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
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ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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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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