Geodesic
On the existence of a cycle of length at least 7 in a (1,≤ 2)-twin-free graph
Discussiones Mathematicae. Graph Theory, Tome 30 (2010) no. 4, pp. 591-609 Cet article a éte moissonné depuis la source Library of Science

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We consider a simple, undirected graph G. The ball of a subset Y of vertices in G is the set of vertices in G at distance at most one from a vertex in Y. Assuming that the balls of all subsets of at most two vertices in G are distinct, we prove that G admits a cycle with length at least 7.
Keywords: undirected graph, twin subsets, identifiable graph, distinguishable graph, identifying code, maximum length cycle 
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Auger, David; Charon, Irène; Hudry, Olivier; Lobstein, Antoine. On the existence of a cycle of length at least 7 in a (1,≤ 2)-twin-free graph. Discussiones Mathematicae. Graph Theory, Tome 30 (2010) no. 4, pp. 591-609. http://geodesic.mathdoc.fr/item/DMGT_2010_30_4_a5/

[1] D. Auger, Induced paths in twin-free graphs, Electron. J. Combinatorics 15 (2008) N17.

[2] C. Berge, Graphes (Gauthier-Villars, 1983).

[3] C. Berge, Graphs (North-Holland, 1985).

[4] I. Charon, I. Honkala, O. Hudry and A. Lobstein, Structural properties of twin-free graphs, Electron. J. Combinatorics 14 (2007) R16.

[5] I. Charon, O. Hudry and A. Lobstein, On the structure of identifiable graphs: results, conjectures, and open problems, in: Proceedings 29th Australasian Conference in Combinatorial Mathematics and Combinatorial Computing (Taupo, New Zealand, 2004) 37-38.

[6] R. Diestel, Graph Theory (Springer, 3rd edition, 2005).

[7] S. Gravier and J. Moncel, Construction of codes identifying sets of vertices, Electron. J. Combinatorics 12 (2005) R13.

[8] I. Honkala, T. Laihonen and S. Ranto, On codes identifying sets of vertices in Hamming spaces, Designs, Codes and Cryptography 24 (2001) 193-204, doi: 10.1023/A:1011256721935.

[9] T. Laihonen, On cages admitting identifying codes, European J. Combinatorics 29 (2008) 737-741, doi: 10.1016/j.ejc.2007.02.016.

[10] T. Laihonen and J. Moncel, On graphs admitting codes identifying sets of vertices, Australasian J. Combinatorics 41 (2008) 81-91.

[11] T. Laihonen and S. Ranto, Codes identifying sets of vertices, in: Lecture Notes in Computer Science, No. 2227 (Springer-Verlag, 2001) 82-91.

[12] A. Lobstein, Bibliography on identifying, locating-dominating and discriminating codes in graphs, http://www.infres.enst.fr/~lobstein/debutBIBidetlocdom.pdf.

[13] J. Moncel, Codes identifiants dans les graphes, Thèse de Doctorat, Université de Grenoble, France, 165 pages, June 2005.