Separability criterion for graph-manifold groups

Some recent work in the theory of 3-manifolds and immersed surfaces indicates that the class of graph manifolds contains a large number of compact 3-manifolds whose fundamental groups are not subgroup separable (LERF). Rubinstein and Wang have given a criterion to determine whether or not a given horizontal surface in a graph manifold is separable (i.e., lifts to an embedded surface in some finite cover of the manifold). This paper extends the criterion to include some nonhorizontal surfaces in graph manifolds.