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:


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

Category-Theoretic Definition
Let $\mathbf{Set}$ be the category of sets.

A digraph is an arrangement of the following form in $\mathbf{Set}$:


 * $\begin{xy}

<0em,0em>*{E} = "E", <5em,0em>*{V} = "V",

"E"+/^.3em/+/r1em/;"V"+/^.3em/+/l1em/ **@{-} ?>*@{>} ?*!/_.6em/{s}, "E"+/_.3em/+/r1em/;"V"+/_.3em/+/l1em/ **@{-} ?>*@{>} ?*!/^.6em/{t}, \end{xy}$

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.