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