  In this paper we present a new approach to visualize directed graphs and
their hierarchies that completely departs from the classical four-phase
framework of Sugiyama and computes readable hierarchical visualizations that
contain the complete reachability information of a graph. Additionally, our
approach has the advantage that only the necessary edges are drawn in the
drawing, thus reducing the visual complexity of the resulting drawing.
Furthermore, most problems involved in our framework require only polynomial
time. Our framework offers a suite of solutions depending upon the
requirements, and it consists of only two steps: (a) the cycle removal step (if
the graph contains cycles) and (b) the channel decomposition and hierarchical
drawing step. Our framework does not introduce any dummy vertices and it keeps
the vertices of a channel vertically aligned. The time complexity of the main
drawing algorithms of our framework is $O(kn)$, where $k$ is the number of
channels, typically much smaller than $n$ (the number of vertices).
