Definition:Orientation (Graph Theory)

Definition

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

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

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

This page or section has statements made on it that ought to be extracted and proved in a Theorem page. In particular: Extract these into statements on their own pages. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by creating any appropriate Theorem pages that may be needed. To discuss this page in more detail, feel free to use the talk page.

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

Sources

There are no source works cited for this page. Source citations are highly desirable, and mandatory for all definition pages. Definition pages whose content is wholly or partly unsourced are in danger of having such content deleted. To discuss this page in more detail, feel free to use the talk page.