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 non-negative numbers. $$c\in {\mathbb {R}}^{+}-\{0\}$$ number, and let G graph. A function $$f:V(G)\rightarrow is c-self-dominating if for every $$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)$$ defined as ^{c}(G):=\min \{\sum _{u\in V(G)}f(u):f$$ $$G\}$$ Then ^{1}(G)$$ ^{\infty }(G)$$ ^{\frac{1}{2}}(G)$$ are equal to domination total half Roman G, respectively. Our main aim continuously fill in gaps among such three invariants. give sharp upper bound all $$c\ge \frac{1}{2}$$