Maximal and maximum independent sets in graphs with at most r cycles

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. (C) 2006 Wiley Periodicals, Inc.