2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/75076Let 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.30 pages, 9 figures, Latex, see related papers at http://www.math.msu.edu/~saganCombinatorics05C35 (Primary) 05C69 (Secondary)text