Identifying codes in ( random ) geometric networks ∗ ( Technical report version )

It is important for networks built from wireless sensors technology to have a functional location detection system. Identifying codes, first introduced to model fault diagnosis of multi-processor systems, have recently proved to be useful to address this question. We are interested in the situation where the area of communication of each sensor is modelled by a disk: thus we consider identifying codes for the class of unit disk graphs. Minimising the size of an identifying code is NP-complete even for bipartite graphs. First, we improve this result by showing that the problem remains NP-complete for bipartite planar unit disk graphs. Then, we address the question of the existence of an identifying code for random unit disk graphs. From a practical point of view, this corresponds to the case when sensors are randomly thrown on a plane. We derive the probability that there exists an identifying code as a function of the radius of the disks. The results obtained are in sharp contrast with those concerning random graphs in the Erdős and Rényi model. ∗This work was partially supported by the European project ist fet Aeolus. †European Institute for Stochastics (EURANDOM), Technical University Eindhoven, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. E-mail: [email protected]. ‡Institute for Theoretical Computer Science (ITI), Faculty of Mathematics and Physics, Charles University, Malostranské Náměst́ı 25, 118 00 Prague, Czech Republic. This author is supported by the European project ist fet Aeolus. E-mail: [email protected].