Chordal- (k,ℓ)and strongly chordal- (k,ℓ)graph sandwich problems

BackgroundIn this work, we consider the graph sandwich decision problem for property Π, introduced by Golumbic, Kaplan and Shamir: given two graphs G1=(V,E1) and G2=(V,E2), the question is to know whether there exists a graph G=(V,E) such that E1⊆E⊆E2 and G satisfies property Π. Particurlarly, we are interested in fully classifying the complexity of this problem when we look to the following properties Π: `G is a chordal- (k,l)-graph' and `G is a strongly chordal- (k,l)-graph', for all k,ℓ.MethodsIn order to do that, we consider each pair of positive values of k and ℓ, exhibiting correspondent polynomial algorithms, or NP-complete reductions.ResultsWe prove that the strongly chordal- (k,ℓ) graph sandwich problem is NP-complete, for k≥1 and ℓ≥1, and that the chordal- (k,ℓ) graph sandwich problem is NP-complete, for positive integers k and ℓ such that k+ℓ ≥ 3. Moreover, we prove that both problems are in P when k or ℓ is zero and k+ℓ ≤ 2.ConclusionsTo complete the complexity dichotomy concerning these problems for all nonnegative values of k and ℓ, there still remains the open question of settling the complexity for the case k+ℓ ≥ 3 and one of them is equal to zero.