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)-polar graphs. On the other hand, both problems can be solved in polynomial time for (1, I)-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 (alpha, beta)-polar graphs and study the computational complexity of the problems on polar graphs of special types. (c) 2006 Elsevier B.V. All rights reserved.