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.