Discrete Mathematics - Topics on domination
Unit disk graph recognition is NP-hard
Computational Geometry: Theory and Applications - Special issue on geometric representations of graphs
New versions of Suen's correlation inequality
proceedings of the eighth international conference on Random structures and algorithms
Wireless sensor networks for habitat monitoring
WSNA '02 Proceedings of the 1st ACM international workshop on Wireless sensor networks and applications
Orthogonal Drawing of High Degree Graphs with Small Area and Few Bends
WADS '97 Proceedings of the 5th International Workshop on Algorithms and Data Structures
2-Visibility Drawings of Planar Graphs
GD '96 Proceedings of the Symposium on Graph Drawing
Minimizing the size of an identifying or locating-dominating code in a graph is NP-hard
Theoretical Computer Science
On robust and dynamic identifying codes
IEEE Transactions on Information Theory
Joint Monitoring and Routing in Wireless Sensor Networks Using Robust Identifying Codes
Mobile Networks and Applications
Locating and identifying codes in circulant networks
Discrete Applied Mathematics
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We model a problem about networks built from wireless devices using identifying and locating–dominating codes in unit disk graphs. It is known that minimizing the size of an identifying code is -complete even for bipartite graphs. First, we improve this result by showing that the problem remains -complete for bipartite planar unit disk graphs. Then, we address the question of the existence of an identifying code for random unit disk graphs. We derive the probability that there exists an identifying code as a function of the radius of the disks, and we find that for all interesting ranges of r this probability is bounded away from one. The results obtained are in sharp contrast to those concerning random graphs in the Erdős–Rényi model. Another well-studied class of codes is that of locating–dominating codes, which are less demanding than identifying codes. A locating–dominating code always exists, but minimizing its size is still -complete in general. We extend this result to our setting by showing that this question remains -complete for arbitrary planar unit disk graphs. Finally, we study the minimum size of such a code in random unit disk graphs, and we prove that with probability tending to one, it is of size (n/r)2/3+o(1) if r ≤ /2−ϵ is chosen such that nr2 → ∞, and of size n1+o(1) if nr2 ≪ lnn.