A Roman Domination Chain

Abstract

For a graph $G=(V,E)$, a Roman dominating function $f:V\rightarrow \{0,1,2\}$ has the property that every vertex $v\in V$ with $f(v)=0$ has a neighbor $u$ with $f(u)=2$. The weight of a Roman dominating function $f$ is the sum $f(V)=\sum \nolimits _{v\in V}f(v)$, and the minimum weight of a Roman dominating function on $G$ is the Roman domination number of $G$. In this paper, we define the Roman independence number, the upper Roman domination number and the upper and lower Roman irredundance numbers, and then develop a Roman domination chain parallel to the well-known domination chain. We also develop sharpness, strictness and bounds for the Roman domination chain inequalities.