n-partite graphs with partially conjoint sets of nodes? As a non-mathematician, I'm looking for the term for an n-partite graph whose nodes fall into different "parties" (i.e., are n-partite if I understand the term correctly) which, however, do not form disjoint sets (which I guess makes them non-n-partite). I'll try to give an example in layman speak. $G$ is a directed graph $= (U, V, W, E)$. $U$, $V$, and $W$ are what I have tried to described as nodes falling into different "parties", i.e., they are of different types/classes. $E$ can contain edges $\\{u,v\\}$, and edges $\\{v,w\\}$ (at this stage we'd have a tri-partite graph, right?), but also edges $\\{u,u\\}$. The three "types" of edges also represent different (real world) relations between the nodes. Please excuse any misuse of terminology and notation. Is there a term for these types of graph, or are they just directed graphs, nevermind the specific properties of nodes and edges, etc.?

If we were to analyze your example, it would probably be sufficient to note that the following characterize it:

(1) The subgraph induced by $U$ can be **any** directed graph

(2) If we contract all the edges connecting nodes of $U,$ then the resulting graph is a graded directed graph (a specific type of bipartite graph).

Since condition (1) is fully general, I'm doubtful there will be a named class smaller than the class of all directed graphs which you could use to study this type of graph.

Also, note that there is no guarantee that decomposing your graph like I did with (1) and (2) will suffice to prove what you want. You'd also have to prove that whatever you're trying to show is preserved by that decomposition.