On graphs with maximal independent sets of few sizes, minimum degree at least 2, and girth at least 7

We prove that for integers r and D with r≥2 and D≥3, there are only finitely many connected graphs of minimum degree at least 2, maximum degree at most D, and girth at least 7 that have maximal independent sets of at most r different sizes. Furthermore, we prove several results restricting the degrees of such graphs. Our contributions generalize known results on well-covered graphs.