# Definition:Adjacent (Graph Theory)

## 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).