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

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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/