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In [1] it is said that a graph homomorphism is a mapping between two graphs, that is, between their vertices, where the edges are preserved.

A mapping is a specific binary relation where any vertex in a graph has a unique image in the other graph.

Why must the binary relation be a mapping in the graph homomorphism definition ? Cannot it be simply a binary relation ?

In the image below is depicted what would happen if a binary relation $\mathcal{R}$ is not a mapping, which I find sound.

Graph homomorphism with the relation $\mathcal{R}$

[1] https://www.proofwiki.org/wiki/Definition:Homomorphism_%28Graph_Theory%29

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You can define things differently. Nothing wrong with that. Just realize that you get different theorems. And some definitions are less equal than others.

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