Structural properties of twin-free 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 admits at least one r-identifying code if and only if it is r-twin-free, that is, the sets B_r(v), v ∈ V, are all different. We study some structural problems in r-twin-free graphs, such as the existence of the path with 2r + 1 vertices as a subgraph, or the consequences of deleting one vertex.