On the Complexity of (k, l)-Graph Sandwich Problems

Abstract

A graph G is (k, l) if its vertex set can be partitioned into at most k independent sets and l cliques. The (k, l)-Graph Sandwich Problem asks, given two graphs G1 = (V,E1) and G2 = (V, E2), whether there exists a graph G = (V, E) such that E1 ? E ? E2 and G is (k, l). In this paper, we prove that the (k, l)-Graph Sandwich Problem is NP- complete for the cases k = 1 and l = 2; k = 2 and l = 1; or k = l = 2. This completely classifies the complexity of the (k, l)-Graph Sandwich Problem as follows: the problem is NP-complete, if k+l > 2; the problem is polynomial otherwise. In addition, we consider the degree ? constraint subproblem and completely classifies the problem as follows: the problem is polynomial, for k ? 2 or ? ? 3; the problem is NP-complete otherwise.