The neighborhood graph of a given graph from a vertex v is the subgraph induced by the neighborhood of a graph from vertex v, most commonly including v itself. Such graphs are sometimes also known in more recent literature as ego graphs or ego-centered networks. A graph G for which the neighborhood graph at each point excluding the point itself is isomorphic to a graph H is said to be a local H graph, or simply "locally H." Neighborhood graphs are implemented in the Wolfram Language as NeighborhoodGraph[g, v].
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