Sandwiches missing two ingredients of order four

Abstract

For a set F of graphs, an instance of the F -free Sandwich Problem is a pair (G 1 , G 2 ) consisting of two graphs G 1 and G 2 with the same vertex set such that G 1 is a subgraph of G 2 , and the task is to determine an F -free graph G containing G 1 and contained in G 2 , or to decide that such a graph does not exist. Initially motivated by the graph sandwich problem for trivially perfect graphs, which are the {P 4 , C 4 }-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,