u)\geq c$ or $\max\{f(v):v\in N_{G}(u)\}\geq 1$. The
$c$-self-domination number $\gamma ^{c}(G)$ of $G$ is defined as $\gamma
^{c}(G):=\min\{\sum_{u\in V(G)}f(u):f$ is a $c$-self-dominating function of
$G\}$. Then $\gamma ^{1}(G)$, $\gamma ^{\infty }(G)$ and $\gamma
^{\frac{1}{2}}(G)$ are equal to the domination number, the total domination
number and the half of the Roman domination number of $G$, respectively. Our
main aim is to continuously fill in the gaps among such three invariants. In
this paper, we give a sharp upper bound of the $c$-self-domination number for
all $c\geq \frac{1}{2}$.
