TITLE: A name for layered directed graph as in a fully-connected neural network
QUESTION [6 upvotes]: Is there a common name for the type of directed acyclic graph structure used in fully-connected neural networks?
E.g.

To be specific, I'm speaking of a directed graph, $G=(V,E)$, whose vertices can be partitioned into $k$ non-empty disjoint sets $V_1, ..., V_k$ such that $V = \displaystyle\bigcup_{i=1}^k V_i$ and $E = \displaystyle\bigcup_{i=1}^{k-1} V_i\times V_{i+1}$.

REPLY [2 votes]: For any digraphs $A$ and $B$ a map $f$ is referred to as a full homomorphism from $A$ to $B$ iff there is a homomorphism from $A$ to $B$ satisfying $\small \forall u,v\in V(A)((a,b)\not\in E(A)\implies (f(u),f(v))\not\in E(B))$, this is fairly common notation (e.g. here are several papers using this terminology 1,2,3), so naturally when there exists a full homomorphism from one digraph to another digraph then the first digraph is said to be fully homomorphic to the second one. Now with that said the digraphs you are describing are exactly those that are fully epimorphic to directed paths. 
To see why note if any digraph $G=(V,E)$ has a surjective full homomosphism to any directed path $P=(U,R)$ where we have $U=\{1,2,3,\ldots n\}$ and $R=\{(1,2),(2,3),(3,4),\ldots (n-1,n)\}$ then there must exist a surjection $f:\{1,2,3,\ldots n\}\to V$ satisfying $(i,j)\in R\iff (f(i),f(j))\in E$ thus we know $\small E=\{(f(i),f(j)):(i,j)\in R\}=\bigcup_{(i,j)\in R}f^{-1}[\{i\}]\times f^{-1}[\{j\}]=\bigcup_{k=1}^{n-1}f^{-1}[\{k\}]\times f^{-1}[\{k+1\}]$ where $f$ is a surjection and all the fibers $\small \{f^{-1}[\{1\}],f^{-1}[\{2\}],\ldots f^{-1}[\{n\}]\}=\text{coim}(f)$ must partition $V$.