Graph sandwich problem

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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

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Additional reading

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