Independent sets of maximum weight in ( p, q)-colorable graphs

Abstract

A graph is said to be ( p, q)-colorable if its vertex set can be partitioned into at most p cliques and q independent sets. In particular, (0,2)-colorable graphs are bipartite, and (1,1)-colorable are the split graphs. For both of these classes, the problem of finding a maximum weight independent set is known to be solvable in polynomial time. In the present note, we give a complete classification of the family of ( p, q)-colorable graphs with respect to time complexity of this problem. Specifically, we show that the problem has a polynomial time solution in the class of ( p, q)-colorable graphs if and only if q⩽2 (assuming P≠ NP).