  We study the algorithmic aspect of edge bundling. A bundled crossing in a
drawing of a graph is a group of crossings between two sets of parallel edges.
The bundled crossing number is the minimum number of bundled crossings that
group all crossings in a drawing of the graph.
  We show that the bundled crossing number is closely related to the orientable
genus of the graph. If multiple crossings and self-intersections of edges are
allowed, the two values are identical; otherwise, the bundled crossing number
can be higher than the genus.
  We then investigate the problem of minimizing the number of bundled
crossings. For circular graph layouts with a fixed order of vertices, we
present a constant-factor approximation algorithm. When the circular order is
not prescribed, we get a $\frac{6c}{c-2}$ approximation for a graph with $n$
vertices having at least $cn$ edges for $c>2$. For general graph layouts, we
develop an algorithm with an approximation factor of $\frac{6c}{c-3}$ for
graphs with at least $cn$ edges for $c > 3$.
