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Consider a connected graph G and let e = wv € E(G) such that G' = G-e is disconnected. Let G₁ and G₂ be two induced subgraphs of G such that V(G₁) = {rG: G' contains an ru-path} and V (G₂) = {re G: G' contains an ru-path}. (1) Show that G₁ and G₂ are both connected. (2) Show that V(G) = V(G₁) UV(G₂). (3) Deduce that G' contains only two connected components.

Consider a connected graph G and let e = wv € E(G) such that G' = G-e is disconnected. Let G₁ and G₂ be two induced subgraphs of G such that V(G₁) = {rG: G' contains an ru-path} and V (G₂) = {re G: G' contains an ru-path}. (1) Show that G₁ and G₂ are both connected. (2) Show that V(G) = V(G₁) UV(G₂). (3) Deduce that G' contains only two connected components.

Advanced Engineering Mathematics
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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part 1 2 e graph theory
s. Show that AUB‡V(G) Exercice 3. Consider a connected graph G and let e = uv € E(G) such that G' = G-e is disconnected. Let G₁ and G₂ be two induced subgraphs of G such that V(G₁) = {x € G: G' contains an ru-path} and V(G₂) = {x € G: G' contains an ru-path}. (1) Show that G₁ and G₂ are both connected. (2) Show that V(G) = V(G₁) UV(G₂). (3) Deduce that G' contains only two connected components.
Transcribed Image Text:s. Show that AUB‡V(G) Exercice 3. Consider a connected graph G and let e = uv € E(G) such that G' = G-e is disconnected. Let G₁ and G₂ be two induced subgraphs of G such that V(G₁) = {x € G: G' contains an ru-path} and V(G₂) = {x € G: G' contains an ru-path}. (1) Show that G₁ and G₂ are both connected. (2) Show that V(G) = V(G₁) UV(G₂). (3) Deduce that G' contains only two connected components.
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