The neighborhood graph N(G) of a simple undirected graph G = (V,E) is the graph where = 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 of G. In particular we investigate conditions for G and k such that becomes a complete graph.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1610,
author = {Martin Sonntag and Hanns-Martin Teichert},
title = {Iterated neighborhood graphs},
journal = {Discussiones Mathematicae Graph Theory},
volume = {32},
year = {2012},
pages = {403-417},
zbl = {1257.05143},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1610}
}
Martin Sonntag; Hanns-Martin Teichert. Iterated neighborhood graphs. Discussiones Mathematicae Graph Theory, Tome 32 (2012) pp. 403-417. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1610/
[000] [1] B.D. Acharya and M.N. Vartak, Open neighborhood graphs, Indian Institute of Technology, Department of Mathematics, Research Report No. 7 (Bombay 1973).
[001] [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. | Zbl 0696.05021
[002] [3] R.C. Brigham and R.D. Dutton, On neighborhood graphs, J. Combin. Inform. System Sci. 12 (1987) 75-85. | Zbl 0668.05056
[003] [4] R. Diestel, Graph Theory, Second Edition, (Springer, 2000).
[004] [5] G. Exoo and F. Harary, Step graphs, J. Combin. Inform. System Sci. 5 (1980) 52-53.
[005] [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. | Zbl 0631.05044
[006] [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.
[007] [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.
[008] [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. | Zbl 0821.05045
[009] [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. | Zbl 0821.05024
[010] [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. | Zbl 0790.05023
[011] [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. | Zbl 0790.05036
[012] [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. | Zbl 0977.05111
[013] [14] M. Pfützenreuter, Konkurrenzgraphen von ungerichteten Graphen (Bachelor thesis, University of Lübeck, 2006).
[014] [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. | Zbl 0924.05032
[015] [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. | Zbl 1258.05064
[016] [17] P. Schweitzer (Max-Planck-Institute for Computer Science, Saarbrücken, Germany), unpublished script (2010).