We describe TopoLayout, a novel framework to draw undirected graphs based on the topological features they contain. Topological features are detected recursively, and their subgraphs are collapsed into single nodes, forming a graph hierarchy. The final layout is drawn using an appropriate algorithm for each topological feature. A more general goal is to partition the graph into features for which there exist good layout algorithms, so in addition to strictly topological features such as trees, connected components, biconnected components, and clusters, we have a detector function to determine when High-Dimensional Embedder is an appropriate choice for subgraph layout. Our framework is the first multi-level approach to provide a phase for reducing the number of node-edge and edge-edge crossings and a phase to eliminate all node-node overlaps. The runtime and layout visual quality of TopoLayout depend on the number and types of topological features present in the graph. We show experimental results comparing speed and visual quality for TopoLayout against four other multi-level algorithms on ten datasets with a range of connectivities and sizes, including real-world graphs of web sites, social networks, and Internet routers. TopoLayout is frequently faster or has results of higher visual quality, and sometimes, it has both. For example, the router dataset of about 140,000 nodes which contains many large tree subgraphs is drawn an order of magnitude faster with improved visual quality.
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