  In this paper, we investigate a family of graphs associated to collections of
arcs on surfaces. These {\it multiarc graphs} naturally interpolate between arc
graphs and flip graphs, both well studied objects in low dimensional geometry
and topology. We show a number of rigidity results, namely showing that, under
certain complexity conditions, that simplicial maps between them only arise in
the "obvious way". We also observe that, again under necessary complexity
conditions, subsurface strata are convex. Put together, these results imply
that certain simplicial maps always give rise to convex images.
