In graph theory and computer science, the graph sandwich problem is a problem of finding a graph that belongs to a particular family of graphs and is "sandwiched" between two other graphs, one of which must be a subgraph and the other of which must be a supergraph of the desired graph.
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]
Problem statement
More precisely, given a vertex set V, a mandatory edge set E1,
and a (potentially) larger edge set E2,
a graph G = (V, E) is called a sandwich graph for the pair
G1 = (V, E1), G2 = (V, E2) if
E1 ⊆ E ⊆ E2.
The graph sandwich problem for property Π is defined as follows:[2][3]
Graph Sandwich Problem for Property Π:
Instance: Vertex set V and edge sets E1 ⊆ E2 ⊆ V × V.
Question: Is there a graph G = (V, E) such that E1 ⊆ E ⊆ E2 and G satisfies property Π ?
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.
^Dantas, Simone; de Figueiredo, Celina M.H.; Maffray, Frédéric; Teixeira, Rafael B. (2013), "The complexity of forbidden subgraph sandwich problems and the skew partition sandwich problem", Discrete Applied Mathematics, 182: 15–24, doi:10.1016/j.dam.2013.09.004.
Further reading
Dantas, Simone; de Figueiredo, Celina M.H.; da Silva, Murilo V.G.; Teixeira, Rafael B. (2011), "On the forbidden induced subgraph sandwich problem", Discrete Applied Mathematics, 159 (16): 1717–1725, doi:10.1016/j.dam.2010.11.010.
