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Introduction. In my (non-mathematical) research I'm regularly encountering graphs, and the need to 'refine' graphs by mapping them to other graphs. Sometimes the mapping is injective on vertices, sometimes it isn't. Often a few edges in one graph $G$ are 'refined' into a path in the graph $G'$ that $G$ is mapped to, as in this example.

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Sometimes the whole graph is 'refined' as in these two examples:

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Sometimes a vertex is 'refined' into a graph like so:

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The main requirement is that if there is a part from vertex $v$ to vertex $w$ in graph $G$, then there must also be a path from whatever $v$ refines to to whatever $w$ refines to.

Questions. I'm interested in algebraic properties of such graph refinements in a general sense: for example what happens to the refinements when I form products, sums, function-spaces etc of graphs? What kinds of paths in graphs and their refinements do we get when applying algebraic operations? I have my own definitions in my specific use case. Now I want to compare what I have done with existing mathematical work. Since refining graphs is a natural operation, I'm sure related matters have been studied in other contexts. I wonder where.

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