**5. Concluding Remarks**

In this paper, we provided a lower bound and an upper bound, each of them being tight, of the determining number of general graphs. We also showed that our lower bound is independent from the one obtained in [14]. The main tool that we used is the twin graph, defined in [22] to study the metric dimension of graphs, and which has proven to be also useful for obtaining determining sets and for computing the determining number. Indeed, as an application of our bounds, we computed the exact value of the determining number of cographs. In the case of unit interval graphs, we placed this parameter in an set of two consecutive integers. In both cases, the obtained values depend only on the number of vertices of both graphs *G* and its twin graph *G*1.

We think that our bounds could be useful to deal with other graph families (e.g., distancehereditary graphs or parity graphs) in order to obtain the exact value of their determining numbers, or at least to bound the range of possible values. Actually, we could find other techniques, different from twin deletion, to provide new bounds of the determining number of a graph: addition of vertices or edges, vertex contraction, etc. Furthermore, it could be of interest to apply all those techniques to other types of sets different from determining sets such as dominating sets, cut sets, and independent sets.

**Author Contributions:** All authors contributed equally to this work. Conceptualization, A.G. and M.L.P.; methodology, A.G. and M.L.P.; formal analysis, A.G. and M.L.P.; validation, A.G. and M.L.P.; writing—original draft preparation, A.G. and M.L.P.; writing-review and editing, A.G. and M.L.P.

**Funding:** The second author is partially supported by grants MTM2015-63791-R (MINECO/FEDER) and RTI2018-095993-B-100.

**Conflicts of Interest:** The authors declare no conflict of interest.
