# Iterated neighborhood graphs

• Volume: 32, Issue: 3, page 403-417
• ISSN: 2083-5892

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## Abstract

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The neighborhood graph N(G) of a simple undirected graph G = (V,E) is the graph $\left(V,{E}_{N}\right)$ 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 ${N}^{k}\left(G\right):=N\left(N\left(...N\left(G\right)\right)\right)$ of G. In particular we investigate conditions for G and k such that ${N}^{k}\left(G\right)$ becomes a complete graph.

## How to cite

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Martin Sonntag, and Hanns-Martin Teichert. "Iterated neighborhood graphs." Discussiones Mathematicae Graph Theory 32.3 (2012): 403-417. <http://eudml.org/doc/270787>.

@article{MartinSonntag2012,
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.},
author = {Martin Sonntag, Hanns-Martin Teichert},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {neighborhood graph; 2-step graph; neighborhood completeness number},
language = {eng},
number = {3},
pages = {403-417},
title = {Iterated neighborhood graphs},
url = {http://eudml.org/doc/270787},
volume = {32},
year = {2012},
}

TY - JOUR
AU - Martin Sonntag
AU - Hanns-Martin Teichert
TI - Iterated neighborhood graphs
JO - Discussiones Mathematicae Graph Theory
PY - 2012
VL - 32
IS - 3
SP - 403
EP - 417
AB - 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.
LA - eng
KW - neighborhood graph; 2-step graph; neighborhood completeness number
UR - http://eudml.org/doc/270787
ER -

## References

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17. [17] P. Schweitzer (Max-Planck-Institute for Computer Science, Saarbrücken, Germany), unpublished script (2010).

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