**6. Conclusions**

In this note, we studied the ID-stable graphs. Some basic properties of ID-stable graphs were presented and new independent domination stable graphs constructed from an old one. We also characterized all independent domination stable trees and unicyclic graphs. In addition, we proved that for any tree *T* of order *n* ≥ 5, ! *n*3 " ≤ *i*(*T*) ≤ ! *n*−2 2 ", and we characterized all trees attaining the lower and upper bound. An interesting problem is to find sharp lower and upper bounds on the independent domination number of ID-stable graphs. The other problem is to characterize all ID-stable bicyclic graphs. Another problem is to study algorithm running times to decide independent domination graphs.

**Author Contributions:** Z.S. and S.M.S. contributed to the supervision, methodology, validation, project administration, and formal analysis. P.W., H.J., S.N.-M., and L.V. contributed to the investigation, resources, and some computations and wrote the initial draft of the paper, which was investigated and approved by Z.S. S.M.S. wrote the final draft.

**Funding:** This work is supported by the Natural Science Foundation of Guangdong Province under Grant 2018A0303130115, the Science and Technology Program of Guangzhou (No. 201904010493), and the Specialized Fund for Science and Technology Platform and Talent Team Project of Guizhou Province (No. QianKeHePingTaiRenCai [2016]5609).

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