  A pseudo-triangle is a simple polygon with exactly three convex vertices, and
all other vertices (if any) are distributed on three concave chains. A
pseudo-triangulation~$\mathcal{T}$ of a point set~$P$ in~$\mathbb{R}^2$ is a
partitioning of the convex hull of~$P$ into pseudo-triangles, such that the
union of the vertices of the pseudo-triangles is exactly~$P$. We call a size-4
pseudo-triangle a dart. For a fixed $k\geq 1$, we study $k$-dart
pseudo-triangulations ($k$-DPTs), that is, pseudo-triangulations in which
exactly $k$ faces are darts and all other faces are triangles. We study the
flip graph for such pseudo-triangulations, in which a flip exchanges the
diagonals of a pseudo-quadrilatral. Our results are as follows. We prove that
the flip graph of $1$-DPTs is generally not connected, and show how to compute
its connected components. Furthermore, for $k$-DPTs on a point configuration
called the double chain we analyze the structure of the flip graph on a more
fine-grained level.
