A continuous generalization of domination-like invariants

In this paper, we define a new domination-like invariant of graphs. Let $${\mathbb {R}}^{+}$$ R + be the set of non-negative numbers. Let $$c\in {\mathbb {R}}^{+}-\{0\}$$ c ∈ R + - { 0 } be a number, and let G be a graph. A function $$f:V(G)\rightarrow {\mathbb {R}}^{+}$$ f : V ( G ) → R + is a c-self-dominating function of G if for every $$u\in V(G)$$ u ∈ V ( G ) , $$f(u)\ge c$$ f ( u ) ≥ c or $$\max \{f(v):v\in N_{G}(u)\}\ge 1$$ max { f ( v ) : v ∈ N G ( u ) } ≥ 1 . The c-self-domination number $$\gamma ^{c}(G)$$ γ c ( G ) of G is defined as $$\gamma ^{c}(G):=\min \{\sum _{u\in V(G)}f(u):f$$ γ c ( G ) : = min { ∑ u ∈ V ( G ) f ( u ) : f is a c-self-dominating function of $$G\}$$ G } . Then $$\gamma ^{1}(G)$$ γ 1 ( G ) , $$\gamma ^{\infty }(G)$$ γ ∞ ( G ) and $$\gamma ^{\frac{1}{2}}(G)$$ γ 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 $$c\ge \frac{1}{2}$$ c ≥ 1 2 .