Topological transitivity, weak mixing and non-wandering are definitions used in topological dynamics to describe the ways in which open sets feed into each other under iteration. These definitions are generalised using finite directed graphs to obtain topological mapping properties. We consider the extent to which these mapping properties are logically distinct. As it turns out, there are only three distinct properties which entail ``interesting" dynamics. Two of these, transitivity and weak mixing, are already well known. The third does not appear in the literature. The remaining properties comprise a countably infinite collection of properties entailing somewhat less interesting dynamics. This latter collection includes non-wandering.