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## Discussiones Mathematicae Graph Theory

2012 | 32 | 3 | 403-417
Tytuł artykułu

### Iterated neighborhood graphs

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Słowa kluczowe
EN
Kategorie tematyczne
Wydawca
Czasopismo
Rocznik
Tom
Numer
Strony
403-417
Opis fizyczny
Daty
wydano
2012
otrzymano
2011-01-12
poprawiono
2011-07-14
zaakceptowano
2011-07-18
Twórcy
autor
• Faculty of Mathematics and Computer Science, Technische Universität Bergakademie Freiberg, D-09599 Freiberg, Germany
• Institute of Mathematics, University of Lübeck, D-23560 Lübeck, Germany
Bibliografia
• [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).
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Bibliografia
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