Definition:Directed Graph

From ProofWiki
Jump to navigation Jump to search

Definition

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

Digraph.png

In the above graph, the vertices are $v_1, v_2, v_3$ and $v_4$.


Arc

Let $G = \struct {V, E}$ be a digraph.

The arcs are the elements of $E$.

Informally, an arc is a line that joins one vertex to another.


In the above graph, the arcs are $v_1 v_2, v_2 v_4, v_4 v_3, v_4 v_1$ and $v_1 v_4$.


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.


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

Let $D = \struct {V, E}$ be a digraph such that the relation $E$ in $D$ is symmetric.

Then $D$ is called a symmetric digraph.


Simple Digraph

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

That is, in a simple digraph there are no pairs of arcs (like there are between $v_1$ and $v_4$ in the diagram above) which go in both directions between two vertices.


Examples

Arbitrary Digraph of Order 6

Let $D = \struct {V, E}$ be a directed graph such that:

$V = \set {v_1, v_2, v_3, v_4, v_5, v_6}$
$E = \set {\tuple {v_1, v_3}, \tuple {v_2, v_3}, \tuple {v_3, v_4}, \tuple {v_4, v_1}, \tuple {v_4, v_3}, \tuple {v_5, v_6} }$


Then $D$ can be presented in diagram form as:

Chartrand-exercise-1-5-32.png


Sources