On identifying vertices of tournament digraphs

An identifying code in a graph is a subset of its vertices where the neighbours' intersections with the subset are nonempty and different for every pair of vertices. After their introduction in 1998 by Karpovsky et al., the interest in this domain has never ended. This growing interest comes from, on the one hand, the theoretical aspect of this concept, and on the other hand, its applications, especially the indoor location and faulty processor network. In this work, we study the identifying code on tournament digraphs which is probably the most studied class of digraphs. Hence, we give some minimum cardinality of special tournaments and show that only transitive tournaments can admit an $r$-identifying code when $r\geq 2$. We also obtain an upper bound for the quadratic residue tournament. Moreover, we study how to reach an optimal code when adding a vertex or inverting an arc in a transitive tournament.