### Abstract

In this paper, we introduce the problem of drawing with "fat" edges. Traditionally, graph drawing algorithms represent vertices as circles and edges as closed curves connecting the vertices. In this paper we consider the problem of drawing graphs 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 polynomial time algorithm that uses the model. We focus on a restricted class of graphs that occur in VLSI wire routing and show how to extend the algorithm to general planar graphs. We show how to take an arbitrary wire routing and convert it into a homotopic equivalent routing such that the distance between any two wires is maximized. Moreover, the routing uses the minimum length wires. Maximizing the distance between wires is equivalent to finding the drawing in which the edges are drawn as thick as possible. To the best of our knowledge this is the first algorithm that finds the maximal distance between any two wires and allows for wires of variable thickness. The previous best known result for the corresponding decision problem with unit wire thickness is the algorithm of Gao et al, which runs in O(kn^{2} log(kn)) time and uses O(kn^{2}) space, where n is the number of wires and k is the maximum of the input and output complexities. The running time of our algorithm is O(kn + n^{3}) and the space required is O(k+n). The algorithm generalizes naturally to general planar graphs as well.

Original language | English (US) |
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Title of host publication | Graph Drawing - 9th International Symposium, GD 2001, Revised Papers |

Editors | Petra Mutzel, Michael Junger, Sebastian Leipert |

Publisher | Springer-Verlag |

Pages | 162-177 |

Number of pages | 16 |

ISBN (Print) | 3540433090, 9783540433095 |

DOIs | |

State | Published - Jan 1 2002 |

Event | 9th International Symposium on Graph Drawing, GD 2001 - Vienna, Austria Duration: Sep 23 2001 → Sep 26 2001 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 2265 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 9th International Symposium on Graph Drawing, GD 2001 |
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Country | Austria |

City | Vienna |

Period | 9/23/01 → 9/26/01 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*Graph Drawing - 9th International Symposium, GD 2001, Revised Papers*(pp. 162-177). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 2265 LNCS). Springer-Verlag. https://doi.org/10.1007/3-540-45848-4_14