Let G and H be graphs. A homomorphism φ : G → H is a map φ : V(G) → V(H) which preserves edges, that is, {x, y} ∈ E(G) ⇒ {φ(x), φ(y)} ∈ E(H). We write G → H if there is a homomorphism φ : G → H.
Let G and H be graphs. The graph H is a retract of G if there exist homomorphisms ρ : G → H and γ : H → G such that ρ ◦ γ = idH, which is the identity map V(H) → V(H). The functions ρ and γ are called the retraction and co-retraction, respectively.
How can we show that when H is a retract of G then H is isomorphic to an induced subgraph of G.