Strong Identification Codes for Graphs

For any graph~$G,$ a set of vertices~${\cal V}$ is said to be dominating if every vertex of~$G$ contains at least one node of~$G$ and separating if each vertex~$v$ contains a unique neighbour~$u_v \in {\cal V}$ that is adjacent to no other vertex of~$G.$ If~${\cal V}$ is both dominating and separating, then~${\cal V}$ is defined to be an identification code. In this paper, we study strong identification codes with an index~$r,$ by imposing the constraint that each vertex of~$G$ contains at least~$r$ unique neighbours in~${\cal V}.$ We use the probabilistic method to study both the minimum size of strong identification codes and the existence of graphs that allow an identification code with a given index.