Graph sandwich problems generalize the problem of testing whether a given graph belongs to a family of graphs, and have attracted attention because of their applications and as a natural generalization of recognition problems.
[1] More precisely, given a vertex set V, a mandatory edge set E1, and a larger edge set E2, a graph G = (V, E) is called a sandwich graph for the pair G1 = (V, E1), G2 = (V, E2) if
The graph sandwich problem for property Π is defined as follows:[2][3] The recognition problem for a class of graphs (those satisfying a property Π) is equivalent to the particular graph sandwich problem where E1 = E2, that is, the optional edge set is empty.
[2][4] It can be solved in polynomial time for split graphs,[2][5] threshold graphs,[2][5] and graphs in which every five vertices contain at most one four-vertex induced path.
[6] The complexity status has also been settled for the H-free graph sandwich problems for each of the four-vertex graphs H.[7]