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

Authors:
Simone Dantas;Celina M. Herraera de Figueiredo;Luerbio Faria
Affiliations:
-;-;-
Venue:
WG '02 Revised Papers from the 28th International Workshop on Graph-Theoretic Concepts in Computer Science
Year:
2002

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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 驴 E 驴 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 驴 constraint subproblem and completely classifies the problem as follows: the problem is polynomial, for k 驴 2 or 驴驴 3; the problem is NP-complete otherwise.