Abstract
In this paper, we unify the Markov theory of a variety of different types of
graphs used in graphical Markov models by introducing the class of loopless
mixed graphs, and show that all independence models induced by $m$-separation
on such graphs are compositional graphoids. We focus in particular on the
subclass of ribbonless graphs which as special cases include undirected graphs,
bidirected graphs, and directed acyclic graphs, as well as ancestral graphs and
summary graphs. We define maximality of such graphs as well as a pairwise and a
global Markov property. We prove that the global and pairwise Markov properties
of a maximal ribbonless graph are equivalent for any independence model that is
a compositional graphoid.