Polar graphs and maximal independent sets

Abstract

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 , 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 ) -polar graphs and study the computational complexity of the problems on polar graphs of special types.