Discrete Mathematics
Fault-tolerant locating-dominating sets
Discrete Mathematics
Identifying and locating-dominating codes on chains and cycles
European Journal of Combinatorics
European Journal of Combinatorics
On locating-dominating sets in infinite grids
European Journal of Combinatorics
Locating sensors in paths and cycles: The case of 2-identifying codes
European Journal of Combinatorics
On a new class of codes for identifying vertices in graphs
IEEE Transactions on Information Theory
Identifying codes and locating-dominating sets on paths and cycles
Discrete Applied Mathematics
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Assume that G=(V,E) is a simple undirected graph, and C is a nonempty subset of V. For every vV, we define Ir(v)={uC| dG(u,v)≤r}, where dG(u,v) denotes the number of edges on any shortest path between u and v. If the sets Ir(v) for are pairwise different, and none of them is the empty set, we say that C is an r-locating-dominating set in G. It is shown that the smallest 2-locating-dominating set in a path with n vertices has cardinality [(n+1)/3], which coincides with the lower bound proved earlier by Bertrand, Charon, Hudry and Lobstein. Moreover, we give a general upper bound which improves a result of Bertrand, Charon, Hudry and Lobstein.