Distinct sizes of maximal independent sets on graphs with restricted girth.

Let $G$ be a graph. If $G$ has exactly $r$ distinct sizes of maximal independent sets, $G$ belongs to a collection called $\mathcal{M}_r$. If $G \in \mathcal{M}_{r}$ and the distinct values of its maximal independent sets are consecutive, then $G$ belongs to $\mathcal{I}_{r}$. The independence gap of $G$ is the difference between the maximum and the minimum sizes of a maximal independent set in $G$. In this paper, we show that recognizing graphs in $\mathcal{I}_r$ is $\mathcal{NP}$-complete, for every integer $r \geq 3$. On the other hand, we show that recognizing trees in $\mathcal{M}_r$ can be done in polynomial time, for every $r \geq 1$. Also, we present characterizations of some graphs with girth at least 6 with independence gap at least 1, including graphs with independence gap $r-1$, for $r\geq 2$, belonging to $\mathcal{I}_r$. Moreover, we present a characterization of some trees in $\mathcal{I}_3$.