The usual definition of a vertex-induced subgraph goes like this:
A vertex-induced subgraph1 is a subset of the vertices of a graph $G$ together with any edges with both endpoints in this subset.
For itself a subgraph is just a graph which can be embedded into some other graphs which it is a subgraph of.
I wonder if the following weaker (and more general) definition of a subgraph has been investigated before (and been found fruitful):
A vertex-induced subgraph2 is a subset of the vertices of a graph $G$ together with any edges with at least one endpoint in this subset.
In the case of directed graphs, this definition gives rise to interesting features a subgraph2 $G_2$ can have, but not a a subgraph1. These features have to do with the existence of input and output nodes:
$i$ is an input node of $G_2$ when there is an edge $(n,i)$ in $G_2$ with node $n$ not in $G_2$
$o$ is an output node of $G_2$ when there is an edge $(o,n)$ in $G_2$ with node $n$ not in $G_2$.
An interesting feature might be being a source/sink, i.e. not containing an input/output node.
Of course, a subgraph2 is not a graph according to the standard definition of a graph. But it would be equivalent to an "input/output graph" which can be defined as a graph together with two functions $in$ and $out$ assigning to each vertex a number of (virtual) input and output edges.
Once again my question:
Has the above definition of subgraphs2 been investigated before?
Alternatively: Has the above definition of input/output graphs been investigated before?
If so, under which name?