Consider an undirected graph. It's obvious what is a vertex (≙ object) and what is an edge (≙ fact).
Now "objectify" the edges (≙ facts) by adding an extra vertex along every edge.
In general, in the "(edge-)objectified" graph it's not determined anymore which vertices correspond to "objects" and which vertices correspond to "facts", e.g. in objectified cycle graphs. But sometimes it is, e.g. in objectified path or star graphs.
In the case of cycle graphs, the objectified graph can nevertheless be "un-objectified" uniquely.
How can graphs be characterized
for which "objects" and "facts" can be distinguished in their "objectified" graph (like in path and star graphs)
that can be "un-objectified" uniquely (like cycle graphs)
Where can I learn more about this approach? Under which term is it filed?