On the number of maximal independent sets: From Moon–Moser to Hujter–Tuza

We connect two classical results in extremal graph theory concerning the number of maximal independent sets. The maximum number (Figure presented.) of maximal independent sets in an (Figure presented.) -vertex graph was determined by Miller and Muller and independently by Moon and Moser. The maximum number (Figure presented.) of maximal independent sets in an (Figure presented.) -vertex triangle-free graph was determined by Hujter and Tuza. We give a common generalization of these results by determining the maximum number (Figure presented.) of maximal independent sets in an (Figure presented.) -vertex graph containing no induced triangle matching of size (Figure presented.). This also improves a stability result of Kahn and Park on (Figure presented.). Our second result is a new (short) proof of a second stability result of Kahn and Park on the maximum number (Figure presented.) of maximal independent sets in (Figure presented.) -vertex triangle-free graphs containing no induced matching of size (Figure presented.).