Will removing one support vector affect others?

Supppose we have SVM trained on a dataset and the support vectors are $$SV=\{x_1,x_2,\cdots,x_n\}$$. Then, we know that the decision plan is decided by $$SV$$. My question is that if we remove one support vector (say $$x_1$$) from the dataset and train the model again. Will the other support vectors $$\{x_2,\cdots,x_n\}$$ still be the new support vectors after we training on the new dataset?