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An independent set is a useful structure because, in some situations, it defines a set of mutually compatible operations, i.e., operations that can be executed simultaneously. We design a fault-containing self-stabilizing algorithm that finds a maximal independent set for an asynchronous distributed system. Our algorithm is an improvement on the self-stabilizing algorithm in Shukla et al. . In the single-fault situation, the worst-case stabilization time of Shukla's algorithm is Ω(n), where n is the number of nodes in the system, whereas the worst-case stabilization time of our algorithm is O(Δ), where Δ is the maximum node degree in the system. Compared also with the fault-containing algorithm that is induced from applying the general transformer in Ghosh et al.  to Shukla's algorithm, our algorithm is also seen to be faster in stabilization time, in the single-fault situation. Therefore, our algorithm can be considered to be the most efficient fault-containing self-stabilizing algorithm for the maximal independent set finding up to this point.