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. ...

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

Problem 1RQ

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