# Definition: Vertex Degrees for Digraphs

Let $$D=(V,E,\alpha,\omega)$$ be a digraph, $$v\in V$$ be a vertex of $$D$$.

The number $$d_D^+(v)$$ of edges outgoing from $$v$$, i.e. the number of elements of the set $$\delta_D^+(v):=\{e\in E: \alpha(e)=v\}$$, is called the outer degree of $$v$$, formally $$d_D^+(v):=|\delta_D^+(v)|$$.

The number $$d_D^-(v)$$ of edges incoming to $$v$$, i.e. the number of elements of the set $$\delta_D^-(v):=\{e\in E: \omega(e)=v\}$$ is called the inner degree of $$v$$, formally $$d_D^-(v):=|\delta_D^-(v)|$$.

The number $$d_D(v):=d_D^+(v)+d_D^-(v)$$ is called the degree of $$v$$.

Note: In the above definitions, the index $$D$$ can be omitted in the notation, if it is clear from the context, which digraph $$D$$ is concerned.

### Example:

The values of the degrees of vertices in the above graph are:

Vertex $$v$$ Degree $$d(v)$$ Inner Degree $$d^-(v)$$ Outer Degree $$d^+(v)$$
$$a$$ $$5$$ $$2$$ $$3$$
$$b$$ $$6$$ $$4$$ $$2$$
$$c$$ $$4$$ $$1$$ $$3$$
$$d$$ $$3$$ $$1$$ $$2$$
$$e$$ $$4$$ $$3$$ $$1$$
$$f$$ $$0$$ $$0$$ $$0$$

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