Bounds on the co-Roman domination number in graphs.

Let G = (V , E) be a graph and let f : V (G) → { 0 , 1 , 2 } be a function. A vertex v is protected with respect to f , if f (v) > 0 or f (v) = 0 and v is adjacent to a vertex of positive weight. The function f is a co-Roman dominating function, abbreviated CRDF if: (i) every vertex in V is protected, and (ii) each u ∈ V with positive weight has a neighbor v ∈ V with f (v) = 0 such that the function f u v : V → { 0 , 1 , 2 } , defined by f u v (v) = 1 , f u v (u) = f (u) − 1 and f u v (x) = f (x) for x ∈ V ∖ { v , u } , has no unprotected vertex. The weight of f is ω (f) = Σ v ∈ V f (v). The co-Roman domination number of a graph G , denoted by γ c r (G) , is the minimum weight of a co-Roman dominating function on G. In this paper, we present some new sharp bounds on γ c r (G). Some of our results improve the previous bounds. [ABSTRACT FROM AUTHOR]