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 EG: G' contains an cu-path} and V(G₂) = {x EG: G' contains an xv-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 = 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 EG: G' contains an cu-path} and V(G₂) = {x EG: G' contains an xv-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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![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 EG: 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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb1d00d4f-96a4-49ef-a4f2-6e0b4ecbefa5%2Fab87646f-3367-41c4-8f7f-6e0ede6f78a3%2Fop1fdzm_processed.jpeg&w=3840&q=75)
Transcribed Image Text: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 EG: 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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