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 vertices of V . A graph G admits a t-set-ID code if and only if N [X] 6= N [Y ] for all pairs X and Y which are distinct subsets of at most t vertices of V . Graphs admitting identifying codes is a recent topic. In this paper, we show that for G1 admitting a t1-set-ID code, and G2 admitting a t2set-ID code, the cartesian product G1 G2 admits a max{t1, t2}-set-ID code, and we show that this result is the best possible. We also study the extremal question of minimizing the number of vertices of a graph admitting a t-set-ID code. Asymptotically, this number is Ω(t), and we give an explicit construction of an infinite family of t-regular graphs attaining this bound. The construction uses so-called distance-regular graphs. ∗ Research supported by the Academy of Finland under grant 111940. 82 TERO LAIHONEN AND JULIEN MONCEL 1 Codes identifying sets of vertices Let G = (V,E) be an undirected and connected graph and t ≥ 1 an integer. Denote by d(u, v) the graphic distance between the vertices u and v, i.e., the number of edges in any shortest path between u and v. We say that a vertex u covers a vertex v if d(u, v) ≤ 1. Let us denote by N [X] the closed neighbourhood of X ⊆ V , that is, the union of X with the set of vertices which are adjacent to an element of X. 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) if the sets N [X] ∩ C are distinct for all X ⊆ V with |X| ≤ t. A graph may not admit a t-set-ID code (that is, there does not exist any C ⊆ V such that all of the sets N [X]∩C were different), for example, in the complete graph Kn on n ≥ 2 vertices we have N [x] = N [y] for any two vertices x 6= y, so Kn does not admit a t-set-ID code for any t ≥ 1. It is easy to see (by choosing C = V ) that G admits a t-set-ID code if and only if N [X] 6= N [Y ] for all pairs of distinct subsets X and Y of at most t vertices of V . The notion of identifying codes was introduced in [10] to model a fault-detection problem in multiprocessor systems. For another application to sensor networks consult [16]. Identifying codes are closely related to other types of codes, like covering codes (which are frequently used to construct identifying codes in Hamming spaces, see e.g. [1, 2, 10]) or superimposed codes (see [6]). There is a large and fast-growing bibliography on identifying codes, which can be found from Antoine Lobstein’s webpage [17]. In [4], structural properties of graphs admitting t-set-ID codes are derived for t = 1, but very little is known on these graphs for the general case t ≥ 1. In particular, only few constructions of graphs admitting t-set-ID codes are known (see [8, 14] for some constructions). In the next section, we show that if Gi admits a ti-set-ID code, for i = 1, 2, then the cartesian product of G1 and G2 admits a max{t1, t2}-set-ID code. We also prove that this result is the best possible. It should be noticed that, for example, multiprocessor systems such as binary hypercubes and certain grids are obtained using the cartesian product. A question (posed in [8]) of finding the minimum number of vertices of a graph admitting a t-set-ID code is studied in Section 3. Asymptotically, the number of vertices of a graph admitting a t-set-ID code is Ω(t) (for the usual notations Ω(.), O(.) and Θ(.), we refer to [5]). In this paper, we give an infinite family of graphs attaining this bound. This family has the additional property that all its graphs are t-regular (that is, they satisfy an extremal degree property as well), which improves a construction given in [11]. 2 Structural properties and the cartesian product First we derive two lemmas dealing with structural properties of graphs admitting t-set-ID codes, which will be useful in the sequel. Let N(x) denote the set of vertices ON GRAPHS ADMITTING CODES 83