# A commonly-used symbol for the set of all heads of directed edges?

Let $G = (V, E)$ be a digraph, and consider the set $\left\{ v \in V \; \middle| \; \left(\{v\} \times V \right) \cap E \neq \varnothing \right\}$, the set of vertices which form edges with other vertices; or if you will, the set of "heads of edges".

Is there a commonly-used term for this set?

In Graphentheoretische Konzepte und Algorithmen by S. Krumke and H. Noltemeier, they introduce for an edge $e = (u,v)$ two functions

$\qquad\alpha(e) = u$ and
$\qquad\omega(e) = v$.

The inspiration is probably alpha = beginning and omega = ending. I have no idea if they invented it, or where they took it from, or how widely adapted this notation is.

But then, which symbol or name you use for the function does not matter. Just define it -- I doubt anybody would reference something that basic.

Using the standard lifting of functions to sets, we get

$\qquad\displaystyle \alpha(E) = \bigcup_{e \in E} \alpha(e)$

for any set $E$ of edges; similarly for $\omega$.

• So, +1, but with a hand over your heart: Would you submit a paper using $\alpha(e)$ without first defining it? – einpoklum Aug 18 '17 at 20:23
• @einpoklum Of course not! I would define it if Knuth used it, for that matter. You always define your notation! – Raphael Aug 18 '17 at 20:28