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

Abstract

We connect two classical results in extremal graph theory concerning the number of maximal independent sets. The maximum number mis$(n)$ of maximal independent sets in an $n$-vertex graph was determined by Moon and Moser. The maximum number mis${}_{△}(n)$ of maximal independent sets in an $n$-vertex triangle-free graph was determined by Hujter and Tuza. We determine the maximum number mis${}_t(n)$ of maximal independent sets in an $n$-vertex graph containing no induced triangle matching of size $t+1$. We also reprove a stability result of Kahn and Park on the maximum number mis${}_{△,t}(n)$ of maximal independent sets in an $n$-vertex triangle-free graphs containing no induced matching of size $t+1$.