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Let’s assume that a graph in question is a DAG (directed acyclic graph).
Question: can I conclude that such graph will have a unique topological sort if, and only if, only one of its vertices has no incoming edges?
In other words, is having only one vertex with no incoming edges necessary (but not sufficient) to generate a unique topological sort?
Haaaaa, ok. sorry for the misunderstanding.
In this case I assume that you are right, here is a sketch of proof:
We have a unique topological sort => We have only one vertex that it is legal to put in the first place => For every vertex, except one, it is not legal to put in the first place => For every vertex, except one, we have incoming edges.
Hope that now I answered your question….