i i “Couto-Faria-Klein-Protti-Nogueira˙MC” — 2014/4/9 — 19:04 — page 17 — #1 i i i i i i Matem´ atica Contemporˆ anea, Vol. 42, 17–26 c ?2014, Sociedade Brasileira de Matem´ atica (k,?)-Sandwich Problems: why not ask for special kinds of bread? F. Couto L. Faria S. Klein F. Protti L. T. Nogueira Abstract In this work, we consider the Golumbic, Kaplan, and Shamir de- cision sandwich problem for a property Π: given two graphs G 1 = (V,E 1 ) and G 2 =(V,E 2 ), the question is: Is there a graph G = (V,E) such that E 1 ⊆ E ⊆ E 2 and G satisfies Π? The graph G is called sandwich graph. Note that what matters here is just the “filling” of the sandwich. Our proposal is to try different kinds of “bread” for each chosen special sandwich filling. In other words, we focus on the complexity of sandwich problems when, beforehand, it is known that G i satisfies a property Π i , i =1, 2. Let (Π 1 , Π, Π 2 )-sp denote the sandwich problem for property Π when G i satisfies Π i , called sandwich problem with boundary conditions. When G i is not required to satisfy any special property, Π i is denoted by ∗. A graph G is (k,?) if there is a partition of V (G) into k independent sets and ? cliques. It is known that (∗, (k,?), ∗)-sp is NP-complete, for all k + ? greater than 2. In order to motivate this new work proposal, in this paper we describe polynomial-time algorithms for three sand- wich problems with boundary conditions: (perfect, (k,?), poly- nomial number of maximal cliques)-sp for all k,? ∈ N,(∗, 2000 AMS Subject Classification: 60K35, 60F05, 60K37. Key Words and Phrases: Graph Sandwich Problems, Boundary Conditions, (k, ?)- graphs. *Partially Supported by CNPq, CAPES and FAPERJ.