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.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.5: Graphs Of Functions
Problem 57E
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