Abstract
Let G be a simple, undirected graph with vertex set V. For v ∈ V and r ≥ 1, we denote by B G, r (v) the ball of radius r and centre v. A set \({\mathcal C} \subseteq V\) is said to be an r-identifying code in G if the sets \(B_{G,r}(v)\cap {\mathcal C}\), v ∈ V, are all nonempty and distinct. A graph G admitting an r-identifying code is called r-twin-free, and in this case the size of a smallest r-identifying code in G is denoted by γ r (G). We study the following structural problem: let G be an r-twin-free graph, and G ∗ be a graph obtained from G by adding or deleting an edge. If G ∗ is still r-twin-free, we compare the behaviours of γ r (G) and γ r (G ∗), establishing results on their possible differences and ratios.
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References
Berge, C.: Graphes, Gauthier-Villars: Paris, 1983. English translation: Graphs. North-Holland Publishing Co., Amsterdam (1985)
Bertrand, N.: Codes Identifiants et Codes Localisateurs-Dominateurs sur Certains Graphes, Mémoire de Stage de Maîtrise, 28 pp. ENST, Paris (2001)
Bertrand, N., Charon, I., Hudry, O., Lobstein, A.: Identifying and locating-dominating codes on chains and cycles. Eur. J. Comb. 25, 969–987 (2004)
Charon, I., Honkala, I., Hudry, O., Lobstein, A.: Structural properties of twin-free graphs. Electron J. Comb. 14(1), R16 (2007)
Charon, I., Honkala, I., Hudry, O., Lobstein, A.: Minimum sizes of identifying codes in graphs differing by one vertex. Cryptogr. Commun.–Discret. Struct. Boolean Funct. Seq. 5, 119–136 (2013)
Charon, I, Hudry, O, Lobstein, A: Extremal cardinalities for identifying and locating-dominating codes in graphs, Technical Report Télécom Paris-2003D006, 18 pp. Paris (2003)
Charon, I., Hudry, O., Lobstein, A.: On the structure of identifiable graphs. Electron. Notes Discret. Math. 22, 491–495 (2005)
Charon, I., Hudry, O., Lobstein, A.: Possible cardinalities for identifying codes in graphs. Australas. J. Comb. 32, 177–195 (2005)
Charon, I., Hudry, O., Lobstein, A.: Extremal cardinalities for identifying and locating-dominating codes in graphs. Discret. Math. 307, 356–366 (2007)
Chen, C., Lu, C., Miao, Z.: Identifying codes and locating-dominating sets on paths and cycles. Discret. Appl. Math. 159, 1540–1547 (2011)
Diestel, R: Graph Theory. Springer, Berlin (2005)
Foucaud, F., Guerrini, E., Kovše, M., Naserasr, R., Parreau, A., Valicov, P.: Extremal graphs for the identifying code problem. Eur. J. Comb. 32, 628–638 (2011)
Frick, M., Fricke, G.H., Mynhardt, C.M., Skaggs, R.D.: Critical graphs with respect to vertex identification. Util. Math. 76, 213–227 (2008)
Gravier, S., Moncel, J.: On graphs having a V ∖ {x} set as an identifying code. Discret. Math. 307, 432–434 (2007)
Honkala, I.: An optimal edge-robust identifying code in the triangular lattice. Ann. Comb. 8, 303–323 (2004)
Honkala, I.: On 2-edge-robust r-identifying codes in the king grid. Australas. J. Comb. 36, 151–165 (2006)
Honkala, I., Laihonen, T.: On identifying codes that are robust against edge changes. Inf. Comput. 205, 1078–1095 (2007)
Karpovsky, M.G., Chakrabarty, K., Levitin, L.B.: On a new class of codes for identifying vertices in graphs. IEEE Trans. Inf. Theory IT-44, 599–611 (1998)
Laihonen, T.: Optimal t-edge-robust r-identifying codes in the king lattice. Graphs Comb. 22, 487–496 (2006)
Laihonen, T.: On edge-robust (1, ≤ ℓ)-identifying codes in binary Hamming spaces. Int. J. Pure Appl. Math. 36, 87–102 (2007)
Lobstein, A.: A bibliography on watching systems, identifying, locating-dominating and discriminating codes in graphs. http://perso.telecom-paristech.fr/~lobstein/debutBIBidetlocdom.pdf
Roberts, D.L., Roberts, F.S.: Locating sensors in paths and cycles: the case of 2-identifying codes. Eur. J. Comb. 29, 72–82 (2008)
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Charon, I., Honkala, I., Hudry, O. et al. Minimum sizes of identifying codes in graphs differing by one edge. Cryptogr. Commun. 6, 157–170 (2014). https://doi.org/10.1007/s12095-013-0094-x
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DOI: https://doi.org/10.1007/s12095-013-0094-x