The chain graph sandwich problem

Abstract

The chain graph sandwich problem asks: Given a vertex set V, a mandatory edge set E 1, and a larger edge set E 2, is there a graph G=(V,E) such that E 1⊆E⊆E 2 with G being a chain graph (i.e., a 2K 2-free bipartite graph)? We prove that the chain graph sandwich problem is NP-complete. This result stands in contrast to (1) the case where E 1 is a connected graph, which has a linear-time solution, (2) the threshold graph sandwich problem, which has a linear-time solution, and (3) the chain probe graph problem, which has a polynomial-time solution.