The chain graph sandwich problem
Résumé
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 \subseteq E \subseteq 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.