Sets of all ω-limit points for one-dimensional maps

Let f be a continuous map from a graph G to itself and m the maximum of orders of all points of G. The main result of this paper is that a point c is an element of G lies in the limit set of some point of G if and only if every open neighborhood of c contains at least (m + 1) points of some trajectory. This shows that every set of all omega-limit points for every graph map f satisfies the analogue to the Birkhoff theorem.