Geodesic
Iterated neighborhood graphs
Discussiones Mathematicae. Graph Theory, Tome 32 (2012) no. 3, pp. 403-417

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The neighborhood graph N(G) of a simple undirected graph G = (V,E) is the graph (V,E_N) where E_N = a,b | a ≠ b, x,a ∈ E and x,b ∈ E for some x ∈ V. It is well-known that the neighborhood graph N(G) is connected if and only if the graph G is connected and non-bipartite.
Keywords: neighborhood graph, 2-step graph, neighborhood completeness number 
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Sonntag, Martin; Teichert, Hanns-Martin. Iterated neighborhood graphs. Discussiones Mathematicae. Graph Theory, Tome 32 (2012) no. 3, pp. 403-417. http://geodesic.mathdoc.fr/item/DMGT_2012_32_3_a1/