Discriminating codes in (bipartite) planar graphs

Consider a connected undirected bipartite graph G=(V=I∪A,E)G=(V=I∪A,E), with no edges inside II or AA. For any vertex v∈Vv∈V, let N(v)N(v) be the set of neighbours of vv. A code C⊆AC⊆A is said to be discriminating if all the sets N(i)∩CN(i)∩C, i∈Ii∈I, are nonempty and distinct.We study some properties of discriminating codes in particular classes of bipartite graphs, namely trees and, more generally, (bipartite) planar graphs.