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a b s t r a c tA maximal independent set is an independent set that is not a proper subset of any other independent set. In this paper, we determine the second largest number of maximal independent sets among all trees and forests of order n ≥ 4. We also characterize those extremal graphs achieving these values.

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[1] found the maximum number of maximal independent sets in connected graphs which contain at most two cycles. In this paper, we give an alternative proof to determine the largest number of maximal independent sets among all connected graphs of order n ≥ 12, which contain at most two cycles. We also characterize the extremal graph achieving this maximum value.