Out-neighborhood

In graph theory, the out-neighborhood ^[1] of a vertex $v$ in a directed graph $G$ is the edge induced subgraph on the set of vertices with a directed edge out of $v$. Its counterpart is the in-neighborhood, the set of vertices with a directed edge into $v$.

Definition

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Definition 1 (Out-neighbor)

Let $G=(V,E)$ be a directed graph. A vertex $u\in V$ is an out-neighbor of a vertex $v\in V$ if

$$(v,u)\in E$$

Definition 2 (Out-neighborhood)

Let $G=(V,E)$ be a directed graph and $v\in V$ a vertex. The out-neighborhood of $v$ is

$$N^{+}(v):=\{u\in V:(v,u)\in E\}$$

Notation

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The out-neighborhood of $v$ is denoted $N^{+}(v)$.

References

1. [1]

   “Directed graph”, Wikipedia, Available: https://en.wikipedia.org/wiki/Directed_graph, Accessed: 2026-07-31 ↩