**Selected open problems in graph drawing**.

F. J. Brandenburg, D. Eppstein, M. T. Goodrich, S. G. Kobourov, G. Liotta, and P. Mutzel.

11th Int. Symp. Graph Drawing, Perugia, Italy, 2003.

Springer,*Lecture Notes in Comp. Sci.*2912, 2004, pp. 515–539.We survey a number of open problems in theoretical and applied graph drawing.

**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.

*Proc. 19th Int. Symp. Graph Drawing*, Eindhoven, The Netherlands, 2011.

Springer,*Lecture Notes in Comp. Sci.*7034, 2012, pp. 308–319.

arXiv:1109.0345.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.

**Balanced circle packings for planar graphs**.

M. J. Alam, D. Eppstein, M. T. Goodrich, S. Kobourov, and S. Pupyrev.

arXiv:1408.4902.

*22nd Int. Symp. Graph Drawing*, Würzburg, Germany, 2014.

Springer,*Lecture Notes in Comp. Sci.*8871, 2014, pp. 125–136.The balanced circle packings of the title are systems of interior-disjoint circles, whose tangencies represent the given graph, and whose radii are all within a polynomial factor of each other. We show that these packings always exist for trees, cactus graphs, outerpaths, k-outerplanar graphs of bounded degree when k is at most logarithmic, and planar graphs of bounded treedepth. The treedepth result uses a new construction of inversive-distance circle packings.

**Contact graphs of circular arcs**.

M. J. Alam, D. Eppstein, M. Kaufmann, S. Kobourov, S. Pupyrev A. Schulz, and T. Ueckerdt.

arXiv:1501.00318.

14th Algorithms and Data Structures Symp. (WADS 2015), Victoria, BC.

Springer,*Lecture Notes in Comp. Sci.*9214 (2015), pp. 1–13.We study the graphs formed by non-crossing circular arcs in the plane, having a vertex for each arc and an edge for each point where an arc endpoint touches the interior of another arc.

(Slides)

**On the planar split thickness of graphs**.

D. Eppstein, P. Kindermann, S. G. Kobourov, G. Liotta, A. Lubiw, A. Maignan, D. Mondal, H. Vosoughpour, S. Whitesides, and S. Wismath.

arXiv:1512.04839.

*Proc. 12th Latin American Theoretical Informatics Symposium (LATIN 2016)*, Ensenada, Mexico.

Springer,*Lecture Notes in Comp. Sci.*9644 (2016), pp. 403–415.

*Algorithmica*, to appear.We study the problem of splitting the vertices of a given graph into a bounded number of sub-vertices (with each edge attaching to one of the sub-vertices) in order to make the resulting graph planar. It is NP-complete, but can be approximated to within a constant factor, and is fixed-parameter tractable in the treewidth.

(Slides)

Co-authors – Publications – David Eppstein – Theory Group – Inf. & Comp. Sci. – UC Irvine

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