Definition:Adjacent (Graph Theory)

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Definition

Vertices

Undirected Graph

Let $G = \struct {V, E}$ be an undirected graph.

Two vertices $u, v \in V$ of $G$ are adjacent if there exists an edge $e = \set {u, v} \in E$ of $G$ to which they are both incident.


Digraph

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

Two vertices $u, v \in V$ of $G$ are adjacent if there exists an arc $e = \left({u, v}\right) \in E$ of $G$ to which they are both incident.


Edges

Undirected Graph

Let $G = \struct {V, E}$ be an undirected graph.

Two edges $e_1, e_2 \in E$ of $G$ adjacent if there exists a vertex $v \in V$ to which they are both incident.


Digraph

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

Two arcs $e_1, e_2 \in E$ of $G$ adjacent if there exists a vertex $v \in V$ to which they are both incident.


Faces

Let $G = \left({V, E}\right)$ be a planar graph.

Two faces of $G$ are adjacent if and only if they are both incident to the same edge (or edges).