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

In 1995 Golumbic, Kaplan and Shamir defined graph sandwich problems as follows: graph sandwich problem for property Π (Π-sp) Input: Two graphs G = (V,E) and G = (V,E) such that E ⊆ E. Question: Is there a graph G = (V,E) satisfying the property Π such that E ⊆ E ⊆ E? The graphG is called sandwich graph for the pair (G, G). Note that making E = E = E we have the recognition problem for the property Π. Thus, we can easily see that graph sandwich problems are natural generalizations of recognition problems and, because of that, if the recognition problem for some property Π is NP-complete, so it will be the corresponding graph sandwich problem. In this work we will deal with two special properties: ‘to be a strongly chordal-(k, `) graph ” and “to be a chordal-(k, `) graph”. A graph is (k, `) if its vertex set can be partitioned into at most k independent sets and into at most ` cliques. Brandstadt et al. have proved that this problem is NP-complete for k ≥ 3 and ` ≥ 3, and polynomially solvable otherwise. It is also known that the sandwich problem for (k, `) graphs is NP-complete for k+` ≥ 3 [Dantas et al.] and polynomial otherwise [Golumbic et al.]. A graph is chordal if it has no Ck as an induced subraph, k ≥ 4. A graph is strongly chordal if it is chordal and if every even cycle of length at least 6 has an odd chord, i.e.,