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Following the lines of this question Need help with Graph notation for a subgraph , I am trying to define the subset of vertices connected to a specific vertex $v_i$ and need some help working through this process.

In that example the question and answer are using: $$ W_{v_i} = \{v \in V(G) \mid \{v,v_j\} \in E(G)\} $$

First, is it possible to use $V_{v_i}$ instead of $W_{v_i}$ since it is a subset of V based on $v_i$?

Second, for a directed graph, the ordering of the vertices in an edge matter, so would it be correct to use $v_j$ explicitly in set $V(G)$, and $v_i$ to be the first value in the set $E(G)$ so it corresponds with $V_{v_i}$: $$ V_{v_i} = \{v_j \in V(G) \mid \{v_i,v_j\} \in E(G)\} $$

Finally, the other question also defines the edges of a subgraph, although the answer also states there is no convention. From what I can find, the way to define a subgraph is $G[S]$.

Then can there be an edge induced subgraph $G_{v_i}$ defined as:

$G_{v_i} = G[E_{v_i}]$ where $E_{v_i} = \{e \in E(G) \mid v_i \in e\}$

Meaning the subgraph is composed of only the vertices and edge connected to $v_i$. Then to make a subgraph of only the directed edges that move away from $v_i$ (meaning $v_i$ is the starting vertex):

$G_{v_i} = G[E_{v_i}]$ where $E_{v_i} = \{e_{ij} \in E(G) \mid (v_i,v_j) \in e_{ij}\}$

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$V_a$ = { v : {a,v} in E } is the set of vertices directly connected to a.
For directed graphs, there are three such sets,
$V_a$ = { v : (a,v) in E }, $W_a$ = { v : (v,a) in E }, $V_a \cup W_a.$

$G_a$ = ({ v : {a,v} in E }, { {a,v} : {a,v} in E })
The outgoing from a graph is
$G_a$ = ({ v : (a,v) in E }, { (a,v) : (a,v) in E })

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