Given a digraph G, prove that the first vertex in the reverse postorder of Gis in a strong...

Given a digraph G, prove that the first vertex in the reverse postorder of Gis in a strong component that is a sourceof G’skernel DAG. Then, prove that the first vertex in the reverse postorder of GRis in a strong component that is a sinkof G’skernel DAG. Hint: Apply exercises 4.2.14 and 4.2.15.

Exercises4.2.14 :

Let Cbe a strong component in a digraph Gand let v be any vertex not in C. Prove that if there is an edge epointing from v to any vertex in C, then vertex v appears before everyvertex in Cin the reverse postorder of G.

Solution: If v is visited before every vertex in C, then every vertex in Cwill be visited and finished before v finishes (because every vertex in Cis reachable from v via edge e). If some vertex in Cis visited before v, then all vertices in Cwill be visited and finished before v is visited (because v is not reachable from any vertex in C—if it were, such a path when combined with edge ewould be part of a directed cycle, implying that v is in C).

Exercises4.2.15:

Let Cbe a strong component in a digraph Gand let v be any vertex not in C. Prove that if there is an edge epointing from any vertex in Cto v, then vertex v appears before everyvertex in Cin the reverse postorder of GR.

Solution: Apply exercise 4.2.14 to GR.

Exercise4.2.14:

Let Cbe a strong component in a digraph Gand let v be any vertex not in C. Prove that if there is an edge epointing from v to any vertex in C, then vertex v appears before everyvertex in Cin the reverse postorder of G.

Solution: If v is visited before every vertex in C, then every vertex in Cwill be visited and finished before v finishes (because every vertex in Cis reachable from v via edge e). If some vertex in Cis visited before v, then all vertices in Cwill be visited and finished before v is visited (because v is not reachable from any vertex in C—if it were, such a path when combined with edge ewould be part of a directed cycle, implying that v is in C).