  A subcomplex $\mathcal{X}$ of a cell complex $\mathcal{C}$ is called rigid
with respect to another cell complex $\mathcal{C}'$ if every injective
simplicial map $\lambda:\mathcal{X} \rightarrow \mathcal{C}'$ has a unique
extension to an injective simplicial map $\phi:\mathcal{C}\rightarrow
\mathcal{C}'$. Given a surface with marked points, its flip graph and arc
complex are simplicial complexes indexing the triangulations and the arcs
between marked points respectively. We relate two recent results of the second
author on the existence of finite rigid subcomplexes in the flip graph and in
the arc complex. The second author employed distinct approaches and proof
methods to these two complexes. In this paper, we leverage the fact that the
flip graph can be embedded in the arc complex as its dual, relating the second
author's past rigidity results. Further, this allows us to prove a new
variation of arc complex rigidity for surfaces with boundary.
