# Graph sandwich problem

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 larger edge set E2, a graph G = (VE) is called a sandwich graph for the pair G1 = (VE1), G2 = (VE2) if E1EE2. 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 E1E2V × V.
Question: Is there a graph G = (V, E) such that E1EE2 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.

## Computational complexity

The graph sandwich problem is NP-complete when Π is the property of being a chordal graph, comparability graph, permutation graph, chordal bipartite graph, or chain graph.[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]

## References

1. ^ Golumbic, Martin Charles; Trenk, Ann N. (2004), "Chapter 4. Interval probe graphs and sandwich problems", Tolerance Graphs, Cambridge, pp. 63–83.
2. ^ a b c d Golumbic, Martin Charles; Kaplan, Haim; Shamir, Ron (1995), Graph sandwich problems, J. Algorithms 19: 449–473, doi:10.1006/jagm.1995.1047.
3. ^ Golumbic, Martin Charles (2004), Algorithmic Graph Theory and Perfect Graphs, Annals of Discrete Mathematics 57 (2nd ed.), Elsevier, p. 279.
4. ^ de Figueiredo, C. M. H.; Faria, L.; Klein, S.; Sritharan, R. (2007), On the complexity of the sandwich problems for strongly chordal graphs and chordal bipartite graphs, Theoretical Computer Science 381 (1-3): 57–67, doi:10.1016/j.tcs.2007.04.007, MR 2347393.
5. ^ a b Mahadev, N.V.R.; Peled, Uri N. (1995), Threshold Graphs and Related Topics, Annals of Discrete Mathematics 57, North-Holland, pp. 19–22.
6. ^ Dantas, S.; Klein, S.; Mello, C.P.; Morgana, A. (2009), The graph sandwich problem for P4-sparse graphs, Discrete Mathematics 309: 3664–3673, doi:10.1016/j.disc.2008.01.014.
7. ^ 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: Available online 11 October 2013, doi:10.1016/j.dam.2013.09.004.