The usual definition of a vertex-induced subgraph goes like this:

A vertex-induced subgraph

_{1}is a subset of the vertices of a graph $G$ together with any edgeswith both endpointsin 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 subgraph

_{2}is a subset of the vertices of a graph $G$ together with any edgeswith at least one endpointin this subset.

In the case of directed graphs, this definition gives rise to interesting features a subgraph_{2} $G_2$ can have, but not a a subgraph_{1}. 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 subgraph_{2} 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 subgraphs

_{2}been investigated before?Alternatively: Has the above definition of input/output graphs been investigated before?

If so, under which name?

boundariesof vertex sets, in the sense of sets of edges withexactlyone end in the given vertex set, and of course you can cobble together a "definition" from this concept of 'boundary' and the usual 'ordinary subgraph'. (2) when working with e.g. string diagrams and operads, a concept ofhalf-edgesexists. I have not seen your idea addressed as such. $\endgroup$ – Peter Heinig Aug 10 '17 at 18:22isa graph, and a problem which is not completely trivial involving that definition". * Define a "1-indirect vertex induced subgraph of graph $G = (V,E)$ induced by $U\subset V$ to have edge-set $F$ of exactly those edges with at least one vertex in $U$, and vertex set $W$ consisting of those vertices touched by a member of $F$. * The nontrivial problem is: Find a graph $G$ and subset $U$ of its vertices such that the 1-indirect vertex-induced subgraph of $G$ by $U$ has greater chromatic number than that of the vertex-induced subgraph of $G$ by $U$. $\endgroup$ – Mark Fischler Aug 10 '17 at 18:50