On the Structure of Identifiable Graphs

Consider a connected undirected graph G = (V, E), a subset of vertices C ⊆ V, and an integer r ≥ 1; for any vertex v ∈ V, let B_r (v) denote the ball of radius r centered at v, i.e., the set of all vertices linked to v by a path of at most r edges. If for all vertices v ∈ V, the sets B_r (v) ∩ C are all nonempty and different, then we call C an r-identifying code. A graph is said to be r-identifiable if it admits at least one r-identifying code. We prove the following structural properties of r-identifiable graphs. For any r ≥ 1, any r-identifiable graph must have at least 2 r + 1 vertices. For r = 1 and for any r-identifiable graph G with at least 2 r + 2 vertices, or for any r ≥ 1 and for any r-identifiable tree G with at least 2 r + 2 vertices, there always exists at least one vertex such that its removing from G leaves an r-identifiable graph. This property is not true for r ≥ 3 in general. The case r = 2 remains open for general graphs.