Definition:Directed Graph

Informal Definition
A directed graph or digraph is a graph each of whose edges has a direction:



In the above graph, the vertices are $$A, B, C$$ and $$D$$.

Arc
In a directed graph, the lines connecting the vertices are called directed edges or arcs.

In the above graph, the arcs are $$AB, BD, DC, DA$$ and $$AD$$.

As can be seen, in this general definition it is allowable for an arc to go in both directions between a given pair of vertices.

Formal Definition
A directed graph or digraph $$D$$ is a non-empty set $$V$$ together with an antireflexive relation $$E$$ on $$V$$.

The elements of $$E$$ are the arcs.

Thus the above digraph can be defined as:


 * $$D = \left({V, E}\right): V = \left\{{A, B, C, D}\right\}, E = \left\{{\left({A, B}\right), \left({B, D}\right), \left({D, C}\right), \left({D, A}\right), \left({A, D}\right)}\right\}$$

Symmetric Digraph
If the relation $$E$$ in $$D$$ is also symmetric, then $$D$$ is called a symmetric digraph.

It follows from the definition of a (simple) graph that a symmetric digraph whose relation $$E$$ is symmetric is in fact the same thing as an undirected graph.

Simple Digraph
If the relation $$E$$ in $$D$$ is also asymmetric, then $$D$$ is called a simple digraph.

That is, in a simple digraph there are no pairs of arcs (like there are between $$A$$ and $$D$$ in the diagram above) which go in both directions between two vertices.