4. Let G = (V, E) be a directed graph and let S1,, Sm be its strongly connected components. Define a new graph G' = (V', E') as follows. V' = {1, -. ,m}. Place an edge from i to j in G' if there is a vertex u E S; and a vertex vE S; such that (u, v) E E. ... • Show that G' is a DAG. • Let n1, n2, . nm be a topological ordering of vertices in G'. Suppose that there is no edge between n; and ni+1 in G', then show that for every pair of vcrtices u, v, with u E Si and v E Si+1, there is no path from u to v in G. ...

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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4. Let G = (V, E) be a directed graph and let S1,. , Sm be its strongly connected components.
Define a new graph G' = (V', E') as follows. V'= {1,.….. , m}. Place an edge from i to j in
G' if there is a vertex u E S; and a vertex v E S; such that (u, v) E E.
• Show that G' is a DAG.
• Let n1, n2, ...Nm be a topological ordering of vertices in G'. Suppose that there is no
edge between n; and ni+1 in G', then show that for every pair of vertices u, v, with
u E S; and v E Si+1, there is no path from u to v in G.
Transcribed Image Text:4. Let G = (V, E) be a directed graph and let S1,. , Sm be its strongly connected components. Define a new graph G' = (V', E') as follows. V'= {1,.….. , m}. Place an edge from i to j in G' if there is a vertex u E S; and a vertex v E S; such that (u, v) E E. • Show that G' is a DAG. • Let n1, n2, ...Nm be a topological ordering of vertices in G'. Suppose that there is no edge between n; and ni+1 in G', then show that for every pair of vertices u, v, with u E S; and v E Si+1, there is no path from u to v in G.
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