David Eppstein - Publications

Publications with Christian A. Duncan

• The geometric thickness of low degree graphs.
  C. Duncan, D. Eppstein, and S. Kobourov.
  arXiv:cs.CG/0312056.
  20th ACM Symp. Comp. Geom., Brooklyn, 2004, pp. 340–346.

  We show that graphs with maximum degree four have geometric thickness at most two, by partitioning them into degree two subgraphs and applying simultaneous embedding techniques.

• Lombardi drawings of graphs.
  C. Duncan, D. Eppstein, M. T. Goodrich, S. Kobourov, and M. Nöllenburg.
  Proc. 18th Int. Symp. Graph Drawing, Konstanz, Germany, 2010.
  Springer, Lecture Notes in Comp. Sci. 6502, 2011, pp. 195–207.
  arXiv:1009.0579.
  Invited talk at 7th Dutch Computational Geometry Day, Eindhoven, the Netherlands, 2010.
  J. Graph Algorithms and Applications 16 (1): 85–108, 2012 (special issue for GD 2010).

  In honor of artist Mark Lombardi, we define a Lombardi drawing to be a drawing of a graph in which the edges are drawn as circular arcs and at each vertex they are equally spaced around the vertex so as to achieve the best possible angular resolution. We describe algorithms for constructing Lombardi drawings of regular graphs, 2-degenerate graphs, graphs with rotational symmetry, and several types of planar graphs. A program for the rotationally symmetric case, the Lombardi Spirograph, is available online.

• Drawing trees with perfect angular resolution and polynomial area.
  C. Duncan, D. Eppstein, M. T. Goodrich, S. Kobourov, and M. Nöllenburg.
  Proc. 18th Int. Symp. Graph Drawing, Konstanz, Germany, 2010.
  Springer, Lecture Notes in Comp. Sci. 6502, 2011, pp. 183–194.
  arXiv:1009.0581.
  Discrete Comput. Geom. 49 (2): 157–182, 2013.

  We consider balloon drawings of trees, in which each subtree of the root is drawn recursively within a disk, and these disks are arranged radially around the root, with the edges at each node spaced equally around the node so as to achieve the best possible angular resolution. If we are allowed to permute the children of each node, then we show that a drawing of this type can be made in which all edges are straight line segments and the area of the drawing is a polynomial multiple of the shortest edge length. However, if the child ordering is prescribed, exponential area might be necessary. We show that, if we relax the requirement of straight line edges and draw the edges as circular arcs (a style we call Lombardi drawing) then even with a prescribed child ordering we can achieve polynomial area.

• Planar and poly-arc Lombardi drawings.
  C. Duncan, D. Eppstein, M. T. Goodrich, S. Kobourov, and M. Löffler, and M. Nöllenburg.
  Proc. 19th Int. Symp. Graph Drawing, Eindhoven, The Netherlands, 2011.
  Springer, Lecture Notes in Comp. Sci. 7034, 2012, pp. 308–319.
  arXiv:1109.0345.
  J. Computational Geometry 9 (1): 328–355, 2018.

  We extend Lombardi drawing (in which each edge is a circular arc and the edges incident to a vertex must be equally spaced around it) to drawings in which edges are composed of multiple arcs, and we investigate the graphs that can be drawn in this more relaxed framework.