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 BG,r(v) the ball of radius r and centre v. A set C⊆V$C \subseteq V$ is said to be an r-identifying code if the sets BG,r(v)?C$B_{G,r}(v)\cap C$, 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 ?r(G). We study the ensemble of all the different optimalr-identifying codes C, i.e., such that |C| = ?r(G). We show that, given any collection A$\mathcal {A}$ of k-subsets of V1={1,2,?,n}, there is a positive integer m, a graph G = (V, E) with V=V1?V2$V=V_{1}\cup V_{2}$, where V2 = {n + 1 ,?, n + m}, and a set S⊆V2$S\subseteq V_{2}$ such that C⊆V$C\subseteq V$ is an optimal r-identifying code in G if, and only if, C=A?S$C=A\cup S$ for some A?A$A\in \mathcal {A}$. This result gives a direct connection with induced subgraphs of Johnson graphs, which are graphs with vertex set a collection of k-subsets of V1, with edges between any two vertices sharing k?1 elements.