A maximal independent set is an independent set that is not a proper subset of any other independent set. A connected graph (respectively, graph) G with vertex set ( ) In this paper, we survey on the large numbers of maximal independent sets among all trees, forests, quasi-trees and quasi-forests. In addition, we further look into the problem of determining the third largest number of maximal independent sets among all quasi-trees and quasi-forests. Extremal graphs achieving these values are

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... o given. Keywords Maximal Independent Set, Quasi-Tree Graph, Quasi-Forest Graph, Extremal Graph ( ) mi G among all graphs of order n; Jou and Lin [4] further explored the same problem for trees and forests; Jin and Yan [5] solved the third largest number of How to cite this paper: Lin,