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

Authors:
Vladimir E. Alekseev;Vadim V. Lozin
Affiliations:
University of Nizhny Novgorod, Gagarina 23, Nizhny Novgorod 603950, Russia;RUTCOR, Rutgers University, 640 Bartholomew Rd., Piscataway, NJ
Venue:
Discrete Mathematics
Year:
2003

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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).