Codes Identifying Sets of Vertices

Connecting identifying codes and fundamental bounds

“…The works in [2,[15][16][17] derive upper/lower bounds on size of the minimum identifying codes, with some providing graph constructions based on relating identifying codes to superimposed codes. The work in [17] focuses on random graphs, providing probabilistic conditions for existence together with bounds.…”

“…Many variants of identifying codes are defined and studied in the literature: a robust identifying code [3,10] is resilient to changes in the underlying graph, a (1, l ≥ 0)-identifying code [2,15] uniquely identifies any subset of at most l vertices, a ρ radius identifying code [1] uniquely identifies every vertex using the set of all codewords within distance ρ or less from the vertex, and a dynamic identifying code [3] is a walk whose vertices form an identifying code.…”

“…Identifying codes are also linked to superimposed codes [1,2,[15][16][17], dominating sets [21], locating dominating sets [22], the set cover [19,23] and the test cover problem [19,23] and r-robust identifying codes are linked to error correcting codes with minimum Hamming distance 2r + 1 [10] and the set rmulti-cover problem [23].…”

Localization and identification in networks using robust identifying codes

“…The intimate relationship between identifying codes and super-imposed codes was initially pointed out in [5] and further developed in [12,13], providing optimal constructions and tight bounds on code size. Locating-dominating sets are very similar to identifying codes with the subtle difference that only the vertices not in the locating-dominating set are required to have unique identifying sets [11].…”

“…In radius (ρ > 1) identifying codes [5], the neighborhood of v is redefined to include all vertices at distance ≤ ρ from v. The (1, ≤ l)-identifying codes [12] can simultaneously identify any subset of at most l vertices. Dynamic identifying codes are identifying codes whose codewords form a walk through the graph.…”