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4. Let G = (V, E) and G' = (V', E') be simple graphs. We say that a graph G is homomorphic to a graph G' if there exists a mapping : V → V' such that for each edge {x,y} € E of the graph G it follows that {o(r), (y)} € E'. Such a mapping is called a homomorphism of the graph G onto the graph G', and is denoted by ø: G → G'. Show that for any three graphs G₁, G2, G3, if there exists a homomorphism f: G₁ → G₂ and a homomorphism g: G₂ → G3, then there also exists a homomorphism h : G₁ → G3. Using the proof, then find the corresponding homomorphisms for the following graphs: P y 9 Ž G₁ I U 21 t E G₂ 7 B a G₁

4. Let G = (V, E) and G' = (V', E') be simple graphs. We say that a graph G is homomorphic to a graph G' if there exists a mapping : V → V' such that for each edge {x,y} € E of the graph G it follows that {o(r), (y)} € E'. Such a mapping is called a homomorphism of the graph G onto the graph G', and is denoted by ø: G → G'. Show that for any three graphs G₁, G2, G3, if there exists a homomorphism f: G₁ → G₂ and a homomorphism g: G₂ → G3, then there also exists a homomorphism h : G₁ → G3. Using the proof, then find the corresponding homomorphisms for the following graphs: P y 9 Ž G₁ I U 21 t E G₂ 7 B a G₁

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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4. Let G = (V, E) and G' = (V', E') be simple graphs. We say that a graph G is homomorphic to a graph G' if there exists a mapping : V → V' such that for each edge {x,y} € E of the graph G it follows that {o(r), (y)} € E'. Such a mapping is called a homomorphism of the graph G onto the graph G', and is denoted by ø: G → G'. Show that for any three graphs G₁, G2, G3, if there exists a homomorphism f: G₁ → G₂ and a homomorphism g: G₂ → G3, then there also exists a homomorphism h : G₁ → G3. Using the proof, then find the corresponding homomorphisms for the following graphs: P y 9 Ž G₁ I U V t E G₂ 7 B a G₁
Transcribed Image Text:4. Let G = (V, E) and G' = (V', E') be simple graphs. We say that a graph G is homomorphic to a graph G' if there exists a mapping : V → V' such that for each edge {x,y} € E of the graph G it follows that {o(r), (y)} € E'. Such a mapping is called a homomorphism of the graph G onto the graph G', and is denoted by ø: G → G'. Show that for any three graphs G₁, G2, G3, if there exists a homomorphism f: G₁ → G₂ and a homomorphism g: G₂ → G3, then there also exists a homomorphism h : G₁ → G3. Using the proof, then find the corresponding homomorphisms for the following graphs: P y 9 Ž G₁ I U V t E G₂ 7 B a G₁
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