*Appendix A.1. Trees*

By applying the constructive method as above, we obtain all ID-stable trees with order up to 12, and the statistics of the number of trees with different orders is presented in Table A1.

We list all the independent domination stable trees with orders from 5 to 12 in Figure A1.

### *Appendix A.2. Unicyclic Graphs*

By applying the constructive method as above, we obtain all independent domination stable unicyclic graphs with order from 3 to 10, and the statistics of the number of unicyclic graphs with different orders is presented in Table A2.

We here list all the independent domination stable unicyclic graphs with orders from 3 to 10 in Figure A2.

**Figure A1.** Independent domination stable trees with orders from 5 to 12.

**Figure A2.** All independent domination stable unicyclic graphs of orders from 3 to 10.


**Table A1.** The number of independent domination stable trees with different orders.

**Table A2.** The number of independent domination stable unicyclic graphs with different orders.

