On the ensemble of optimal identifying codes in a twin-free graph

Abstract

Let G = (V, E) be a graph. For v ∈ V and r ≥ 1, we denote by B _G,r (v) the ball of radius r and centre v. A set $C \subseteq V$ is said to be an r-identifying code if the sets $B_{G,r}(v)\cap C$, v ∈ V, are all nonempty and distinct. A graph G which admits an r-identifying code is called r -twin-free, and in this case the smallest size of an r-identifying code is denoted by γ _r (G). We study the ensemble of all the different optimal r-identifying codes C, i.e., such that |C| = γ _r (G). We show that, given any collection $\mathcal {A}$ of k-subsets of V ₁={1,2,…,n}, there is a positive integer m, a graph G = (V, E) with $V=V_{1}\cup V_{2}$, where V ₂ = {n + 1 ,…, n + m}, and a set $S\subseteq V_{2}$ such that $C\subseteq V$ is an optimal r-identifying code in G if, and only if, $C=A\cup S$ for some $A\in \mathcal {A}$. This result gives a direct connection with induced subgraphs of Johnson graphs, which are graphs with vertex set a collection of k-subsets of V ₁, with edges between any two vertices sharing k−1 elements.