The graph sandwich problem for 1-join composition is NP-complete

A graph is a 1-join composition if its vertex set can be partitioned into four nonempty sets and SR such that: every vertex of AL is adjacent to every vertex of AR; no vertex of SL is adjacent to vertex of AR∪SR; no vertex of SR is adjacent to a vertex of AL∪SL. The graph sandwich problem for 1-join composition is defined as follows: Given a vertex set V, a forced edge set E1, and a forbidden edge set E3, is there a graph G=(V,E) such that E1⊆E and E∩E3=∅, which is a 1-join composition graph? We prove that the graph sandwich problem for 1-join composition is NP-complete. This result stands in contrast to the case where SL=∅ (SR=∅), namely, the graph sandwich problem for homogeneous set, which has a polynomial-time solution.