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

Voir la notice de l'article provenant de la source Library of Science

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 
@article{DMGT_2012_32_3_a1,
     author = {Sonntag, Martin and Teichert, Hanns-Martin},
     title = {Iterated neighborhood graphs},
     journal = {Discussiones Mathematicae. Graph Theory},
     pages = {403--417},
     publisher = {mathdoc},
     volume = {32},
     number = {3},
     year = {2012},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMGT_2012_32_3_a1/}
}
TY  - JOUR
AU  - Sonntag, Martin
AU  - Teichert, Hanns-Martin
TI  - Iterated neighborhood graphs
JO  - Discussiones Mathematicae. Graph Theory
PY  - 2012
SP  - 403
EP  - 417
VL  - 32
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DMGT_2012_32_3_a1/
LA  - en
ID  - DMGT_2012_32_3_a1
ER  - 
%0 Journal Article
%A Sonntag, Martin
%A Teichert, Hanns-Martin
%T Iterated neighborhood graphs
%J Discussiones Mathematicae. Graph Theory
%D 2012
%P 403-417
%V 32
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DMGT_2012_32_3_a1/
%G en
%F DMGT_2012_32_3_a1
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/

[1] B.D. Acharya and M.N. Vartak, Open neighborhood graphs, Indian Institute of Technology, Department of Mathematics, Research Report No. 7 (Bombay 1973).

[2] J.W. Boland, R.C. Brigham and R.D. Dutton, Embedding arbitrary graphs in neighborhood graphs, J. Combin. Inform. System Sci. 12 (1987) 101-112.

[3] R.C. Brigham and R.D. Dutton, On neighborhood graphs, J. Combin. Inform. System Sci. 12 (1987) 75-85.

[4] R. Diestel, Graph Theory, Second Edition, (Springer, 2000).

[5] G. Exoo and F. Harary, Step graphs, J. Combin. Inform. System Sci. 5 (1980) 52-53.

[6] H.J. Greenberg, J.R. Lundgren and J.S. Maybee, The inversion of 2-step graphs, J. Combin. Inform. System Sci. 8 (1983) 33-43.

[7] S.R. Kim, The competition number and its variants, in: Quo Vadis, Graph Theory?, J. Gimbel, J.W. Kennedy, L.V. Quintas (Eds.), Ann. Discrete Math. 55 (1993) 313-326.

[8] J.R. Lundgren, Food webs, competition graphs, competition-common enemy graphs and niche graphs, in: Applications of Combinatorics and Graph Theory to the Biological and Social Sciences, F. Roberts (Ed.) (Springer, New York 1989) IMA 17 221–243.

[9] J.R. Lundgren, S.K. Merz, J.S. Maybee and C.W. Rasmussen, A characterization of graphs with interval two-step graphs, Linear Algebra Appl. 217 (1995) 203-223, doi: 10.1016/0024-3795(94)00173-B.

[10] J.R. Lundgren, S.K. Merz and C.W. Rasmussen, Chromatic numbers of competition graphs, Linear Algebra Appl. 217 (1995) 225-239, doi: 10.1016/0024-3795(94)00227-5.

[11] J.R. Lundgren and C. Rasmussen, Two-step graphs of trees, Discrete Math. 119 (1993) 123-139, doi: 10.1016/0012-365X(93)90122-A.

[12] J.R. Lundgren, C.W. Rasmussen and J.S. Maybee, Interval competition graphs of symmetric digraphs, Discrete Math. 119 (1993) 113-122, doi: 10.1016/0012-365X(93)90121-9.

[13] M.M. Miller, R.C. Brigham and R.D. Dutton, An equation involving the neighborhood (two step) and line graphs, Ars Combin. 52 (1999) 33-50.

[14] M. Pfützenreuter, Konkurrenzgraphen von ungerichteten Graphen (Bachelor thesis, University of Lübeck, 2006).

[15] F.S. Roberts, Competition graphs and phylogeny graphs, in: Graph Theory and Combinatorial Biology, Proceedings of International Colloquium Balatonlelle (1996), Bolyai Society of Mathematical Studies, L. Lovász (Ed.) (Budapest, 1999) 7, 333–362.

[16] I. Schiermeyer, M. Sonntag and H.-M. Teichert, Structural properties and hamiltonicity of neighborhood graphs, Graphs Combin. 26 (2010) 433-456, doi: 10.1007/s00373-010-0909-x.

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