On the ensemble of optimal dominating and locating-dominating codes in a graph

Let G be a simple, undirected graph with vertex set V. For every v∈V, we denote by N(v) the set of neighbours of v, and let N[v]=N(v)∪{v}. A set C⊂ EV is said to be a dominating code in G if the sets N[v]∩C, v∈V, are all nonempty. A set C⊂E V is said to be a locating-dominating code in G if the sets N[v]∩C, v∈V\C, and distinct. The smallest size of a dominating (resp., locating-dominating) code in G is denoted by d(G) (resp., ℓ(G)). We study the ensemble of all the different optimal dominating (resp., locating-dominating) codes C, i.e., such that |C|=d(G) (resp., |C|=ℓ(G)) in a graph G, and strongly link this problem to that of induced subgraphs of Johnson graphs.