# Definition:Orientation (Graph Theory)

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## Definition

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

Let $H = \struct {V, A}$ be a directed graph.

Then $H$ is an **orientation** of $G$ if and only if both of the following hold:

- $(1): \quad H$ is a simple digraph. That is, $A$ is antisymmetric.
- $(2): \quad \forall x, y \in V: \paren {\set {x, y} \in E \iff \tuple {x, y} \in A \lor \tuple {y, x} \in A}$

That is, $H$ is formed from $G$ by replacing every **edge** of $G$ with an **arc**.

## Also see

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Note that every simple digraph is an **orientation** of exactly one simple graph, but a simple graph may have more than one **orientation**.

## Sources

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