Co-Roman domination in trees

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_uv:V→{0,1,2}‎, ‎defined by f_uv (v)=1‎, ‎f_uv (u)=f(u)-1 and f_uv (x)=f(x)for x∈V-{v,u}‎, ‎has no unprotected vertex‎. ‎The weight of f is ω(f)=∑_(v∈V)▒〖f(v)〗‎. ‎The co-Roman domination number of a graph G ‎, ‎denoted by γ_cr G)‎, ‎is the minimum weight of a co-Roman dominating function on G ‎. ‎In this paper, we first present an upper bound on the co-Roman domination number of trees in terms of order, the number of leaves and supports‎. Then we find bounds on the co-Roman domination number of a graph and its other dominating parameters .