Chordal- (k,ℓ)and strongly chordal- (k,ℓ)graph sandwich problems - Couto, Fernanda; Faria, Luerbio; Klein, Sulamita (Journal of the Brazilian Computer Society, 2014)
Background In this work, we consider the graph sandwich decision problem for property Π , introduced by Golumbic, Kaplan and Shamir: given two graphs G 1 =( V , E 1 ) and G 2 =( V , E 2 ), the question is to know whether there exists a graph G =( V , E ) such that E 1 ⊆ E ⊆ E 2 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 . Methods In order to do that, we consider each pair of positive values of k and ℓ , exhibiting correspondent polynomial algorithms, or NP-complete reductions. Results We 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. Conclusions To 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.