Defining subgraph and subset of directed graph

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

$$V_a$$ = { v : {a,v} in E } is the set of vertices directly connected to a.
$$V_a$$ = { 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 })
$$G_a$$ = ({ v : (a,v) in E }, { (a,v) : (a,v) in E })