Drawing with fat edges

Traditionally, graph drawing algorithms represent vertices as circles and edges as curves connecting the vertices. We introduce the problem of drawing with "fat" edges, i.e., with edges of variable thickness. The thickness of an edge is often used as a visualization cue, to indicate importance, or to convey some additional information. We present a model for drawing with fat edges and a corresponding efficient polynomial time algorithm that uses the model. We first focus on a restricted class of graphs that occur in VLSI wire routing and then show how to extend the algorithm to general planar graphs. We show how to convert an arbitrary wire routing into a homotopically equivalent routing that maximizes the distance between any two wires, which is a desired property in VLSI design. Among such, we obtain the routing with minimum total wire length. A homotopically equivalent routing that maximizes the distance between any two wires yields a graph drawing which maximizes edge thickness. Our algorithm does not require unit edge thickness but can be applied as well in the presence of different edge weights.