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 AL, AR, SL and SR such that: every vertex of AL is adjacent to every vertex of AR; no vertex of SL is adjacent to a vertex of AR USR; no vertex of SR is adjacent to a vertex of AL U SL; The graph sandwich problem for 1-join composition is denned as follows: Given a vertex set V, a forced edge set E1, and a forbidden edge set E13, is there a graph G = (V, E) such that El⊆ E and E ∩ E3 = φ, 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 = φ (SR = φ), namely, the graph sandwich problem for homogeneous set, which has a polynomial-time solution.