Let G be a graph and f: G → G be continuous. Denote by R(f) and Ω(f) the set of recurrent points and the set of non-wandering points of f respectively. Let Ω0 (f) = G and Ωn (f) =Ω(f|Ωn-1(f)) for all n ∈ N. The minimal m ∈ NU {∞} such that Ωm(f) = Ωm+1(f) is called the depth of f. In this paper, we show that Ω2(f) = R(f) and the depth of f is at most 2. Furthermore, we obtain some properties of non-wandering points of f.