Construction of Codes Identifying Sets of Vertices

In this paper the problem of constructing graphs having a (1,≤ `)-identifying code of small cardinality is addressed. It is known that the cardinality of such a code is bounded by Ω ( `2 log ` log n ) . Here we construct graphs on n vertices having a (1,≤ `)-identifying code of cardinality O (`4 log n) for all ` ≥ 2. We derive our construction from a connection between identifying codes and superimposed codes, which we describe in this paper. 1 Codes identifying sets of vertices Let G = (V, E) be a simple, non-oriented graph. For a vertex v ∈ V , let us denote by N [v] the closed neighborhood of v : N [v] = N(v)∪{v}. Let C ⊆ V be a subset of vertices of G, and for all nonempty subset of at most ` vertices X ⊆ V , let us denote