Locating dominating codes: Bounds and extremal cardinalities

In this work, two types of codes such that they both dominate and locate the vertices of a graph are studied. Those codes might be sets of detectors in a network or processors controlling a system whose set of responses should determine a malfunctioning processor or an intruder. Here, we present our contributions on @l-codes and @h-codes concerning bounds, extremal values and realization theorems.

[1]  Frédéric Maffray,et al.  Locating-domination and identifying codes in trees , 2007, Australas. J Comb..

[2]  Peter J. Slater,et al.  Open neighborhood locating-dominating in trees , 2011, Discret. Appl. Math..

[3]  G. Chartrand,et al.  Graphs & Digraphs , 1986 .

[4]  W. Imrich,et al.  Handbook of Product Graphs, Second Edition , 2011 .

[5]  Antoine Lobstein,et al.  Possible cardinalities for locating-dominating codes in graphs , 2006, Australas. J Comb..

[6]  Tero Laihonen,et al.  Improved Upper Bounds on Binary Identifying Codes , 2007, IEEE Transactions on Information Theory.

[7]  David R. Wood,et al.  Extremal Graph Theory for Metric Dimension and Diameter , 2007, Electron. J. Comb..

[8]  Michael A. Henning,et al.  Metric-Locating-Dominating Sets in Graphs , 2004, Ars Comb..

[9]  Iiro S. Honkala,et al.  An optimal locating-dominating set in the infinite triangular grid , 2006, Discret. Math..

[10]  Changhong Lu,et al.  Identifying codes and locating-dominating sets on paths and cycles , 2011, Discret. Appl. Math..

[11]  Nathalie Bertrand,et al.  Identifying and locating-dominating codes on chains and cycles , 2004, Eur. J. Comb..

[12]  Mercè Mora,et al.  Nordhaus-Gaddum bounds for locating domination , 2014, Eur. J. Comb..

[13]  Mark G. Karpovsky,et al.  On a New Class of Codes for Identifying Vertices in Graphs , 1998, IEEE Trans. Inf. Theory.

[14]  Antoine Lobstein,et al.  Extremal cardinalities for identifying and locating-dominating codes in graphs , 2007, Discret. Math..

[15]  Tero Laihonen,et al.  Locating-dominating codes in paths , 2011, Discret. Math..

[16]  Mark G. Karpovsky,et al.  On robust and dynamic identifying codes , 2006, IEEE Transactions on Information Theory.

[17]  Ari Trachtenberg,et al.  Identifying Codes and Covering Problems , 2008, IEEE Transactions on Information Theory.

[18]  Peter J. Slater.  Domination and location in acyclic graphs , 1987, Networks.

[19]  S. Lane,et al.  Trees with Unique Minimum Locating-Dominating Sets. , 2006 .

[20]  Peter J. Slater,et al.  Fundamentals of domination in graphs , 1998, Pure and applied mathematics.

[21]  N. Duncan.  Leaves on trees , 2014 .