Roman dominating influence parameters

A function f:V(G)→{0,1,2}f:V(G)→{0,1,2} is a Roman dominating function for a graph G=(V,E)G=(V,E) if for every vertex vv with f(v)=0f(v)=0, there exists a vertex w∈N(v)w∈N(v) with f(w)=2f(w)=2. Emperor Constantine had the requirement that an army or legion could be sent from its home to defend a neighboring location only if there was a second army which would stay and protect the home. Thus, there are two types of armies, stationary and traveling. Each vertex with no army must have a neighboring vertex with a traveling army. Stationary armies then dominate their own vertices, and a vertex with two armies is dominated by its stationary army, and its open neighborhood is dominated by the traveling army. In this paper, we introduce Roman dominating influence parameters in which the interest is in dominating each vertex exactly once.