Polar graphs and maximal independent sets

We study two central problems of algorithmic graph theory: finding maximum and minimum maximal independent sets. Both problems are known to be NP-hard in general. Moreover, they remain NP-hard in many special classes of graphs. For instance, the problem of finding minimum maximal independent sets has been recently proven to be NP-hard in the class of so-called (1,2)(1,2)-polar graphs. On the other hand, both problems can be solved in polynomial time for (1,1)(1,1)-polar, also known as split   graphs. In this paper, we address the question of distinguishing new classes of graphs admitting polynomial-time solutions for the two problems in question. To this end, we extend the hierarchy of (α,β)(α,β)-polar graphs and study the computational complexity of the problems on polar graphs of special types.