On cages admitting identifying codes

Graphs Admitting (1, ≤ 2)-identifying Codes

A (1,≤ 2)-identifying code is a subset of the vertex set C of a graph such that each pair of vertices intersects C in a distinct way. This has useful applications in locating errors in multiprocessor networks and threat monitoring. At the time of writing, there is no simply-stated rule that will indicate if a graph is (1,≤ 2)-identifiable. As such, we discuss properties that must be satisfied b...

Regular parcial linear spaces admitting (1;≤ k)-identifying codes

Let (P ,L, I) be a partial linear space and X ⊆ P ∪ L. Let us denote by (X)I = ⋃ x∈X{y : yIx} and by [X ] = (X)I ∪ X . With this terminology a partial linear space (P ,L, I) is said to admit a (1,≤ k)-identifying code if and only if the sets [X ] are mutually different for all X ⊆ P ∪L with |X | ≤ k. In this paper we give a characterization of k-regular partial linear spaces admitting a (1,≤ k)...

On graphs admitting t-ID codes

Let G = (V,E) be a graph and N [X] denote the closed neighbourhood of X ⊆ V , that is to say, the union of X with the set of vertices which are adjacent to X. Given an integer t ≥ 1, a subset of vertices C ⊆ V is said to be a code identifying sets of at most t vertices of G—or, for short, a t-set-ID code of G—if the sets N [X] ∩ C are all distinct, when X runs through subsets of at most t verti...

On strongly identifying codes

On Dynamic Identifying Codes

A walk c1, c2, . . . , cM in an undirected graph G = (V, E) is called a dynamic identifying code, if all the sets I(v) = {u ∈ C : d(u, v) ≤ 1} for v ∈ V are nonempty and no two of them are the same set. Here d(u, v) denotes the number of edges on any shortest path from u to v, and C = {c1, c2, . . . , cM}. We consider dynamic identifying codes in square grids, triangular grids, hexagonal meshes...