In some approaches to operads and properads, categories of trees and graphs are used as "indexing categories" for this structure (see, for instance, https://arxiv.org/abs/0902.1954 or https://arxiv.org/abs/1705.08546). An essential class of morphisms in these categories are face and degeneracy maps which generalize the face and degeneracy maps of the simplex category $\Delta$. Just as with the simplex category, these generate all the morphisms of these tree and graph categories. Moreover, they are precisely maps which describe edge contraction and vertex "insertion."
A simple example is to think about a degeneracy map of the simplex category $\delta_i\colon\{0<1<...<n\}\to \{0<1<...<n<n+1\}$ which is the order preserving injection that does not have $i$ in its image. If you think about $\{0<1<...<n\}$ as a graph, this "looks like" inserting vertex between $i-1$ and $i$. In the other direction, the face maps $\sigma_i\colon\{0<1<\ldots<n+1\}\to\{0<\ldots<n\}$ are given by taking $i$ and $i+1$ to $i$, which "looks like" contracting $i$ and $i+1$ into a single vertex.
In graph theory, if I understand correctly, there are similar notions of "edge contraction" and "vertex cleaving." It appears to me that these don't give sensible graph homomorphisms (in the usual sense) because they don't preserve incidence. Nonetheless, they are a relatively natural way to think about going from one graph to another.
Is there a notion of a category of graphs, that graph theorists think about, that has these things as morphisms? The tree and graph categories I mentioned above don't seem to quite fit the bill, as they allow, for instance, edges with no vertices. They also seem to have a lot of labels flying around of both edges and vertices.
EDIT: To clarify, I mean that the category should have composites of such things as its morphisms, not only these kinds of things.
