What is the name of this digraph created from other digraphs?

Question

Let $D_1, D_2, ..., D_n$ be digraphs of various sizes and let $C$ be a diagraph with $n$ vertices $\{1,...,n\}$. Construct a new digraph, $D$, whose vertex set is $V(D) = V(D_1) \cup \cdots \cup V(D_n)$ and whose edge set is defined by $u \to v$ if and only if one of the two conditions hold:

$\quad$(a) $u,v \in V(D_i) \text{ and } u \to v \text{ in } D_i$

$\quad$(b) $\exists i \neq j$ such that $u \in V(D_i), v\in V(D_j)\text{ and } i \to j \text{ in } C$.

In other words, $D$ is the digraph obtained by replacing the vertices of $C$ with the diagraphs $D_1,D_2,...,D_n$.

Question: Is there a canonical name/notation for this process? I was digging through the Wikipedia article Graph operations to find something similar, but to no avail (perhaps I just missed it). In particular, I'm interested in when $D_1, D_2, ..., D_n$ are strongly connected and $C$ is transitive. Then the condensation of $D$ is $C$. My idea is to "reverse" the process of condensation.

Answer 1

I finally found a canonical name for this process in this article: Hamiltonian tournaments with the least number of $3$-cycles. The author calls $D = C(D_1,D_2,...,D_n)$ the composition of the $n$ components $M_1,M_2,...,M_n$ with the quotient $C$.

Answer 2

This can be phrased as a "partial graph join" (the graph join is a well-known operation) described by $C$. Indeed, suppose $C$ has the edge $1 \to 2$. Then $D$ has an edge $u \to v$ if $u \in D_1$ and $v \in D_2$; i.e., join all vertices of $D_1$ to vertices of $D_2$, respecting the orientation $1 \to 2$.

Thus the adjacency matrix $A$ of $D$ will be of the following block form, writing $IJ$ for the $ij$th block of the adjacency matrix, $A_i$ for the adjacency matrix of $D_i$, $\mathbf{1}$ for the all-ones matrix, and $\mathbf{0}$ for the all-zeroes matrix:

$$(A)_{IJ} = \begin{cases} A_i &\text{if } I = J = i \\ \mathbf{1} &\text{if } I \to J \in C\\ \mathbf{0} &\text{else }\end{cases}$$

Based off a comment, if we denote the graph join as $G \leftrightarrow H$, then one can extend this by induction to $\leftrightarrow_{j=1}^n G_j$, which is clearly associative. Then if we index the graph created by the procedure in the question as $C(D_1,\dots,D_n)$ one can see why I thought "partial" graph join was a suitable name; on the underlying graphs $G_i$ of the $D_i$:

$$K_n(G_1,\dots G_n) = \hphantom{ }\leftrightarrow_{j=1}^n G_j$$

It is also partial in the sense that this algorithm acts on digraphs, which could be considered "partial graphs" insofar as we can identify (nonloop) $2$-cycles with edges.