A DAG is a directed graph that contains no directed cycles. Define G = (V, E)...

A DAG is a directed graph that contains no directed cycles.

Define G = (V, E) in which V is the set of all nodes as {v₁, v₂, ..., v_i , ...v_n} and E is the set of edges E = {e_i,j = (v_i , v_j ) | v_i , v_j ∈ V} .

A topological order of a directed graph G = (V, E) is an ordering of its nodes as {v₁, v₂, ..., v_i , ...v_n} so that for every edge (v_i , v_j ) we have i < j.

There are several lemmas can be inferred from the definition of DAG. One lemma is: if G is a DAG, then G has a node with no incoming edges. Given the above lemma, prove the following two lemmas:

If the graph G is a DAG, then G has a topological ordering.

If the graph G has a topological order, then G is a DAG.

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venereology answered 9 months ago

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