A continuous generalization of domination-like invariants

Abstract

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 $f:V(G)\to {\mathbb{R}}^{+}$ is a $c$-self-dominating function of $G$ if for every $u\in V(G)$, $f(u)\ge c$ or $\max \{f(v):v\in N_G(u)\}\ge 1$. 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 $c\ge \frac{1}{2}$.

18 pages, 1 figure