, Volume 25, Issue 3, pp 405-418

Drawing planar graphs with circular arcs

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Abstract

In this paper we address the problem of drawing planar graphs with circular arcs while maintaining good angular resolution and small drawing area. We present a lower bound on the area of drawings in which edges are drawn using exactly one circular arc. We also give an algorithm for drawing n-vertex planar graphs such that the edges are sequences of two continuous circular arcs. The algorithm runs in O(n) time and embeds the graph on the O(n) × O(n) grid, while maintaining Θ(1/d(v)) angular resolution, where d(v) is the degree of vertex v. Since in this case we use circular arcs of infinite radius, this is also the first algorithm that simultaneously achieves good angular resolution, small area, and at most one bend per edge using straight-line segments. Finally, we show how to create drawings in which edges are smooth C 1-continuous curves, represented by a sequence of at most three circular arcs.

A preliminary version of this paper appeared in the Proceedings of the 7th Annual Symposium on Graph Drawing, 1999. The first author was partially supported by ONR Grant N00014-96-1-0829, the other three authors were partially supported by NSF Grant CCR-9732300 and ARO Grant DAAH04-96-1-0013.