Defining outgoing edge set of a graph This is pretty much the same question as Is there a standard notation for the sets of incoming and outgoing edges from a vertex? which does not have an answer, but does have comments.
As I understand, one comment is referring to the degree, and $\delta^+(v)$ is the number of edges, not the set of edges. The other comment mentions to use $E^+(v)$, which I have not been able to find/see in any textbooks, compared to say $N^+_G(v)$ to represent the neighborhood of $v$.  This seems like there is a standard notation for the set of vertices connected to the outgoing edges of $v$, but not one for the edges themselves?
I was trying to formulate the notation for the set of outgoing edges as:
(Existing basic definitions)
$G := (V,E,w)$
$V := \{v_i,\dots,v_n \}$
$E := \{e_{ij}, \dots,e_k ~| ~e_{ij} = (v_i,v_j) \}$
(Then outgoing edge set from neighborhood)
$ E^+_G(v_i) := \{ e_{ij} | ~v_j \in N^+_G(v_i) \}$
Question; is this definition reasonable as is, or does it seem too verbose if the symbol is already well enough defined (or is there an error)?
Side question; Is the use of $N^+(v)$ special in any way, or if $E^+(v)$ meant outgoing edges, why not use $V^+(v)$ for outgoing vertices?
 A: For my graph theory class, we used $\delta^+(X)$ to be the set of edges leaving $X$ and $\delta^-(X)$ to be the set of edges coming into $X$ ($X\subseteq V(D)$ and can be a singleton). For your side question, I'm not sure what you mean by "outgoing vertices". I have never seen direction added to vertices.
A: In "Modern Graph Theory" by Bollabas, the chapter on "Flows in Directed Graphs" gives a helpful notation for this:
Given two subsets X, Y of V, we write $\vec{E}(X,Y)$ for the set of directed X-Y edges:
$\vec{E}(X,Y) = \{\vec{xy} \in \vec{E} : x\in X, y\in Y \}$
If $S$ is a subset of $V$ containing $s$ but not $t$ then $\vec{E}(S,\bar{S})$ is called a cut separating $s$ from $t$.... If we delete the edges of a cut then no positive-valued flow from $s$ to $t$ can be defined on the remainder.
From the original and linked question, the idea of using $\delta^+$ was only questionable because I had seen it defined as the "minimum out-degree".  Based on all of these references:

*

*"Graduate Texts in Mathematics: Graph Theory, J.A Bondy, U.S.R Murty"

*"A first course in Graph Theory. Gary Chartrand and ping Zhang"

*"Classes of Directed Graphs. Jorgen Bang-Jensen, Gregory Gutin"

*"Graph Theory by Diestel"

*"Modern Graph Theory by Bollabas"

I don't think it would be good to redefine a ubiquitous term within the same context.
