Directed edge

In graph theory, a directed edge (or arc) ^[1][2] is an edge that connects an ordered pair of vertices, from a tail to a head. Directed edges are the defining feature of a directed graph. Its counterpart is the undirected edge, which connects the same two vertices without assigning a direction.

Definition

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Definition 1 (Directed edge)

Let $G=(V,E)$ be a graph with vertex set $V$ and edge set $E$. A directed edge (or arc) is an ordered pair $(u,v)\in E$ of vertices $u,v\in V$, where $u$ is called the tail and $v$ the head of the edge.

Notation

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A directed edge from $u$ to $v$ is written $(u,v)$ or $u\to v$. Unlike the undirected edge $\{u,v\}$, order matters: $(u,v)\ne (v,u)$ unless $u=v$.

References

1. [1]

   “Directed graph”, Wikipedia, Available: https://en.wikipedia.org/wiki/Directed_graph, Accessed: 2026-07-14 ↩

2. [2]

   “Directed Edge”, Wolfram MathWorld, Available: https://mathworld.wolfram.com/DirectedEdge.html, Accessed: 2026-07-14 ↩