  We consider visibility graphs involving bars and arcs in which lines of sight
can pass through up to k objects. We prove a new edge bound for arc
k-visibility graphs, provide maximal constructions for arc and semi-arc
k-visibility graphs, and give a complete characterization of semi-arc
visibility graphs. We show that the family of arc i-visibility graphs is never
contained in the family of bar j-visibility graphs for any i and j, and that
the family of bar i-visibility graphs is not contained in the family of bar
j-visibility graphs for $i \neq j$. We also give the first thickness bounds for
arc and semi-arc k-visibility graphs. Finally, we introduce a model for random
semi-bar and semi-arc k-visibility graphs and analyze its properties.
