On the structure of $ \alpha $-limit sets of backward trajectories for graph maps

In the paper we study what sets can be obtained as $\alpha$-limit of backward trajectories in graph maps. We show that case mixing maps, all those are $\omega$-limit and for but finitely many points $x$, obtain every $\omega$-limits set a trajectory starting $x$. For zero entropy is minimal set. maps with positive entropy, partial characterization which very close to complete picture possible situations.