Algorithms and Bounds for Drawing Directed Graphs

TL;DR

This paper introduces a novel algorithmic framework for visualizing directed graphs that preserves reachability, reduces visual complexity by drawing only necessary edges, and operates efficiently in polynomial time.

Contribution

It proposes a new approach that departs from classical methods, avoids dummy vertices, and provides polynomial-time algorithms for hierarchical graph visualization.

Findings

Complete reachability information is preserved in visualizations.

The approach reduces visual complexity by drawing only essential edges.

Main algorithms operate in $O(kn)$ time, with $k$ much smaller than $n$.

Abstract

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).