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Let $G,H$ be graphs with at most one edge between vertices, with loops allowed. A homomorphism of graphs from $G$ to $H$ is a vertex mapping $f\colon V(G)\to V(H)$ such that if $\{u,v\}$ is an edge of $G$, then $\{f(u),f(v)\}$ is an edge of $H$.

I my research, I need a name for a somewhat more general notion: a vertex mapping $f\colon V(G)\to V(H)$ such that if $\{u,v\}$ is an edge of $G$, then either $f(u)=f(v)$ or $\{f(u),f(v)\}$ is an edge. So every edge of $G$ is either mapped to an edge or contracted to a vertex.

Question: Is there (at least a semi-standard) name of such mappings between graphs?

I am aware that this terminological problem commonly solved/handwaved by putting loops on every vertex. However, in my case I need to consider both types of morphisms at the same time, so I cannot do this.

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