What is the name of this graph-theoretic construction?

Question

I have been thinking about a graph-theoretic construction and was wondering if anyone knew what it was called.

I suspect that this is a construction that is

• already named or closely related to a named construction; and
• valid for all directed acyclic graphs (DAGs).

I'll give an example first and then try to formulate the construction.

Original DAG

Constructed DAG

Nonrigorous description of construction

Below is a description of the construction I am asking about.

Given a DAG, the construction (vaguely) consists of

1. possibly adding one or more new vertices (in the example above, the vertex $X$)

2. possibly modifying the edges such that

   1. the constructed graph is still a DAG
   2. reachability and non-reachability between all original vertices are preserved
   3. the underlying undirected graph of the constructed DAG is a tree (unsure about this one — as @Dániel G. points out, this would be impossible if, say, the subgraph of reachable vertices from some vertex in the original DAG contained a cycle in its underlying graph)

I have not verified that this construction is well-defined (existent/unique) for all DAGs.

The motivation is, in the context where a directed edge from a vertex points to a direct ancestor or a direct dependency, abstracting/inferring "hidden" ancestors or dependencies that "simplify" the overall ancestry or dependence structure.

Related concepts

Based on my search so far, I suspect these concepts may be related to my construction as (vaguely) described above.

1. polytree
2. semilattice