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"

canbe 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?

subdivide. (This is already mentioned in Tony Huynh's answer, but should be more conspicuous in this thread.) Also, this graph-theoretic kind of 'subdivide' is an instance of the concept ofbarycentric subdivisionin topology. Philosophical note: assuming 'naive realism', the reconstruction-issue is moot: then 'objects' are of different sort than the 'facts' (which I imagine as texts); on the other hand, assuming 'everything is language', there's a point to the above: then, each fact is of the same sort as the objects. $\endgroup$ – Peter Heinig Dec 11 '17 at 8:12