# Graph Isomerism and Isomorphism

Two graphs $G$ and $G^{\prime}$ are said to be graph isomeric if the share the same number of vertices and edges. If there is a graph homomorphism $h \colon G \to G^{\prime}$ between graph isomers which preserves vertex degree, can one conclude that $G$ and $G^{\prime}$ are graph isomorphic?

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I never know what people's conventions for graph homomorphisms are. Are they just maps from vertices to vertices that preserve the relation given by edges? Are vertices considered related to themselves for the purposes of this relation? – Qiaochu Yuan Aug 6 '12 at 17:06
I'm using this definition of homomorphism. – user02138 Aug 6 '12 at 17:21

$$1-2-1 \quad 1-2-1$$
is not isomorphic with $$1-1 \quad 1-2-2-1$$
Oh, I see. I didn't understand how you were mapping the vertices of degree $1$ but it is clear now. – Qiaochu Yuan Aug 6 '12 at 17:12