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Discussiones Mathematicae Graph Theory 32(3) (2012) 403-417
DOI: https://doi.org/10.7151/dmgt.1610

Iterated Neighborhood Graphs

Martin Sonntag Faculty of Mathematics and Computer Science Technische Universit{ä}t Bergakademie Freiberg D-09599 Freiberg, Germany

Hanns-Martin Teichert Institute of Mathematics University of L{ü}beck D--23560 Lübeck, Germany

Abstract

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.

Keywords: neighborhood graph, 2-step graph, neighborhood completeness number

2010 Mathematics Subject Classification: 05C76, 05C12.

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Received 12 January 2011
Revised 14 July 2011
Accepted 18 July 2011