Graph sandwich problems

A sandwich problem for a graph with respect to a graph property Π is a partially specified graph, i.e., only some of the edges and non-edges are given, and the question to be answered is, can this graph be completed to a graph which has the property Π? The graph sandwich problem was investigated for a large number of families of graphs in a 1995 paper by Golumbic, Kaplan and Shamir, and much subsequent work has taken place since. In some cases, the problem is NP-complete such as for interval graphs, comparability graphs, chordal graphs and others. In other cases, the sandwich problem can be solved in polynomial time such as for threshold graphs, cographs, and split graphs. But since Π may range over all imaginable graph properties, there must be at least 2011 different graph sandwich problems. There are also interesting special cases of the sandwich problem, most notably the probe graph problem where the unspecified edges are confined to be within a subset of the vertices. Similar sandwich problems can also be defined for hypergraphs, matrices and Boolean functions, namely, completing partially specified structures such that the result satisfies a desirable property. In this talk, we will present a survey of results that we and others have obtained in this area during the past several years.