Before revision:  Both algorithms can compute high-quality independent sets over dynamic graphs with O(\Delta^3) time for general graphs and with O( 1 ) time for bounded-degree graphs. Moreover, the lower bound of the size of the solution maintained by our algorithms is derived if there is no swappable vertex in it. Then the algorithms are extended under\textit{semi-external setting to maintained a high-quality independent set with limited memory consumption .
After revision:  Both algorithms can compute high-quality independent sets over dynamic graphs with O(\Delta^3) time for general graphs and with O( 1 \textit{semi-external setting to maintained a high-quality independent set with limited memory consumption .