# Graph homomorphisms allowing contractions of edges

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.

(2) Another option might be to embed the category of graphs without loops $$\mathbb G$$ to the category $$\mathbb G^+$$ of graphs which possibly can have loops (but do not have to). Then, you have a monad $$T\colon \mathbb G^+ \to \mathbb G^+$$ which adds all missing loops (I hope that this is a monad :-)). Then your morphisms are just Kleisli morphisms $$A \to T(B)$$ where $$A$$ and $$B$$ are without loops.
The advantage of the second option is that you can still recover the original subcategory $$\mathbb G$$ within $$\mathbb G^+$$.