For a set F of graphs, an instance of the F-{\sc free
Sandwich Problem} is a pair (G1β,G2β) consisting of two graphs G1β and
G2β with the same vertex set such that G1β is a subgraph of G2β, and the
task is to determine an F-free graph G containing G1β and
contained in G2β, or to decide that such a graph does not exist. Initially
motivated by the graph sandwich problem for trivially perfect graphs, which are
the {P4β,C4β}-free graphs, we study the complexity of the F-{\sc
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,K4β}, {diamond,C4β}, {diamond,paw}, {K4β,K4ββ}, {P4β,C4β}, {P4β,claw}, {P4β,pawβ}, {P4β,diamond},
{paw,C4β}, {paw,claw},
{paw,claw}, {paw,pawβ}, {C4β,C4ββ},
{claw,claw}, and {claw,C4ββ}, then the F-{\sc free Sandwich Problem}
can be solved in polynomial time, and, if F is one of the sets
{C4β,K4β}, {paw,K4β}, {paw,K4ββ}, {paw,C4ββ},
{diamond,C4ββ}, {paw,diamond}, and {diamond,diamond}, then the decision version of the F-{\sc free
Sandwich Problem} is NP-complete