A continuous generalization of domination-like invariants

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 $c\geq \frac{1}{2}$.