Let \(D=(V,E,\alpha,\omega)\) be a digraph. A vertex \(v\in V\) and an edge \(e\in E\) are called incident if \(v\in\{\alpha(e),\omega(e)\}\).
Two different edges \(e,e'\in E\) are called adjacent, if there is at least one vertex incident with these edges, formally \( e\neq e'\) and \(\exists v:~ v\in \{\alpha(e),\omega(e)\} \cap \{\alpha(e'),\omega(e')\} \).
Two different vertices \(v,v'\in V\) are neighbours or are called adjacent, if there is at at least one edge incident with these vertices. \( v\neq v'\) and \(\exists e:~v,v'\in \{\alpha(e),\omega(e)\} \).
Let \(v\in V\) be a vertex of \(D\). (Note: In the following definitions, the index \(D\) can be omitted in the notation, if it is clear from the context, which digraph \(D\) is concerned).
The set \(\delta_D^+(v):=\{e\in E: \alpha(e)=v\}\) is called edges outgoing from v.
The set \(\delta_D^-(v):=\{e\in E: \omega(e)=v\}\) is called edges incoming to v.
The set \(N_D^+(v):=\{\omega(e): e\in\delta_D^+(v)\}\) is called successor vertices of v or successors of v.
The set \(N_D^-(v):=\{\alpha(e): e\in\delta_D^-(v)\}\) is called predecessor vertices of v or predecessors of v.
The set \(N_D(v):=N_D^+(v)\cup N_D^-(v)\) is called the neighbours of v.
The values of the degrees of vertices in the above graph are:
| Vertex \(v\) | Neighbours \(N(v)\) | Predecessors \(N^-(v)\) | Successors \(N^+(v)\) |
|---|---|---|---|
| \(a\) | \(\{b,c,d\}\) | \(\{b\}\) | \(\{b,c,d\}\) |
| \(b\) | \(\{a,c\}\) | \(\{a,c\}\) | \(\{a\}\) |
| \(c\) | \(\{a,b\}\) | \(\{a\}\) | \(\{b\}\) |
| \(d\) | \(\{a,e\}\) | \(\{a\}\) | \(\{e\}\) |
| \(e\) | \(\{d,e\}\) | \(\{d,e\}\) | \(\{e\}\) |
| \(f\) | \(\emptyset\) | \(\emptyset\) | \(\emptyset\) |
