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?
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No, you cannot say that. Check the next image as an example, in which I removed one blue support vector and the new boundary does not use any of the old support vectors. More specifically, on the left example the margin is passing through the yellow area, while on the right one it is passing through the green one.
