  In this paper, we study classes of graphs with three types of edges that
capture the modified independence structure of a directed acyclic graph (DAG)
after marginalisation over unobserved variables and conditioning on selection
variables using the $m$-separation criterion. These include MC, summary, and
ancestral graphs. As a modification of MC graphs, we define the class of
ribbonless graphs (RGs) that permits the use of the $m$-separation criterion.
RGs contain summary and ancestral graphs as subclasses, and each RG can be
generated by a DAG after marginalisation and conditioning. We derive simple
algorithms to generate RGs, from given DAGs or RGs, and also to generate
summary and ancestral graphs in a simple way by further extension of the
RG-generating algorithm. This enables us to develop a parallel theory on these
three classes and to study the relationships between them as well as the use of
each class.
