Abbreviation: DiGraph

A \emph{directed graph} (or \emph{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.

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 $E$: $\langle x,y\rangle\in E^{\mathbf G}\Longrightarrow \langle h(x), h(y)\rangle\in E^{\mathbf H}$

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

$\ldots$ is …: $axiom$

$\ldots$ is …: $axiom$

Example 1:

Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described.

$\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}$

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[[...]] expansion

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[[...]] subreduct