Directed graphs

Abbreviation: DiGraph

Definition

A directed graph (or digraph for short) is a structure $\mathbf{G}=\langle G,E\rangle$ such that

$E$ is binary relation on $G$: $E\subseteq G\times G$

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It is not unusual to give several (equivalent) definitions. Ideally, one of the definitions would give an irredundant axiomatization that does not refer to other classes.

Morphisms

Let $\mathbf{G}$ and $\mathbf{H}$ be directed graphs. A morphism from $\mathbf{G}$ to $\mathbf{H}$ is a function $h:G\rightarrow H$ that preserves $E$: $\langle x,y\rangle\in E^{\mathbf G}\Longrightarrow \langle h(x), h(y)\rangle\in E^{\mathbf H}$

Definition

An is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle ...\rangle$ such that

$...$ is …: $axiom$

$...$ is …: $axiom$

Examples

Example 1:

Basic results

Properties

Finite members

$\begin{array}{lr} f(1)= &1\\ f(2)= &\\ f(3)= &\\ f(4)= &\\ f(5)= &\\ \end{array}$ $\begin{array}{lr} f(6)= &\\ f(7)= &\\ f(8)= &\\ f(9)= &\\ f(10)= &\\ \end{array}$

Subclasses

[[...]] subvariety
[[...]] expansion

Superclasses

[[...]] supervariety
[[...]] subreduct

References