Graphs with the second largest number of maximal independent sets

Let image be a simple undirected graph. Denote by image (respectively, image) the number of maximal (respectively, maximum) independent sets in image. Erdős and Moser raised the problem of determining the maximum value of image among all graphs of order image and the extremal graphs achieving this maximum value. This problem was solved by Moon and Moser. Then it was studied for many special classes of graphs, including trees, forests, bipartite graphs, connected graphs, (connected) triangle-free graphs, (connected) graphs with at most one cycle, and recently, (connected) graphs with at most image cycles. In this paper we determine the second largest value of image and image among all graphs of order image. Moreover, the extremal graphs achieving these values are also determined.