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Let D be a directed graph. (a) Assume that there exists a vertex vЄ V(D) such that d, (v) = 0. Prove or disprove that D must be a directed acyclic graph. (b) Assume that D is a directed acyclic graph. Prove or disprove that there exists a vertex vЄ V(D) such that d (v) = 0. Now consider the directed network (D, w) given by the following drawing, where each arc e E A(D) is labelled by its weight w(e). V1 4 V3 7 VA Сл 2 5 4 1 V5 3 V2 4 V6 (c) Give a topological ordering of D. (d) Give the strongly connected components of D. (e) Use Morávek's algorithm to find a longest directed v₁-v6-path in (D, w). Show your working and give the path and its length. (f) If w(v3v4) was decreased from 7 to 4, would this affect the length of a longest directed v₁-v6-path in (D, w)? Justify your answer.

Let D be a directed graph. (a) Assume that there exists a vertex vЄ V(D) such that d, (v) = 0. Prove or disprove that D must be a directed acyclic graph. (b) Assume that D is a directed acyclic graph. Prove or disprove that there exists a vertex vЄ V(D) such that d (v) = 0. Now consider the directed network (D, w) given by the following drawing, where each arc e E A(D) is labelled by its weight w(e). V1 4 V3 7 VA Сл 2 5 4 1 V5 3 V2 4 V6 (c) Give a topological ordering of D. (d) Give the strongly connected components of D. (e) Use Morávek's algorithm to find a longest directed v₁-v6-path in (D, w). Show your working and give the path and its length. (f) If w(v3v4) was decreased from 7 to 4, would this affect the length of a longest directed v₁-v6-path in (D, w)? Justify your answer.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter3: Matrices
Section3.7: Applications
Problem 69EQ
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can you please answer d and e

Let D be a directed graph. (a) Assume that there exists a vertex vЄ V(D) such that d, (v) = 0. Prove or disprove that D must be a directed acyclic graph. (b) Assume that D is a directed acyclic graph. Prove or disprove that there exists a vertex vЄ V(D) such that d (v) = 0. Now consider the directed network (D, w) given by the following drawing, where each arc e E A(D) is labelled by its weight w(e). V1 4 V3 7 VA Сл 2 5 4 1 V5 3 V2 4 V6 (c) Give a topological ordering of D. (d) Give the strongly connected components of D. (e) Use Morávek's algorithm to find a longest directed v₁-v6-path in (D, w). Show your working and give the path and its length. (f) If w(v3v4) was decreased from 7 to 4, would this affect the length of a longest directed v₁-v6-path in (D, w)? Justify your answer.
Transcribed Image Text:Let D be a directed graph. (a) Assume that there exists a vertex vЄ V(D) such that d, (v) = 0. Prove or disprove that D must be a directed acyclic graph. (b) Assume that D is a directed acyclic graph. Prove or disprove that there exists a vertex vЄ V(D) such that d (v) = 0. Now consider the directed network (D, w) given by the following drawing, where each arc e E A(D) is labelled by its weight w(e). V1 4 V3 7 VA Сл 2 5 4 1 V5 3 V2 4 V6 (c) Give a topological ordering of D. (d) Give the strongly connected components of D. (e) Use Morávek's algorithm to find a longest directed v₁-v6-path in (D, w). Show your working and give the path and its length. (f) If w(v3v4) was decreased from 7 to 4, would this affect the length of a longest directed v₁-v6-path in (D, w)? Justify your answer.
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