The minimum maximal-independent-set conjecture for connected twin-free triangle-free graphs

Let $G$ be a connected, twin-free and triangle-free graph of order $n$. Write $i_{\text{max}}(G)$ for the number of maximal independent sets of $G$. Minimum maximal-independent-set conjecture.

$$i_{\text{max}}(G) \ge \left\lceil\frac{n}{2}\right\rceil+1.$$

Furthermore, if $n$ is even, every graph attaining equality is bipartite. The preceding theorem proves this bound for connected twin-free bipartite graphs, while the extension to triangle-free graphs is presented as unproved; the equality characterization is likewise part of the conjecture.