Sandwiches Missing Two Ingredients of Order Four

X iv :1 70 4. 01 92 2v 1 [ m at h. C O ] 6 A pr 2 01 7 Sandwiches Missing Two Ingredients of Order Four José D. Alvarado Simone Dantas Dieter Rautenbach 1 Instituto de Matemática e Estat́ıstica, Universidade Federal Fluminense, Niterói, Brazil [email protected], [email protected] 2 Institute of Optimization and Operations Research, Ulm University, Ulm, Germany [email protected] Abstract For a set F of graphs, an instance of the F -free Sandwich Problem is a pair (G1, G2) consisting of two graphs G1 and G2 with the same vertex set such that G1 is a subgraph of G2, and the task is to determine an F -free graph G containing G1 and contained in G2, or to decide that such a graph does not exist. Initially motivated by the graph sandwich problem for trivially perfect graphs, which are the {P4, C4}-free graphs, we study the complexity of the F -free Sandwich Problem for sets F containing two non-isomorphic graphs of order four. We show that if F is one of the sets {diamond,K4}, {diamond, C4}, {diamond, paw}, {