Let f : G --> G be a continuous map of a graph and let d(A) denote the deri
ved set (or limit points) of A subset of G. We prove that d(Omega>(*) over
bar * (f)) subset of Lambda>(*) over bar * (f) and the depth of f is at mos
t three. We also prove that if f is piecewise monotone or has zero topologi
cal entropy, then the depth of f is at most two. Furthermore, we obtain som
e results on the topological structure of Omega>(*) over bar * (f). 2000 Ma
thematics subject classification: primary 54H20, 58F03.