  Let m(G) denote the number of maximal independent sets of vertices in a graph
G and let c(n,r) be the maximum value of m(G) over all connected graphs with n
vertices and at most r cycles. A theorem of Griggs, Grinstead, and Guichard
gives a formula for c(n,r) when r is large relative to n, while a theorem of
Goh, Koh, Sagan, and Vatter does the same when r is small relative to n. We
complete the determination of c(n,r) for all n and r and characterize the
extremal graphs. Problems for maximum independent sets are also completely
resolved.
