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Graph Neighborhood


The graph neighborhood of a vertex v in a graph is the set of all the vertices adjacent to v including v itself. More generally, the ith neighborhood of v is the set of all vertices that lie at the distance i from v.

The subgraph induced by the neighborhood of a graph from vertex v is called the neighborhood graph.

Note that while "graph neighborhood" generally includes vertices adjacent to v together with the vertex v itself, the term "graph neighbors" generally means vertices adjacent to v excluding v itself (e.g., Brouwer et al. 1989), so some care is needed when encountering these terms.


See also

Dominating Set, Graph Distance Graph, Irredundant Set, Local Graph, Neighborhood, Neighborhood Graph

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References

Brouwer, A. E.; Cohen, A. M.; and Neumaier, A. Distance-Regular Graphs. New York: Springer-Verlag, 1989.

Referenced on Wolfram|Alpha

Graph Neighborhood

Cite this as:

Weisstein, Eric W. "Graph Neighborhood." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GraphNeighborhood.html

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