  We study an algorithmic problem that is motivated by ink minimization for
sparse set visualizations. Our input is a set of points in the plane which are
either blue, red, or purple. Blue points belong exclusively to the blue set,
red points belong exclusively to the red set, and purple points belong to both
sets. A \emph{red-blue-purple spanning graph} (RBP spanning graph) is a set of
edges connecting the points such that the subgraph induced by the red and
purple points is connected, and the subgraph induced by the blue and purple
points is connected.
  We study the geometric properties of minimum RBP spanning graphs and the
algorithmic problems associated with computing them. Specifically, we show that
the general problem can be solved in polynomial time using matroid techniques.
In addition, we discuss more efficient algorithms for the case in which points
are located on a line or a circle, and also describe a fast $(\frac
12\rho+1)$-approximation algorithm, where $\rho$ is the Steiner ratio.
