If D = (V, A) is a digraph, its competition graph (with loops) CGl(D) has the vertex set V and {u, v} ⊆ V is an edge of CGl(D) if and only if there is a vertex w ∈ V such that (u, w), (v, w) ∈ A. In CGl(D), loops {v} are allowed only if v is the only predecessor of a certain vertex w ∈ V. For several products D1 ⚬ D2 of digraphs D1 and D2, we investigate the relations between the competition graphs of the factors D1, D2 and the competition graph of their product D1 ⚬ D2.
@article{bwmeta1.element.doi-10_7151_dmgt_1851,
author = {Martin Sonntag and Hanns-Martin Teichert},
title = {Products Of Digraphs And Their Competition Graphs},
journal = {Discussiones Mathematicae Graph Theory},
volume = {36},
year = {2016},
pages = {43-58},
zbl = {1329.05252},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_7151_dmgt_1851}
}
Martin Sonntag; Hanns-Martin Teichert. Products Of Digraphs And Their Competition Graphs. Discussiones Mathematicae Graph Theory, Tome 36 (2016) pp. 43-58. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_7151_dmgt_1851/
[1] J. Bang-Jensen and G. Gutin, Digraphs: Theory, Algorithms and Applications (Springer, London, 2001). | Zbl 0958.05002
[2] J.E. Cohen, Interval graphs and food webs: a finding and a problem (Rand Corporation Document 17696-PR, Santa Monica, CA, 1968).
[3] W. Imrich and S. Klavžar, Product Graphs (John Wiley & Sons, Inc., New York, 2000).
[4] S.R. Kim, The competition number and its variants, in: Quo vadis, graph theory?, J. Gimbel, J.W. Kennedy and L.V. Quintas (Eds.), Ann. Discrete Math. 55 (1993) 313–326. doi:10.1016/s0167-5060(08)70396-0[Crossref]
[5] 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.), (IMA 17, Springer, New York, 1989) 221–243. doi:10.1007/978-1-4684-6381-1_9[Crossref]
[6] F.S. Roberts, Competition graphs and phylogeny graphs, in: Graph theory and combinatorial biology, L. Lovász (Ed.), (Proc. Int. Colloqu. Balatonlelle (Hungary) 1996, Bolyai Soc. Math. Studies 7, Budapest, 1999) 333–362. | Zbl 0924.05032
[7] M. Sonntag and H.-M. Teichert, Competition hypergraphs, Discrete Appl. Math. 143 (2004) 324–329. doi:10.1016/j.dam.2004.02.010[Crossref]
[8] M. Sonntag and H.-M. Teichert, Competition hypergraphs of products of digraphs, Graphs Combin. 25 (2009) 611–624. doi:10.1007/s00373-005-0868-9[Crossref] | Zbl 1197.05101