Iterated neighborhood graphs
Discussiones Mathematicae Graph Theory, Tome 32 (2012) no. 3, p. 403.

Voir la notice de l'article dans European Digital Mathematics Library

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. We present some results concerning the k-iterated neighborhood graph N k ( G ) : = N ( N ( . . . N ( G ) ) ) of G. In particular we investigate conditions for G and k such that N k ( G ) becomes a complete graph.
Classification : 05C12, 05C76
Mots-clés : neighborhood graph, 2-step graph, neighborhood completeness number 
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Martin Sonntag; Hanns-Martin Teichert. Iterated neighborhood graphs. Discussiones Mathematicae Graph Theory, Tome 32 (2012) no. 3, p. 403. https://geodesic-test.mathdoc.fr/item/DMGT_2012__32_3_270787/